Geyer, Anna
2016-01-01
Following a general principle introduced by Ehrnstr\\"{o}m et.al. we prove that for an equation modeling the free surface evolution of moderate amplitude waves in shallow water, all symmetric waves are traveling waves.
Conditionally invariant solutions of the rotating shallow water wave equations
Energy Technology Data Exchange (ETDEWEB)
Huard, Benoit, E-mail: huard@dms.umontreal.c [Departement de mathematiques et de statistique, CP 6128, Succc. Centre-ville, Montreal, (QC) H3C 3J7 (Canada)
2010-06-11
This paper is devoted to the extension of the recently proposed conditional symmetry method to first-order nonhomogeneous quasilinear systems which are equivalent to homogeneous systems through a locally invertible point transformation. We perform a systematic analysis of the rank-1 and rank-2 solutions admitted by the shallow water wave equations in (2 + 1) dimensions and construct the corresponding solutions of the rotating shallow water wave equations. These solutions involve in general arbitrary functions depending on Riemann invariants, which allow us to construct new interesting classes of solutions.
Traveling wave solutions of a highly nonlinear shallow water equation
Geyer, A.; Quirchmayr, Ronald
2018-01-01
Motivated by the question whether higher-order nonlinear model equations, which go beyond the Camassa-Holm regime of moderate amplitude waves, could point us to new types of waves profiles, we study the traveling wave solutions of a quasilinear evolution equation which models the propagation of
Diffusive Wave Approximation to the Shallow Water Equations: Computational Approach
Collier, Nathan
2011-05-14
We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, in the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation.
Symmetry reductions and exact solutions of Shallow water wave equations
Clarkson, P A
1994-01-01
In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation u_{xxxt} + \\alpha u_x u_{xt} + \\beta u_t u_{xx} - u_{xt} - u_{xx} = 0,\\eqno(1) where \\alpha and \\beta are arbitrary, nonzero, constants, which is derivable using the so-called Boussinesq approximation. Two special cases of this equation, or the equivalent nonlocal equation obtained by setting u_x=U, have been discussed in the literature. The case \\alpha=2\\beta was discussed by Ablowitz, Kaup, Newell and Segur [{\\it Stud.\\ Appl.\\ Math.}, {\\bf53} (1974) 249], who showed that this case was solvable by inverse scattering through a second order linear problem. This case and the case \\alpha=\\beta were studied by Hirota and Satsuma [{\\it J.\\ Phys.\\ Soc.\\ Japan}, {\\bf40} (1976) 611] using Hirota's bi-linear technique. Further the case \\alpha=\\beta is solvable by inverse scattering through a third order linear problem. In this paper a catalogue of symmetry reductions is obtained using the classical Lie method and th...
Ocean swell within the kinetic equation for water waves
Directory of Open Access Journals (Sweden)
S. I. Badulin
2017-06-01
Full Text Available Results of extensive simulations of swell evolution within the duration-limited setup for the kinetic Hasselmann equation for long durations of up to 2 × 106 s are presented. Basic solutions of the theory of weak turbulence, the so-called Kolmogorov–Zakharov solutions, are shown to be relevant to the results of the simulations. Features of self-similarity of wave spectra are detailed and their impact on methods of ocean swell monitoring is discussed. Essential drop in wave energy (wave height due to wave–wave interactions is found at the initial stages of swell evolution (on the order of 1000 km for typical parameters of the ocean swell. At longer times, wave–wave interactions are responsible for a universal angular distribution of wave spectra in a wide range of initial conditions. Weak power-law attenuation of swell within the Hasselmann equation is not consistent with results of ocean swell tracking from satellite altimetry and SAR (synthetic aperture radar data. At the same time, the relatively fast weakening of wave–wave interactions makes the swell evolution sensitive to other effects. In particular, as shown, coupling with locally generated wind waves can force the swell to grow in relatively light winds.
Mostafa M.A. Khater; Dipankar Kumar
2017-01-01
The aim of the article is to construct exact solutions for the time fractional coupled Boussinesq–Burger and approximate long water wave equations by using the generalized Kudryashov method. The fractional differential equation is converted into ordinary differential equations with the help of fractional complex transform and the modified Riemann–Liouville derivative sense. Applying the generalized Kudryashov method through with symbolic computer maple package, numerous new exact solutions ar...
Shallow Water Waves and Solitary Waves
Hereman, Willy
2013-01-01
Encyclopedic article covering shallow water wave models used in oceanography and atmospheric science. Sections: Definition of the Subject; Introduction and Historical Perspective; Completely Integrable Shallow Water Wave Equations; Shallow Water Wave Equations of Geophysical Fluid Dynamics; Computation of Solitary Wave Solutions; Numerical Methods; Water Wave Experiments and Observations; Future Directions, and Bibliography.
Time adaptivity in the diffusive wave approximation to the shallow water equations
Collier, Nathan
2013-05-01
We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation. © 2011 Elsevier B.V.
Directory of Open Access Journals (Sweden)
Mostafa M.A. Khater
2017-09-01
Full Text Available The aim of the article is to construct exact solutions for the time fractional coupled Boussinesq–Burger and approximate long water wave equations by using the generalized Kudryashov method. The fractional differential equation is converted into ordinary differential equations with the help of fractional complex transform and the modified Riemann–Liouville derivative sense. Applying the generalized Kudryashov method through with symbolic computer maple package, numerous new exact solutions are successfully obtained. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions with the integer and fractional order.
Li, Tatsien
2017-01-01
This book focuses on nonlinear wave equations, which are of considerable significance from both physical and theoretical perspectives. It also presents complete results on the lower bound estimates of lifespan (including the global existence), which are established for classical solutions to the Cauchy problem of nonlinear wave equations with small initial data in all possible space dimensions and with all possible integer powers of nonlinear terms. Further, the book proposes the global iteration method, which offers a unified and straightforward approach for treating these kinds of problems. Purely based on the properties of solut ions to the corresponding linear problems, the method simply applies the contraction mapping principle.
Shatah, Jalal
2000-01-01
This volume contains notes of the lectures given at the Courant Institute and a DMV-Seminar at Oberwolfach. The focus is on the recent work of the authors on semilinear wave equations with critical Sobolev exponents and on wave maps in two space dimensions. Background material and references have been added to make the notes self-contained. The book is suitable for use in a graduate-level course on the topic.
A comparative study of diffraction of shallow-water waves by high-level IGN and GN equations
Energy Technology Data Exchange (ETDEWEB)
Zhao, B.B. [College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin (China); Ertekin, R.C. [Department of Ocean and Resources Engineering, University of Hawai' i, Honolulu, HI 96822 (United States); College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin (China); Duan, W.Y., E-mail: duanwenyangheu@hotmail.com [College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin (China)
2015-02-15
This work is on the nonlinear diffraction analysis of shallow-water waves, impinging on submerged obstacles, by two related theories, namely the classical Green–Naghdi (GN) equations and the Irrotational Green–Naghdi (IGN) equations, both sets of equations being at high levels and derived for incompressible and inviscid flows. Recently, the high-level Green–Naghdi equations have been applied to some wave transformation problems. The high-level IGN equations have also been used in the last decade to study certain wave propagation problems. However, past works on these theories used different numerical methods to solve these nonlinear and unsteady sets of differential equations and at different levels. Moreover, different physical problems have been solved in the past. Therefore, it has not been possible to understand the differences produced by these two sets of theories and their range of applicability so far. We are thus motivated to make a direct comparison of the results produced by these theories by use of the same numerical method to solve physically the same wave diffraction problems. We focus on comparing these two theories by using similar codes; only the equations used are different but other parts of the codes, such as the wave-maker, damping zone, discretion method, matrix solver, etc., are exactly the same. This way, we eliminate many potential sources of differences that could be produced by the solution of different equations. The physical problems include the presence of various submerged obstacles that can be used for example as breakwaters or to represent the continental shelf. A numerical wave tank is created by placing a wavemaker on one end and a wave absorbing beach on the other. The nonlinear and unsteady sets of differential equations are solved by the finite-difference method. The results are compared with different equations as well as with the available experimental data.
Directory of Open Access Journals (Sweden)
Ilhan Özgen
2017-10-01
Full Text Available In urban flood modeling, so-called porosity shallow water equations (PSWEs, which conceptually account for unresolved structures, e.g., buildings, are a promising approach to addressing high CPU times associated with state-of-the-art explicit numerical methods. The PSWE can be formulated with a single porosity term, referred to as the single porosity shallow water model (SP model, which accounts for both the reduced storage in the cell and the reduced conveyance, or with two porosity terms: one accounting for the reduced storage in the cell and another accounting for the reduced conveyance. The latter form is referred to as an integral or anisotropic porosity shallow water model (AP model. The aim of this study was to analyze the differences in wave propagation speeds of the SP model and the AP model and the implications of numerical model results. First, augmented Roe-type solutions were used to assess the influence of the source terms appearing in both models. It is shown that different source terms have different influences on the stability of the models. Second, four computational test cases were presented and the numerical models were compared. It is observed in the eigenvalue-based analysis as well as in the computational test cases that the models converge if the conveyance porosity in the AP model is close to the storage porosity. If the porosity values differ significantly, the AP model yields different wave propagation speeds and numerical fluxes from those of the BP model. In this study, the ratio between the conveyance and storage porosities was determined to be the most significant parameter.
An ADER-type scheme for a class of equations arising from the water-wave theory
Montecinos G.I.; López-Rios J.C.; Lecaros R.; Ortega J.H.; Toro E.F.
2016-01-01
In this work we propose a numerical strategy to solve a family of partial differential equations arising from the water-wave theory. These problems may contain four terms; a source which is an algebraic function of the solution, a convective part involving first order spatial derivatives of the solution, a diffusive part involving second order spatial derivatives and the transient part. Unlike partial differential equations of hyperbolic or parabolic type, where the transient part is the time...
Schlesinger, R. E.; Johnson, D. R.; Uccellini, L. W.
1983-01-01
In the present investigation, a one-dimensional linearized analysis is used to determine the effect of Asselin's (1972) time filter on both the computational stability and phase error of numerical solutions for the shallow water wave equations, in cases with diffusion but without rotation. An attempt has been made to establish the approximate optimal values of the filtering parameter nu for each of the 'lagged', Dufort-Frankel, and Crank-Nicholson diffusion schemes, suppressing the computational wave mode without materially altering the physical wave mode. It is determined that in the presence of diffusion, the optimum filter length depends on whether waves are undergoing significant propagation. When moderate propagation is present, with or without diffusion, the Asselin filter has little effect on the spatial phase lag of the physical mode for the leapfrog advection scheme of the three diffusion schemes considered.
Energy Technology Data Exchange (ETDEWEB)
Chabchoub, A., E-mail: achabchoub@swin.edu.au [Centre for Ocean Engineering Science and Technology, Swinburne University of Technology, Hawthorn, Victoria 3122 (Australia); Kibler, B.; Finot, C.; Millot, G. [Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS, Université de Bourgogne, 21078 Dijon (France); Onorato, M. [Dipartimento di Fisica, Università degli Studi di Torino, Torino 10125 (Italy); Istituto Nazionale di Fisica Nucleare, INFN, Sezione di Torino, Torino 10125 (Italy); Dudley, J.M. [Institut FEMTO-ST, UMR 6174 CNRS- Université de Franche-Comté, 25030 Besançon (France); Babanin, A.V. [Centre for Ocean Engineering Science and Technology, Swinburne University of Technology, Hawthorn, Victoria 3122 (Australia)
2015-10-15
The dynamics of waves in weakly nonlinear dispersive media can be described by the nonlinear Schrödinger equation (NLSE). An important feature of the equation is that it can be derived in a number of different physical contexts; therefore, analogies between different fields, such as for example fiber optics, water waves, plasma waves and Bose–Einstein condensates, can be established. Here, we investigate the similarities between wave propagation in optical Kerr media and water waves. In particular, we discuss the modulation instability (MI) in both media. In analogy to the water wave problem, we derive for Kerr-media the Benjamin–Feir index, i.e. a nondimensional parameter related to the probability of formation of rogue waves in incoherent wave trains.
Water Wave Solutions of the Coupled System Zakharov-Kuznetsov and Generalized Coupled KdV Equations
Seadawy, A. R.; El-Rashidy, K.
2014-01-01
An analytic study was conducted on coupled partial differential equations. We formally derived new solitary wave solutions of generalized coupled system of Zakharov-Kuznetsov (ZK) and KdV equations by using modified extended tanh method. The traveling wave solutions for each generalized coupled system of ZK and KdV equations are shown in form of periodic, dark, and bright solitary wave solutions. The structures of the obtained solutions are distinct and stable. PMID:25374940
Poincar wave equations as Fourier transformations of Galilei wave equations
Gomis Torné, Joaquim; Poch Parés, Agustí; Pons Ràfols, Josep Maria
1980-01-01
The relationship between the Poincar and Galilei groups allows us to write the Poincar wave equations for arbitrary spin as a Fourier transform of the Galilean ones. The relation between the Lagrangian formulation for both cases is also studied.
Relativistic quantum mechanics wave equations
Greiner, Walter
1990-01-01
Relativistic Quantum Mechanics - Wave Equations concentrates mainly on the wave equations for spin-0 and spin-12 particles Chapter 1 deals with the Klein-Gordon equation and its properties and applications The chapters that follow introduce the Dirac equation, investigate its covariance properties and present various approaches to obtaining solutions Numerous applications are discussed in detail, including the two-center Dirac equation, hole theory, CPT symmetry, Klein's paradox, and relativistic symmetry principles Chapter 15 presents the relativistic wave equations for higher spin (Proca, Rarita-Schwinger, and Bargmann-Wigner) The extensive presentation of the mathematical tools and the 62 worked examples and problems make this a unique text for an advanced quantum mechanics course
Wave-equation dispersion inversion
Li, Jing
2016-12-08
We present the theory for wave-equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. The dispersion curves are obtained from Rayleigh waves recorded by vertical-component geophones. Similar to wave-equation traveltime tomography, the complicated surface wave arrivals in traces are skeletonized as simpler data, namely the picked dispersion curves in the phase-velocity and frequency domains. Solutions to the elastic wave equation and an iterative optimization method are then used to invert these curves for 2-D or 3-D S-wave velocity models. This procedure, denoted as wave-equation dispersion inversion (WD), does not require the assumption of a layered model and is significantly less prone to the cycle-skipping problems of full waveform inversion. The synthetic and field data examples demonstrate that WD can approximately reconstruct the S-wave velocity distributions in laterally heterogeneous media if the dispersion curves can be identified and picked. The WD method is easily extended to anisotropic data and the inversion of dispersion curves associated with Love waves.
Evolution of basic equations for nearshore wave field
ISOBE, Masahiko
2013-01-01
In this paper, a systematic, overall view of theories for periodic waves of permanent form, such as Stokes and cnoidal waves, is described first with their validity ranges. To deal with random waves, a method for estimating directional spectra is given. Then, various wave equations are introduced according to the assumptions included in their derivations. The mild-slope equation is derived for combined refraction and diffraction of linear periodic waves. Various parabolic approximations and time-dependent forms are proposed to include randomness and nonlinearity of waves as well as to simplify numerical calculation. Boussinesq equations are the equations developed for calculating nonlinear wave transformations in shallow water. Nonlinear mild-slope equations are derived as a set of wave equations to predict transformation of nonlinear random waves in the nearshore region. Finally, wave equations are classified systematically for a clear theoretical understanding and appropriate selection for specific applications. PMID:23318680
High-order Finite Difference Solution of Euler Equations for Nonlinear Water Waves
DEFF Research Database (Denmark)
Christiansen, Torben Robert Bilgrav; Bingham, Harry B.; Engsig-Karup, Allan Peter
2012-01-01
is discretized using arbitrary-order finite difference schemes on a staggered grid with one optional stretching in each coordinate direction. The momentum equations and kinematic free surface condition are integrated in time using the classic fourth-order Runge-Kutta scheme. Mass conservation is satisfied...
Wave equations for pulse propagation
Energy Technology Data Exchange (ETDEWEB)
Shore, B.W.
1987-06-24
Theoretical discussions of the propagation of pulses of laser radiation through atomic or molecular vapor rely on a number of traditional approximations for idealizing the radiation and the molecules, and for quantifying their mutual interaction by various equations of propagation (for the radiation) and excitation (for the molecules). In treating short-pulse phenomena it is essential to consider coherent excitation phenomena of the sort that is manifest in Rabi oscillations of atomic or molecular populations. Such processes are not adequately treated by rate equations for excitation nor by rate equations for radiation. As part of a more comprehensive treatment of the coupled equations that describe propagation of short pulses, this memo presents background discussion of the equations that describe the field. This memo discusses the origin, in Maxwell's equations, of the wave equation used in the description of pulse propagation. It notes the separation into lamellar and solenoidal (or longitudinal and transverse) and positive and negative frequency parts. It mentions the possibility of separating the polarization field into linear and nonlinear parts, in order to define a susceptibility or index of refraction and, from these, a phase and group velocity. The memo discusses various ways of characterizing the polarization characteristics of plane waves, that is, of parameterizing a transverse unit vector, such as the Jones vector, the Stokes vector, and the Poincare sphere. It discusses the connection between macroscopically defined quantities, such as the intensity or, more generally, the Stokes parameters, and microscopic field amplitudes. The material presented here is a portion of a more extensive treatment of propagation to be presented separately. The equations presented here have been described in various books and articles. They are collected here as a summary and review of theory needed when treating pulse propagation.
Wave equations for pulse propagation
Shore, B. W.
1987-06-01
Theoretical discussions of the propagation of pulses of laser radiation through atomic or molecular vapor rely on a number of traditional approximations for idealizing the radiation and the molecules, and for quantifying their mutual interaction by various equations of propagation (for the radiation) and excitation (for the molecules). In treating short-pulse phenomena it is essential to consider coherent excitation phenomena of the sort that is manifest in Rabi oscillations of atomic or molecular populations. Such processes are not adequately treated by rate equations for excitation nor by rate equations for radiation. As part of a more comprehensive treatment of the coupled equations that describe propagation of short pulses, this memo presents background discussion of the equations that describe the field. This memo discusses the origin, in Maxwell's equations, of the wave equation used in the description of pulse propagation. It notes the separation into lamellar and solenoidal (or longitudinal and transverse) and positive and negative frequency parts. It mentions the possibility of separating the polarization field into linear and nonlinear parts, in order to define a susceptibility or index of refraction and, from these, a phase and group velocity.
Wave equations in higher dimensions
Dong, Shi-Hai
2011-01-01
Higher dimensional theories have attracted much attention because they make it possible to reduce much of physics in a concise, elegant fashion that unifies the two great theories of the 20th century: Quantum Theory and Relativity. This book provides an elementary description of quantum wave equations in higher dimensions at an advanced level so as to put all current mathematical and physical concepts and techniques at the reader’s disposal. A comprehensive description of quantum wave equations in higher dimensions and their broad range of applications in quantum mechanics is provided, which complements the traditional coverage found in the existing quantum mechanics textbooks and gives scientists a fresh outlook on quantum systems in all branches of physics. In Parts I and II the basic properties of the SO(n) group are reviewed and basic theories and techniques related to wave equations in higher dimensions are introduced. Parts III and IV cover important quantum systems in the framework of non-relativisti...
Waves Described By The Nonlinear Fifth Order Partial Differential Equation
Kudryashov, N. A.; Soukharev, M. B.; Siroklin, S. A.
Nonlinear long waves on the water are considered taking into account the general- ization of the Korteveg - de Vries equation that is the fifth order partial differential equation. The following problems are studied. First of all it was shown the Cauchy problem can not be solved for the partial differential equation considered by the in- verse scattering transform. However a number of special solution are found in the form of solitary waves and in the form of cnoidal waves. In the general case the boundary value problem described by this fifth order nonlinear partial differential equation is solved by means of the finite difference method. With this aim the finite - difference approximations of the fifth nonlinear partial differential equations was developed. The stability condition and the convergence of the difference equation are discussed. The possible types of nonlinear waves (kinks, periodical waves and solitary waves) are considered. The stability of nonlinear waves studied is discussed taking into account the numerical approach.
Dutta, Gaurav
2016-10-12
Strong subsurface attenuation leads to distortion of amplitudes and phases of seismic waves propagating inside the earth. The amplitude and the dispersion losses from attenuation are often compensated for during prestack depth migration. However, most attenuation compensation or Qcompensation migration algorithms require an estimate of the background Q model. We have developed a wave-equation gradient optimization method that inverts for the subsurface Q distribution by minimizing a skeletonized misfit function ∈, where ∈ is the sum of the squared differences between the observed and the predicted peak/centroid-frequency shifts of the early arrivals. The gradient is computed by migrating the observed traces weighted by the frequency shift residuals. The background Q model is perturbed until the predicted and the observed traces have the same peak frequencies or the same centroid frequencies. Numerical tests determined that an improved accuracy of the Q model by wave-equation Q tomography leads to a noticeable improvement in migration image quality. © 2016 Society of Exploration Geophysicists.
Pelinovsky, Efim; Chaikovskaia, Natalya; Rodin, Artem
2015-04-01
The paper presents the analysis of the formation and evolution of shock wave in shallow water with no restrictions on its amplitude in the framework of the nonlinear shallow water equations. It is shown that in the case of large-amplitude waves appears a new nonlinear effect of reflection from the shock front of incident wave. These results are important for the assessment of coastal flooding by tsunami waves and storm surges. Very often the largest number of victims was observed on the coastline where the wave moved breaking. Many people, instead of running away, were just looking at the movement of the "raging wall" and lost time. This fact highlights the importance of researching the problem of security and optimal behavior of people in situations with increased risk. Usually there is uncertainty about the exact time, when rogue waves will impact. This fact limits the ability of people to adjust their behavior psychologically to the stressful situations. It concerns specialists, who are busy both in the field of flying activity and marine service as well as adults, young people and children, who live on the coastal zone. The rogue wave research is very important and it demands cooperation of different scientists - mathematicians and physicists, as well as sociologists and psychologists, because the final goal of efforts of all scientists is minimization of the harm, brought by rogue waves to humanity.
Directory of Open Access Journals (Sweden)
Weiguo Rui
2015-01-01
Full Text Available By using Frobenius’ idea together with integral bifurcation method, we study a third order nonlinear equation of generalization form of the modified KdV equation, which is an important water wave model. Some exact traveling wave solutions such as smooth solitary wave solutions, nonsmooth peakon solutions, kink and antikink wave solutions, periodic wave solutions of Jacobian elliptic function type, and rational function solution are obtained. And we show their profiles and discuss their dynamic properties aim at some typical solutions. Though the types of these solutions obtained in this work are not new and they are familiar types, they did not appear in any existing literatures because the equation ut+ux+νuxxt+βuxxx + αuux+1/3να(uuxxx+2uxuxx+3μα2u2ux+νμα2(u2uxxx+ux3+4uuxuxx + ν2μα2(ux2uxxx+2uxuxx2 = 0 is very complex. Particularly, compared with the cited references, all results obtained in this paper are new.
Karczewska, Anna; Rozmej, Piotr; Infeld, Eryk
2015-11-01
It is well known that the Korteweg-de Vries (KdV) equation has an infinite set of conserved quantities. The first three are often considered to represent mass, momentum, and energy. Here we try to answer the question of how this comes about and also how these KdV quantities relate to those of the Euler shallow-water equations. Here Luke's Lagrangian is helpful. We also consider higher-order extensions of KdV. Though in general not integrable, in some sense they are almost so within the accuracy of the expansion.
Wave Equation Inversion of Skeletonized SurfaceWaves
Zhang, Zhendong
2015-08-19
We present a surface-wave inversion method that inverts for the S-wave velocity from the Rayleigh dispersion curve for the fundamental-mode. We call this wave equation inversion of skeletonized surface waves because the dispersion curve for the fundamental-mode Rayleigh wave is inverted using finite-difference solutions to the wave equation. The best match between the predicted and observed dispersion curves provides the optimal S-wave velocity model. Results with synthetic and field data illustrate the benefits and limitations of this method.
Shallow water cnoidal wave interactions
Directory of Open Access Journals (Sweden)
A. R. Osborne
1994-01-01
Full Text Available The nonlinear dynamics of cnoidal waves, within the context of the general N-cnoidal wave solutions of the periodic Korteweg-de Vries (KdV and Kadomtsev-Petvishvilli (KP equations, are considered. These equations are important for describing the propagation of small-but-finite amplitude waves in shallow water; the solutions to KdV are unidirectional while those of KP are directionally spread. Herein solutions are constructed from the 0-function representation of their appropriate inverse scattering transform formulations. To this end a general theorem is employed in the construction process: All solutions to the KdV and KP equations can be written as the linear superposition of cnoidal waves plus their nonlinear interactions. The approach presented here is viewed as significant because it allows the exact construction of N degree-of-freedom cnoidal wave trains under rather general conditions.
Generalized Nonlinear Wave Equation in Frequency Domain
DEFF Research Database (Denmark)
Guo, Hairun; Zeng, Xianglong; Bache, Morten
2013-01-01
We interpret the forward Maxwell equation with up to third order induced polarizations and get so called nonlinear wave equation in frequency domain (NWEF), which is based on Maxwell wave equation and using slowly varying spectral amplitude approximation. The NWEF is generalized in concept...... as it directly describes the electric field dynamics rather than the envelope dynamics and because it concludes most current-interested nonlinear processes such as three-wave mixing, four-wave-mixing and material Raman effects. We give two sets of NWEF, one is a 1+1D equation describing the (approximated) planar...... wave propagation in nonlinear bulk material and the other corresponds to the propagation in a waveguide structure....
Skeletonized wave equation of surface wave dispersion inversion
Li, Jing
2016-09-06
We present the theory for wave equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. Similar to wave-equation travel-time inversion, the complicated surface-wave arrivals in traces are skeletonized as simpler data, namely the picked dispersion curves in the (kx,ω) domain. Solutions to the elastic wave equation and an iterative optimization method are then used to invert these curves for 2D or 3D velocity models. This procedure, denoted as wave equation dispersion inversion (WD), does not require the assumption of a layered model and is less prone to the cycle skipping problems of full waveform inversion (FWI). The synthetic and field data examples demonstrate that WD can accurately reconstruct the S-wave velocity distribution in laterally heterogeneous media.
Electronic representation of wave equation
Energy Technology Data Exchange (ETDEWEB)
Veigend, Petr; Kunovský, Jiří, E-mail: kunovsky@fit.vutbr.cz; Kocina, Filip; Nečasová, Gabriela; Valenta, Václav [University of Technology, Faculty of Information Technology, Božetěchova 2, 612 66 Brno (Czech Republic); Šátek, Václav [IT4Innovations, VŠB Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava-Poruba (Czech Republic); University of Technology, Faculty of Information Technology, Božetěchova 2, 612 66 Brno (Czech Republic)
2016-06-08
The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.
Electronic representation of wave equation
Veigend, Petr; Kunovský, Jiří; Kocina, Filip; Nečasová, Gabriela; Šátek, Václav; Valenta, Václav
2016-06-01
The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.
Nonlinear Electrostatic Wave Equations for Magnetized Plasmas
DEFF Research Database (Denmark)
Dysthe, K.B.; Mjølhus, E.; Pécseli, Hans
1984-01-01
The lowest order kinetic effects are included in the equations for nonlinear electrostatic electron waves in a magnetized plasma. The modifications of the authors' previous analysis based on a fluid model are discussed.......The lowest order kinetic effects are included in the equations for nonlinear electrostatic electron waves in a magnetized plasma. The modifications of the authors' previous analysis based on a fluid model are discussed....
Octonion wave equation and tachyon electrodynamics
Indian Academy of Sciences (India)
The octonion wave equation is discussed to formulate the localization spaces for subluminal and superluminal particles. Accordingly, tachyon electrodynamics is established to obtain a consistent and manifestly covariant equation for superluminal electromagnetic fields. It is shown that the true localization space for ...
Wave Equations in Bianchi Space-Times
Directory of Open Access Journals (Sweden)
S. Jamal
2012-01-01
Full Text Available We investigate the wave equation in Bianchi type III space-time. We construct a Lagrangian of the model, calculate and classify the Noether symmetry generators, and construct corresponding conserved forms. A reduction of the underlying equations is performed to obtain invariant solutions.
Diffusion phenomenon for linear dissipative wave equations
Said-Houari, Belkacem
2012-01-01
In this paper we prove the diffusion phenomenon for the linear wave equation. To derive the diffusion phenomenon, a new method is used. In fact, for initial data in some weighted spaces, we prove that for {equation presented} decays with the rate {equation presented} [0,1] faster than that of either u or v, where u is the solution of the linear wave equation with initial data {equation presented} [0,1], and v is the solution of the related heat equation with initial data v 0 = u 0 + u 1. This result improves the result in H. Yang and A. Milani [Bull. Sci. Math. 124 (2000), 415-433] in the sense that, under the above restriction on the initial data, the decay rate given in that paper can be improved by t -γ/2. © European Mathematical Society.
Water Waves The Mathematical Theory with Applications
Stoker, J J
2011-01-01
Offers an integrated account of the mathematical hypothesis of wave motion in liquids with a free surface, subjected to gravitational and other forces. Uses both potential and linear wave equation theories, together with applications such as the Laplace and Fourier transform methods, conformal mapping and complex variable techniques in general or integral equations, methods employing a Green's function. Coverage includes fundamental hydrodynamics, waves on sloping beaches, problems involving waves in shallow water, the motion of ships and much more.
Wave-equation Qs Inversion of Skeletonized Surface Waves
Li, Jing
2017-02-08
We present a skeletonized inversion method that inverts surface-wave data for the Qs quality factor. Similar to the inversion of dispersion curves for the S-wave velocity model, the complicated surface-wave arrivals are skeletonized as simpler data, namely the amplitude spectra of the windowed Rayleigh-wave arrivals. The optimal Qs model is the one that minimizes the difference in the peak frequencies of the predicted and observed Rayleigh wave arrivals using a gradient-based wave-equation optimization method. Solutions to the viscoelastic wave-equation are used to compute the predicted Rayleigh-wave arrivals and the misfit gradient at every iteration. This procedure, denoted as wave-equation Qs inversion (WQs), does not require the assumption of a layered model and tends to have fast and robust convergence compared to full waveform inversion (FWI). Numerical examples with synthetic and field data demonstrate that the WQs method can accurately invert for a smoothed approximation to the subsurface Qs distribution as long as the Vs model is known with sufficient accuracy.
Skeletonized wave-equation Qs tomography using surface waves
Li, Jing
2017-08-17
We present a skeletonized inversion method that inverts surface-wave data for the Qs quality factor. Similar to the inversion of dispersion curves for the S-wave velocity model, the complicated surface-wave arrivals are skeletonized as simpler data, namely the amplitude spectra of the windowed Rayleigh-wave arrivals. The optimal Qs model is then found that minimizes the difference in the peak frequencies of the predicted and observed Rayleigh wave arrivals using a gradient-based wave-equation optimization method. Solutions to the viscoelastic wave-equation are used to compute the predicted Rayleigh-wave arrivals and the misfit gradient at every iteration. This procedure, denoted as wave-equation Qs tomography (WQs), does not require the assumption of a layered model and tends to have fast and robust convergence compared to Q full waveform inversion (Q-FWI). Numerical examples with synthetic and field data demonstrate that the WQs method can accurately invert for a smoothed approximation to the subsur-face Qs distribution as long as the Vs model is known with sufficient accuracy.
A unifying fractional wave equation for compressional and shear waves.
Holm, Sverre; Sinkus, Ralph
2010-01-01
This study has been motivated by the observed difference in the range of the power-law attenuation exponent for compressional and shear waves. Usually compressional attenuation increases with frequency to a power between 1 and 2, while shear wave attenuation often is described with powers less than 1. Another motivation is the apparent lack of partial differential equations with desirable properties such as causality that describe such wave propagation. Starting with a constitutive equation which is a generalized Hooke's law with a loss term containing a fractional derivative, one can derive a causal fractional wave equation previously given by Caputo [Geophys J. R. Astron. Soc. 13, 529-539 (1967)] and Wismer [J. Acoust. Soc. Am. 120, 3493-3502 (2006)]. In the low omegatau (low-frequency) case, this equation has an attenuation with a power-law in the range from 1 to 2. This is consistent with, e.g., attenuation in tissue. In the often neglected high omegatau (high-frequency) case, it describes attenuation with a power-law between 0 and 1, consistent with what is observed in, e.g., dynamic elastography. Thus a unifying wave equation derived properly from constitutive equations can describe both cases.
Radio wave propagation and parabolic equation modeling
Apaydin, Gokhan
2018-01-01
A thorough understanding of electromagnetic wave propagation is fundamental to the development of sophisticated communication and detection technologies. The powerful numerical methods described in this book represent a major step forward in our ability to accurately model electromagnetic wave propagation in order to establish and maintain reliable communication links, to detect targets in radar systems, and to maintain robust mobile phone and broadcasting networks. The first new book on guided wave propagation modeling and simulation to appear in nearly two decades, Radio Wave Propagation and Parabolic Equation Modeling addresses the fundamentals of electromagnetic wave propagation generally, with a specific focus on radio wave propagation through various media. The authors explore an array of new applications, and detail various v rtual electromagnetic tools for solving several frequent electromagnetic propagation problems. All of the methods described are presented within the context of real-world scenari...
Multicomponent integrable wave equations: II. Soliton solutions
Energy Technology Data Exchange (ETDEWEB)
Degasperis, A [Dipartimento di Fisica, Universita di Roma ' La Sapienza' , and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Rome (Italy); Lombardo, S [School of Mathematics, University of Manchester, Alan Turing Building, Upper Brook Street, Manchester M13 9EP (United Kingdom)], E-mail: antonio.degasperis@roma1.infn.it, E-mail: sara.lombardo@manchester.ac.uk, E-mail: sara@few.vu.nl
2009-09-25
The Darboux-dressing transformations developed in Degasperis and Lombardo (2007 J. Phys. A: Math. Theor. 40 961-77) are here applied to construct soliton solutions for a class of boomeronic-type equations. The vacuum (i.e. vanishing) solution and the generic plane wave solution are both dressed to yield one-soliton solutions. The formulae are specialized to the particularly interesting case of the resonant interaction of three waves, a well-known model which is of boomeronic type. For this equation a novel solution which describes three locked dark pulses (simulton) is introduced.
Space-time fractional Zener wave equation.
Atanackovic, T M; Janev, M; Oparnica, Lj; Pilipovic, S; Zorica, D
2015-02-08
The space-time fractional Zener wave equation, describing viscoelastic materials obeying the time-fractional Zener model and the space-fractional strain measure, is derived and analysed. This model includes waves with finite speed, as well as non-propagating disturbances. The existence and the uniqueness of the solution to the generalized Cauchy problem are proved. Special cases are investigated and numerical examples are presented.
Space–time fractional Zener wave equation
Atanackovic, T.M.; Janev, M.; Oparnica, Lj.; Pilipovic, S.; Zorica, D.
2015-01-01
The space–time fractional Zener wave equation, describing viscoelastic materials obeying the time-fractional Zener model and the space-fractional strain measure, is derived and analysed. This model includes waves with finite speed, as well as non-propagating disturbances. The existence and the uniqueness of the solution to the generalized Cauchy problem are proved. Special cases are investigated and numerical examples are presented. PMID:25663807
Gabor Wave Packet Method to Solve Plasma Wave Equations
Energy Technology Data Exchange (ETDEWEB)
A. Pletzer; C.K. Phillips; D.N. Smithe
2003-06-18
A numerical method for solving plasma wave equations arising in the context of mode conversion between the fast magnetosonic and the slow (e.g ion Bernstein) wave is presented. The numerical algorithm relies on the expansion of the solution in Gaussian wave packets known as Gabor functions, which have good resolution properties in both real and Fourier space. The wave packets are ideally suited to capture both the large and small wavelength features that characterize mode conversion problems. The accuracy of the scheme is compared with a standard finite element approach.
A Modified Wide Angle Parabolic Wave Equation
St. Mary, Donald F.; Lee, Ding; Botseas, George
1987-08-01
We demonstrate the implicit finite difference discretization of a higher order parabolic-like partial differential equation approximating the reduced wave equation in the far field and show that the discretization is unconditionally stable. We discuss a method of associating an angle of dispersion with parabolic approximations, present an example which can be used to compare methods on the basis of dispersion angle, and make comparisons among well-known methods and the new method.
Adomian decomposition method used to solve the gravity wave equations
Mungkasi, Sudi; Dheno, Maria Febronia Sedho
2017-01-01
The gravity wave equations are considered. We solve these equations using the Adomian decomposition method. We obtain that the approximate Adomian decomposition solution to the gravity wave equations is accurate (physically correct) for early stages of fluid flows.
Partial Differential Equations and Solitary Waves Theory
Wazwaz, Abdul-Majid
2009-01-01
"Partial Differential Equations and Solitary Waves Theory" is a self-contained book divided into two parts: Part I is a coherent survey bringing together newly developed methods for solving PDEs. While some traditional techniques are presented, this part does not require thorough understanding of abstract theories or compact concepts. Well-selected worked examples and exercises shall guide the reader through the text. Part II provides an extensive exposition of the solitary waves theory. This part handles nonlinear evolution equations by methods such as Hirota’s bilinear method or the tanh-coth method. A self-contained treatment is presented to discuss complete integrability of a wide class of nonlinear equations. This part presents in an accessible manner a systematic presentation of solitons, multi-soliton solutions, kinks, peakons, cuspons, and compactons. While the whole book can be used as a text for advanced undergraduate and graduate students in applied mathematics, physics and engineering, Part II w...
Tucker, Vance A.
1971-01-01
Capillary and gravity water waves are related to the position, wavelength, and velocity of an object in flowing water. Water patterns are presented for ships and the whirling beetle with an explanation of how the design affects the objects velocity and the observed water wavelengths. (DS)
On the so called rogue waves in nonlinear Schrodinger equations
Directory of Open Access Journals (Sweden)
Y. Charles Li
2016-04-01
Full Text Available The mechanism of a rogue water wave is still unknown. One popular conjecture is that the Peregrine wave solution of the nonlinear Schrodinger equation (NLS provides a mechanism. A Peregrine wave solution can be obtained by taking the infinite spatial period limit to the homoclinic solutions. In this article, from the perspective of the phase space structure of these homoclinic orbits in the infinite dimensional phase space where the NLS defines a dynamical system, we examine the observability of these homoclinic orbits (and their approximations. Our conclusion is that these approximate homoclinic orbits are the most observable solutions, and they should correspond to the most common deep ocean waves rather than the rare rogue waves. We also discuss other possibilities for the mechanism of a rogue wave: rough dependence on initial data or finite time blow up.
New exact travelling wave solutions of bidirectional wave equations
Indian Academy of Sciences (India)
where a, b, c and d are real constants. Here x represents the distance along the channel, t is the elapsed time, the variable v(x, t) is the dimensionless deviation of the water surface from its undisturbed position and u(x, t) is the dimensionless horizontal velocity. This set of equations is used as a model equation for the ...
Skeletonized wave-equation inversion for Q
Dutta, Gaurav
2016-09-06
A wave-equation gradient optimization method is presented that inverts for the subsurface Q distribution by minimizing a skeletonized misfit function ε. Here, ε is the sum of the squared differences between the observed and the predicted peak/centroid frequency shifts of the early-arrivals. The gradient is computed by migrating the observed traces weighted by the frequency-shift residuals. The background Q model is perturbed until the predicted and the observed traces have the same peak frequencies or the same centroid frequencies. Numerical tests show that an improved accuracy of the inverted Q model by wave-equation Q tomography (WQ) leads to a noticeable improvement in the migration image quality.
Second-Order Time-Dependent Mild-Slope Equation for Wave Transformation
Directory of Open Access Journals (Sweden)
Ching-Piao Tsai
2014-01-01
Full Text Available This study is to propose a wave model with both wave dispersivity and nonlinearity for the wave field without water depth restriction. A narrow-banded sea state centred around a certain dominant wave frequency is considered for applications in coastal engineering. A system of fully nonlinear governing equations is first derived by depth integration of the incompressible Navier-Stokes equation in conservative form. A set of second-order nonlinear time-dependent mild-slope equations is then developed by a perturbation scheme. The present nonlinear equations can be simplified to the linear time-dependent mild-slope equation, nonlinear long wave equation, and traditional Boussinesq wave equation, respectively. A finite volume method with the fourth-order Adams-Moulton predictor-corrector numerical scheme is adopted to directly compute the wave transformation. The validity of the present model is demonstrated by the simulation of the Stokes wave, cnoidal wave, and solitary wave on uniform depth, nonlinear wave shoaling on a sloping beach, and wave propagation over an elliptic shoal. The nearshore wave transformation across the surf zone is simulated for 1D wave on a uniform slope and on a composite bar profile and 2D wave field around a jetty. These computed wave height distributions show very good agreement with the experimental results available.
Skeletonized Least Squares Wave Equation Migration
Zhan, Ge
2010-10-17
The theory for skeletonized least squares wave equation migration (LSM) is presented. The key idea is, for an assumed velocity model, the source‐side Green\\'s function and the geophone‐side Green\\'s function are computed by a numerical solution of the wave equation. Only the early‐arrivals of these Green\\'s functions are saved and skeletonized to form the migration Green\\'s function (MGF) by convolution. Then the migration image is obtained by a dot product between the recorded shot gathers and the MGF for every trial image point. The key to an efficient implementation of iterative LSM is that at each conjugate gradient iteration, the MGF is reused and no new finitedifference (FD) simulations are needed to get the updated migration image. It is believed that this procedure combined with phase‐encoded multi‐source technology will allow for the efficient computation of wave equation LSM images in less time than that of conventional reverse time migration (RTM).
Mandal, Birendra Nath
2015-01-01
The theory of water waves is most varied and is a fascinating topic. It includes a wide range of natural phenomena in oceans, rivers, and lakes. It is mostly concerned with elucidation of some general aspects of wave motion including the prediction of behaviour of waves in the presence of obstacles of some special configurations that are of interest to ocean engineers. Unfortunately, even the apparently simple problems appear to be difficult to tackle mathematically unless some simplified assumptions are made. Fortunately, one can assume water to be an incompressible, in viscid and homogeneous
Wave-equation reflection traveltime inversion
Zhang, Sanzong
2011-01-01
The main difficulty with iterative waveform inversion using a gradient optimization method is that it tends to get stuck in local minima associated within the waveform misfit function. This is because the waveform misfit function is highly nonlinear with respect to changes in the velocity model. To reduce this nonlinearity, we present a reflection traveltime tomography method based on the wave equation which enjoys a more quasi-linear relationship between the model and the data. A local crosscorrelation of the windowed downgoing direct wave and the upgoing reflection wave at the image point yields the lag time that maximizes the correlation. This lag time represents the reflection traveltime residual that is back-projected into the earth model to update the velocity in the same way as wave-equation transmission traveltime inversion. No travel-time picking is needed and no high-frequency approximation is assumed. The mathematical derivation and the numerical examples are presented to partly demonstrate its efficiency and robustness. © 2011 Society of Exploration Geophysicists.
Invariance analysis and conservation laws of the wave equation on ...
Indian Academy of Sciences (India)
In this paper we discuss symmetries of classes of wave equations that arise as a consequence of some Vaidya metrics. We show how the wave equation is altered by the underlying geometry. In particular, a range of consequences on the form of the wave equation, the symmetries and number of conservation laws, inter alia ...
Exact traveling wave solutions of some nonlinear evolution equations
Kumar, Hitender; Chand, Fakir
2014-02-01
Using a traveling wave reduction technique, we have shown that Maccari equation, (2+1)-dimensional nonlinear Schrödinger equation, medium equal width equation, (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation, (2+1)-dimensional long wave-short wave resonance interaction equation, perturbed nonlinear Schrödinger equation can be reduced to the same family of auxiliary elliptic-like equations. Then using extended F-expansion and projective Riccati equation methods, many types of exact traveling wave solutions are obtained. With the aid of solutions of the elliptic-like equation, more explicit traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions are found out. It is shown that these methods provide a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. A variety of structures of the exact solutions of the elliptic-like equation are illustrated.
2016-01-01
This volume brings together four lecture courses on modern aspects of water waves. The intention, through the lectures, is to present quite a range of mathematical ideas, primarily to show what is possible and what, currently, is of particular interest. Water waves of large amplitude can only be fully understood in terms of nonlinear effects, linear theory being not adequate for their description. Taking advantage of insights from physical observation, experimental evidence and numerical simulations, classical and modern mathematical approaches can be used to gain insight into their dynamics. The book presents several avenues and offers a wide range of material of current interest. Due to the interdisciplinary nature of the subject, the book should be of interest to mathematicians (pure and applied), physicists and engineers. The lectures provide a useful source for those who want to begin to investigate how mathematics can be used to improve our understanding of water wave phenomena. In addition, some of the...
Travelling Waves in Hyperbolic Chemotaxis Equations
Xue, Chuan
2010-10-16
Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235-248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically. © 2010 Society for Mathematical Biology.
Conservation laws for equations related to soil water equations
Directory of Open Access Journals (Sweden)
Khalique C. M.
2005-01-01
Full Text Available We obtain all nontrivial conservation laws for a class of ( 2+1 nonlinear evolution partial differential equations which are related to the soil water equations. It is also pointed out that nontrivial conservation laws exist for certain classes of equations which admit point symmetries. Moreover, we associate symmetries with conservation laws for special classes of these equations.
Stochastic solution to a time-fractional attenuated wave equation.
Meerschaert, Mark M; Straka, Peter; Zhou, Yuzhen; McGough, Robert J
2012-10-01
The power law wave equation uses two different fractional derivative terms to model wave propagation with power law attenuation. This equation averages complex nonlinear dynamics into a convenient, tractable form with an explicit analytical solution. This paper develops a random walk model to explain the appearance and meaning of the fractional derivative terms in that equation, and discusses an application to medical ultrasound. In the process, a new strictly causal solution to this fractional wave equation is developed.
Saha Ray, S.; Sahoo, S.
2017-01-01
In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely time fractional modified Kawahara equations by using the ( G^'/G)-expansion method via fractional complex transform. As a result, new types of exact analytical solutions are obtained.
Wave-equation Based Earthquake Location
Tong, P.; Yang, D.; Yang, X.; Chen, J.; Harris, J.
2014-12-01
Precisely locating earthquakes is fundamentally important for studying earthquake physics, fault orientations and Earth's deformation. In industry, accurately determining hypocenters of microseismic events triggered in the course of a hydraulic fracturing treatment can help improve the production of oil and gas from unconventional reservoirs. We develop a novel earthquake location method based on solving full wave equations to accurately locate earthquakes (including microseismic earthquakes) in complex and heterogeneous structures. Traveltime residuals or differential traveltime measurements with the waveform cross-correlation technique are iteratively inverted to obtain the locations of earthquakes. The inversion process involves the computation of the Fréchet derivative with respect to the source (earthquake) location via the interaction between a forward wavefield emitting from the source to the receiver and an adjoint wavefield reversely propagating from the receiver to the source. When there is a source perturbation, the Fréchet derivative not only measures the influence of source location but also the effects of heterogeneity, anisotropy and attenuation of the subsurface structure on the arrival of seismic wave at the receiver. This is essential for the accuracy of earthquake location in complex media. In addition, to reduce the computational cost, we can first assume that seismic wave only propagates in a vertical plane passing through the source and the receiver. The forward wavefield, adjoint wavefield and Fréchet derivative with respect to the source location are all computed in a 2D vertical plane. By transferring the Fréchet derivative along the horizontal direction of the 2D plane into the ones along Latitude and Longitude coordinates or local 3D Cartesian coordinates, the source location can be updated in a 3D geometry. The earthquake location obtained with this combined 2D-3D approach can then be used as the initial location for a true 3D wave-equation
Separate P‐ and SV‐wave equations for VTI media
Pestana, Reynam C.
2011-01-01
In isotropic media we use the scalar acoustic wave equation to perform reverse time migration RTM of the recorded pressure wavefleld data. In anisotropic media P- and SV-waves are coupled and the elastic wave equation should be used for RTM. However, an acoustic anisotropic wave equation is often used instead. This results in significant shear wave energy in both modeling and RTM. To avoid this undesired SV-wave energy, we propose a different approach to separate P- and SV-wave components for vertical transversely isotropic VTI media. We derive independent pseudo-differential wave equations for each mode. The derived equations for P- and SV-waves are stable and reduce to the isotropic case. The equations presented here can be effectively used to model and migrate seismic data in VTI media where ε - δ is small. The SV-wave equation we develop is now well-posed and triplications in the SV wavefront are removed resulting in stable wave propagation. We show modeling and RTM results using the derived pure P-wave mode in complex VTI media and use the rapid expansion method REM to propagate the waveflelds in time. © 2011 Society of Exploration Geophysicists.
Initial-boundary value problems for the wave equation
Directory of Open Access Journals (Sweden)
Tynysbek Sh. Kalmenov
2014-02-01
Full Text Available In this work we consider an initial-boundary value problem for the one-dimensional wave equation. We prove the uniqueness of the solution and show that the solution coincides with the wave potential.
Approximate equations at breaking for nearshore wave transformation coefficients
Digital Repository Service at National Institute of Oceanography (India)
Chandramohan, P.; Nayak, B.U.; SanilKumar, V.
Based on small amplitude wave theory approximate equations are evaluated for determining the coefficients of shoaling, refraction, bottom friction, bottom percolation and viscous dissipation at breaking. The results obtainEd. by these equations...
Deep water periodic waves as Hamiltonian relative equilibria
van Groesen, Embrecht W.C.; Lie She Liam, L.S.L.; Lakhturov, I.; Andonowati, A.; Biggs, N.
2007-01-01
We use a recently derived KdV-type of equation for waves on deep water to study Stokes waves as relative equilibria. Special attention is given to investigate the cornered Stokes-120 degree wave as a singular solution in the class of smooth steady wave profiles.
On the strongly damped wave equation and the heat equation with mixed boundary conditions
Directory of Open Access Journals (Sweden)
Aloisio F. Neves
2000-01-01
Full Text Available We study two one-dimensional equations: the strongly damped wave equation and the heat equation, both with mixed boundary conditions. We prove the existence of global strong solutions and the existence of compact global attractors for these equations in two different spaces.
Wang, Zhenyu; Li, Chunyang; Zatianina, Razafizana; Zhang, Pei; Zhang, Yongqiang
2017-11-01
Cloaking is a challenging topic in the field of wave motion, and is of significant theoretical value. In this article, a type of carpet cloak has been theoretically designed for water waves by using the effective medium and transformation theory. This carpet cloak device, created by a three-dimensional printer, is composed of a periodic structure which realizes the equivalent anisotropic water depth. We demonstrate its excellent cloaking performance numerically and experimentally in a wide range of frequencies and angles of incidence, with low wave attenuation characteristics and simple device realization of this carpet cloak illustrating that water wave transformation is a powerful method with which to manipulate water waves.
Wave equations on anti self dual (ASD) manifolds
Bashingwa, Jean-Juste; Kara, A. H.
2017-11-01
In this paper, we study and perform analyses of the wave equation on some manifolds with non diagonal metric g_{ij} which are of neutral signatures. These include the invariance properties, variational symmetries and conservation laws. In the recent past, wave equations on the standard (space time) Lorentzian manifolds have been performed but not on the manifolds from metrics of neutral signatures.
Local energy decay for linear wave equations with variable coefficients
Ikehata, Ryo
2005-06-01
A uniform local energy decay result is derived to the linear wave equation with spatial variable coefficients. We deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data, and its proof is quite simple. This generalizes a previous famous result due to Morawetz [The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961) 561-568]. In order to prove local energy decay, we mainly apply two types of ideas due to Ikehata-Matsuyama [L2-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002) 33-42] and Todorova-Yordanov [Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].
Orbital stability of standing waves for some nonlinear Schroedinger equations
Energy Technology Data Exchange (ETDEWEB)
Cazenave, T.; Lions, P.L.
1982-08-01
We present a general method which enables as to prove the orbital stability of some standing waves in nonlinear Schroedinger equations. For example, we treat the cases of nonlinear Schroedinger equations arising in laser beams, of time-dependent Hartree equations.
Traveling Wave Solutions of Space-Time Fractional Generalized Fifth-Order KdV Equation
Directory of Open Access Journals (Sweden)
Dianchen Lu
2017-01-01
Full Text Available The Korteweg-de Vries (KdV equation, especially the fractional higher order one, provides a relatively accurate description of motions of long waves in shallow water under gravity and wave propagation in one-dimensional nonlinear lattice. In this article, the generalized exp(-Φ(ξ-expansion method is proposed to construct exact solutions of space-time fractional generalized fifth-order KdV equation with Jumarie’s modified Riemann-Liouville derivatives. At the end, three types of exact traveling wave solutions are obtained which indicate that the method is very practical and suitable for solving nonlinear fractional partial differential equations.
Review of water wave kinematics
Energy Technology Data Exchange (ETDEWEB)
Sterndorff, M.J.
1995-03-01
The present report covers a comprehensive review of water wave kinematics carried out by Danish Hydraulic Institute (DHI) in connection with the EFP`93 project: Dynamics of Mono Tower Platforms (ref. EFP`93, 1313/93-0009). This project is carried out in cooperation with Ramboell, Hannemann and Hoejlund A/S. The main objectives of the project are to develop and verify a method for the determination of the non-linear wave load and the dynamic response of mono tower platforms. One of the characteristics of mono tower platforms is that due to the small water plane area the hydrodynamic loading will be very concentrated. Such platforms may therefore respond strongly and in a highly dynamic manner to short waves and high order components of extreme waves having periods corresponding to the first natural period of the platform. A key element in the hydrodynamic load process is the wave kinematics. The present report is a comprehensive review of recent literature concerning wave theories, wave-current interaction, laboratory experiments, and field measurements of water wave kinematics. The review has been concentrated on non-breaking waves on deep to intermediate water depths. Papers concerning shallow water waves have only been reviewed if they present methods which may be applied for deep to intermediate water waves. (au) EFP-93; 30 refs.
Runge-Kutta Integration of the Equal Width Wave Equation Using the Method of Lines
Directory of Open Access Journals (Sweden)
M. A. Banaja
2015-01-01
Full Text Available The equal width (EW equation governs nonlinear wave phenomena like waves in shallow water. Numerical solution of the (EW equation is obtained by using the method of lines (MOL based on Runge-Kutta integration. Using von Neumann stability analysis, the scheme is found to be unconditionally stable. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Accuracy of the proposed method is discussed by computing the L2 and L∞ error norms. The results are found in good agreement with exact solution.
On the Exact Solution of Wave Equations on Cantor Sets
Directory of Open Access Journals (Sweden)
Dumitru Baleanu
2015-09-01
Full Text Available The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholtz and diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora studied diffusion problems in fractal media in Cantor sets. Zhang et al. studied local fractional wave equations under fixed entropy. In this paper, we are concerned with the exact solutions of wave equations by the help of local fractional Laplace variation iteration method (LFLVIM. We develop an iterative scheme for the exact solutions of local fractional wave equations (LFWEs. The efficiency of the scheme is examined by two illustrative examples.
Single integrodifferential wave equation for a Lévy walk
Fedotov, Sergei
2016-02-01
We derive the single integrodifferential wave equation for the probability density function of the position of a classical one-dimensional Lévy walk with continuous sample paths. This equation involves a classical wave operator together with memory integrals describing the spatiotemporal coupling of the Lévy walk. It is valid at all times, not only in the long time limit, and it does not involve any large-scale approximations. It generalizes the well-known telegraph or Cattaneo equation for the persistent random walk with the exponential switching time distribution. Several non-Markovian cases are considered when the particle's velocity alternates at the gamma and power-law distributed random times. In the strong anomalous case we obtain the asymptotic solution to the integrodifferential wave equation. We implement the nonlinear reaction term of Kolmogorov-Petrovsky-Piskounov type into our equation and develop the theory of wave propagation in reaction-transport systems involving Lévy diffusion.
On exact traveling-wave solutions for local fractional Korteweg-de Vries equation
Yang, Xiao-Jun; Tenreiro Machado, J. A.; Baleanu, Dumitru; Cattani, Carlo
2016-08-01
This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.
On exact traveling-wave solutions for local fractional Korteweg-de Vries equation.
Yang, Xiao-Jun; Tenreiro Machado, J A; Baleanu, Dumitru; Cattani, Carlo
2016-08-01
This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.
New exact wave solutions for Hirota equation
Indian Academy of Sciences (India)
Nonlinear partial differential equations (NPDEs) of mathematical physics are major sub- jects in physical science. With the development of soliton theory, many useful methods for obtaining exact solutions of NPDEs have been presented. Some of them are: the (G /G)- expansion method [1–4], the simplest equation method ...
Anisotropic wave-equation traveltime and waveform inversion
Feng, Shihang
2016-09-06
The wave-equation traveltime and waveform inversion (WTW) methodology is developed to invert for anisotropic parameters in a vertical transverse isotropic (VTI) meidum. The simultaneous inversion of anisotropic parameters v0, ε and δ is initially performed using the wave-equation traveltime inversion (WT) method. The WT tomograms are then used as starting background models for VTI full waveform inversion. Preliminary numerical tests on synthetic data demonstrate the feasibility of this method for multi-parameter inversion.
Unified formulation of radiation conditions for the wave equation
DEFF Research Database (Denmark)
Krenk, Steen
2002-01-01
A family of radiation conditions for the wave equation is derived by truncating a rational function approxiamtion of the corresponding plane wave representation, and it is demonstrated how these boundary conditions can be formulated in terms of fictitious surface densities, governed by second...
Travelling waves in a singularly perturbed sine-Gordon equation
Derks, Gianne; Doelman, Arjen; van Gils, Stephanus A.; Visser, T.P.P.
2003-01-01
We determine the linearised stability of travelling front solutions of a perturbed sine-Gordon equation. This equation models the long Josephson junction using the RCSJ model for currents across the junction and includes surface resistance for currents along the junction. The travelling waves
Multicomponent integrable wave equations. I. Darboux-dressing transformation
Degasperis, A.; Lombardo, S.
2007-01-01
The Darboux-dressing transformations are applied to the Lax pair associated with systems of coupled nonlinear wave equations in the case of boundary values which are appropriate to both bŕight' and dárk' soliton solutions. The general formalism is set up and the relevant equations are explicitly
Invariance analysis and conservation laws of the wave equation on ...
Indian Academy of Sciences (India)
pp. 555–570. Invariance analysis and conservation laws of the wave equation on Vaidya manifolds. R NARAIN and A H KARA. ∗. School of Mathematics and Centre for Differential Equations, Continuum Mechanics and. Applications, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050,. South Africa. ∗.
Exact traveling wave solutions for system of nonlinear evolution equations.
Khan, Kamruzzaman; Akbar, M Ali; Arnous, Ahmed H
2016-01-01
In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.
DEFF Research Database (Denmark)
Beels, Charlotte; Troch, Peter; Visch, Kenneth De
2010-01-01
Time-dependent mild-slope equations have been extensively used to compute wave transformations near coastal and offshore structures for more than 20 years. Recently the wave absorption characteristics of a Wave Energy Converter (abbreviated as WEC) of the overtopping type have been implemented...... in a time-dependent mild-slope equation model by using numerical sponge layers. In this paper the developed WEC implementation is applied to a single Wave Dragon WEC and multiple Wave Dragon WECs. The Wave Dragon WEC is a floating offshore converter of the overtopping type. Two wave reflectors focus...... the incident wave power towards a ramp. The focussed waves run up the ramp and overtop in a water reservoir above mean sea level. The obtained potential energy is converted into electricity when the stored water drains back to the sea through hydro turbines. The wave reflectors and the main body (ramp...
New exact travelling wave solutions of bidirectional wave equations
Indian Academy of Sciences (India)
where , , and d are real constants. In general, the exact travelling wave solutions will be helpful in the theoretical and numerical study of the nonlinear evolution systems. In this paper, we obtain exact travelling wave solutions of system (1) using the modiﬁed tanh–coth function method with computerized symbolic ...
New exact travelling wave solutions of some complex nonlinear equations
Bekir, Ahmet
2009-04-01
In this paper, we establish exact solutions for complex nonlinear equations. The tanh-coth and the sine-cosine methods are used to construct exact periodic and soliton solutions of these equations. Many new families of exact travelling wave solutions of the coupled Higgs and Maccari equations are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems.
Scattering of surface waves modelled by the integral equation method
Lu, Laiyu; Maupin, Valerie; Zeng, Rongsheng; Ding, Zhifeng
2008-09-01
The integral equation method is used to model the propagation of surface waves in 3-D structures. The wavefield is represented by the Fredholm integral equation, and the scattered surface waves are calculated by solving the integral equation numerically. The integration of the Green's function elements is given analytically by treating the singularity of the Hankel function at R = 0, based on the proper expression of the Green's function and the addition theorem of the Hankel function. No far-field and Born approximation is made. We investigate the scattering of surface waves propagating in layered reference models imbedding a heterogeneity with different density, as well as Lamé constant contrasts, both in frequency and time domains, for incident plane waves and point sources.
Linear fractional diffusion-wave equation for scientists and engineers
Povstenko, Yuriy
2015-01-01
This book systematically presents solutions to the linear time-fractional diffusion-wave equation. It introduces the integral transform technique and discusses the properties of the Mittag-Leffler, Wright, and Mainardi functions that appear in the solutions. The time-nonlocal dependence between the flux and the gradient of the transported quantity with the “long-tail” power kernel results in the time-fractional diffusion-wave equation with the Caputo fractional derivative. Time-nonlocal generalizations of classical Fourier’s, Fick’s and Darcy’s laws are considered and different kinds of boundary conditions for this equation are discussed (Dirichlet, Neumann, Robin, perfect contact). The book provides solutions to the fractional diffusion-wave equation with one, two and three space variables in Cartesian, cylindrical and spherical coordinates. The respective sections of the book can be used for university courses on fractional calculus, heat and mass transfer, transport processes in porous media and ...
New exact wave solutions for Hirota equation
Indian Academy of Sciences (India)
... integrals in polynomial form with a high accuracy for two-dimensional plane autonomous systems. Exact soliton solution is constructed through the established first integrals. This method is a powerful tool for searching exact travelling solutions of nonlinear partial differential equations (NPDEs) in mathematical physics.
An Object Oriented, Finite Element Framework for Linear Wave Equations
Energy Technology Data Exchange (ETDEWEB)
Koning, Joseph M. [Univ. of California, Berkeley, CA (United States)
2004-03-01
This dissertation documents an object oriented framework which can be used to solve any linear wave equation. The linear wave equations are expressed in the differential forms language. This differential forms expression allows a strict discrete interpretation of the system. The framework is implemented using the Galerkin Finite Element Method to define the discrete differential forms and operators. Finite element basis functions including standard scalar Nodal and vector Nedelec basis functions are used to implement the discrete differential forms resulting in a mixed finite element system. Discretizations of scalar and vector wave equations in the time and frequency domains will be demonstrated in both differential forms and vector calculi. This framework conserves energy, maintains physical continuity, is valid on unstructured grids, conditionally stable and second order accurate. Examples including linear electrodynamics, acoustics, elasticity and magnetohydrodynamics are demonstrated.
Finite element and discontinuous Galerkin methods for transient wave equations
Cohen, Gary
2017-01-01
This monograph presents numerical methods for solving transient wave equations (i.e. in time domain). More precisely, it provides an overview of continuous and discontinuous finite element methods for these equations, including their implementation in physical models, an extensive description of 2D and 3D elements with different shapes, such as prisms or pyramids, an analysis of the accuracy of the methods and the study of the Maxwell’s system and the important problem of its spurious free approximations. After recalling the classical models, i.e. acoustics, linear elastodynamics and electromagnetism and their variational formulations, the authors present a wide variety of finite elements of different shapes useful for the numerical resolution of wave equations. Then, they focus on the construction of efficient continuous and discontinuous Galerkin methods and study their accuracy by plane wave techniques and a priori error estimates. A chapter is devoted to the Maxwell’s system and the important problem ...
Nonlinear electrostatic wave equations for magnetized plasmas - II
DEFF Research Database (Denmark)
Dysthe, K. B.; Mjølhus, E.; Pécseli, H. L.
1985-01-01
For pt.I see ibid., vol.26, p.443-7 (1984). The problem of extending the high frequency part of the Zakharov equations for nonlinear electrostatic waves to magnetized plasmas, is considered. Weak electromagnetic and thermal effects are retained on an equal footing. Direction dependent (electrosta......For pt.I see ibid., vol.26, p.443-7 (1984). The problem of extending the high frequency part of the Zakharov equations for nonlinear electrostatic waves to magnetized plasmas, is considered. Weak electromagnetic and thermal effects are retained on an equal footing. Direction dependent...... (electrostatic) cut-off implies that various cases must be considered separately, leading to equations with rather different properties. Various equations encountered previously in the literature are recovered as limiting cases....
An Unconditionally Stable Method for Solving the Acoustic Wave Equation
Directory of Open Access Journals (Sweden)
Zhi-Kai Fu
2015-01-01
Full Text Available An unconditionally stable method for solving the time-domain acoustic wave equation using Associated Hermit orthogonal functions is proposed. The second-order time derivatives in acoustic wave equation are expanded by these orthogonal basis functions. By applying Galerkin temporal testing procedure, the time variable can be eliminated from the calculations. The restriction of Courant-Friedrichs-Levy (CFL condition in selecting time step for analyzing thin layer can be avoided. Numerical results show the accuracy and the efficiency of the proposed method.
Lecture Notes for the Course in Water Wave Mechanics
DEFF Research Database (Denmark)
Andersen, Thomas Lykke; Frigaard, Peter
The present notes are written for the course in water wave mechanics given on the 7th semester of the education in civil engineering at Aalborg University. The prerequisites for the course are the course in fluid dynamics also given on the 7th semester and some basic mathematical and physical...... knowledge. The course is at the same time an introduction to the course in coastal hydraulics on the 8th semester. The notes cover the following five lectures: 1. Definitions. Governing equations and boundary conditions. Derivation of velocity potential for linear waves. Dispersion relationship. 2. Particle...... paths, velocities, accelerations, pressure variation, deep and shallow water waves, wave energy and group velocity. 3. Shoaling, refraction, diffraction and wave breaking. 4. Irregular waves. Time domain analysis of waves. 5. Wave spectra. Frequency domain analysis of waves. The present notes are based...
Solitary Wave Solutions of the Boussinesq Equation and Its Improved Form
Directory of Open Access Journals (Sweden)
Reza Abazari
2013-01-01
Full Text Available This paper presents the general case study of previous works on generalized Boussinesq equations, (Abazari, 2011 and (Kılıcman and Abazari, 2012, that focuses on the application of G′/G-expansion method with the aid of Maple to construct more general exact solutions for the coupled Boussinesq equations. In this work, the mentioned method is applied to construct more general exact solutions of Boussinesq equation and improved Boussinesq equation, which the French scientist Joseph Valentin Boussinesq (1842–1929 described in the 1870s model equations for the propagation of long waves on the surface of water with small amplitude. Our work is motivated by the fact that the G′/G-expansion method provides not only more general forms of solutions but also periodic, solitary waves and rational solutions. The method appears to be easier and faster by means of a symbolic computation.
Asymptotics of solutions to semilinear stochastic wave equations
Chow, Pao-Liu
2006-01-01
Large-time asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are established. Under appropriate conditions, the existence theorem for a unique global solution is given. Next the questions of bounded solutions and the exponential stability of an equilibrium solution, in mean-square and the almost sure sense, are studied. Then...
Symmetries, Conservation Laws, and Wave Equation on the Milne Metric
Directory of Open Access Journals (Sweden)
Ahmad M. Ahmad
2012-01-01
representing physical systems. For partial differential equation possessing Lagrangians these symmetries are obtained by the invariance of the corresponding action integral. In this paper we provide a systematic procedure for determining Noether symmetries and conserved vectors for a Lagrangian constructed from a Lorentzian metric of interest in mathematical physics. For completeness, we give Lie point symmetries and conservation laws admitted by the wave equation on this Lorentzian metric.
Resolution limits for wave equation imaging
Huang, Yunsong
2014-08-01
Formulas are derived for the resolution limits of migration-data kernels associated with diving waves, primary reflections, diffractions, and multiple reflections. They are applicable to images formed by reverse time migration (RTM), least squares migration (LSM), and full waveform inversion (FWI), and suggest a multiscale approach to iterative FWI based on multiscale physics. That is, at the early stages of the inversion, events that only generate low-wavenumber resolution should be emphasized relative to the high-wavenumber resolution events. As the iterations proceed, the higher-resolution events should be emphasized. The formulas also suggest that inverting multiples can provide some low- and intermediate-wavenumber components of the velocity model not available in the primaries. Finally, diffractions can provide twice or better the resolution than specular reflections for comparable depths of the reflector and diffractor. The width of the diffraction-transmission wavepath is approximately λ at the diffractor location for the diffraction-transmission wavepath. © 2014 Elsevier B.V.
An acoustic wave equation for pure P wave in 2D TTI media
Zhan, Ge
2011-01-01
In this paper, a pure P wave equation for an acoustic 2D TTI media is derived. Compared with conventional TTI coupled equations, the resulting equation is unconditionally stable due to the complete isolation of the SV wave mode. To avoid numerical dispersion and produce high quality images, the rapid expansion method REM is employed for numerical implementation. Synthetic results validate the proposed equation and show that it is a stable algorithm for modeling and reverse time migration RTM in a TTI media for any anisotropic parameter values. © 2011 Society of Exploration Geophysicists.
Study of nonlinear waves described by the cubic Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Walstead, A.E.
1980-03-12
The cubic Schroedinger equation (CSE) is ubiquitous as a model equation for the long-time evolution of finite-amplitude near-monochromatic dispersive waves. It incorporates the effects of the radiation field pressure on the constitutive properties of the supporting medium in a self-consistent manner. The properties of the uniformly transiating periodic wave solutions of the one-dimensional CSE are studied here. These (so-called cnoidal) waves are characterized by the values of four parameters. Whitham's averaged variational principle is used to derive a system of quasilinear evolution equations (the modulational equations) for the values of these parameters when they are slowly varying in space and time. Explicit expressions for the characteristic velocities of the modulational equations are obtained for the full set of cnoidal waves. Riemann invariants are obtained for several limits for the stable case, and growth rates are obtained for several limits, including the solitary wave chain, for the unstable case. The results for several nontrivial limiting cases agree with those obtained by independent methods by others. The dynamics of the CSE generalized to two spatial dimensions are studied for the unstable case. A large class of similarity solutions with cylindrical symmetry are obtained systematically using infinitesimal transformation group techniques. The methods are adapted to obtain the symmetries of the action functional of the CSE and to deduce nine integral invariants. A numerical study of the self-similar solutions reveals that they are modulationally unstable and that singularities dominate the dynamics of the CSE in two dimensions. The CSE is derived using perturbation theory for a specific problem in plasma physics: the evolution of the envelope of a near-monochromatic electromagnetic wave in a cold magnetized plasma. 13 figures, 2 tables.
Plane waves and spherical means applied to partial differential equations
John, Fritz
2004-01-01
Elementary and self-contained, this heterogeneous collection of results on partial differential equations employs certain elementary identities for plane and spherical integrals of an arbitrary function, showing how a variety of results on fairly general differential equations follow from those identities. The first chapter deals with the decomposition of arbitrary functions into functions of the type of plane waves. Succeeding chapters introduce the first application of the Radon transformation and examine the solution of the initial value problem for homogeneous hyperbolic equations with con
The wave equation: From eikonal to anti-eikonal approximation
Directory of Open Access Journals (Sweden)
Luis Vázquez
2016-06-01
Full Text Available When the refractive index changes very slowly compared to the wave-length we may use the eikonal approximation to the wave equation. In the opposite case, when the refractive index highly variates over the distance of one wave-length, we have what can be termed as the anti-eikonal limit. This situation is addressed in this work. The anti-eikonal limit seems to be a relevant tool in the modelling and design of new optical media. Besides, it describes a basic universal behaviour, independent of the actual values of the refractive index and, thus, of the media, for the components of a wave with wave-length much greater than the characteristic scale of the refractive index.
Estimation of freak wave occurrence in shallow water regions
Kashima, Hiroaki
2014-05-01
In the last two decades, freak waves have become an important topic in engineering and science and are sometimes featured by a single and steep crest causing severe damage to offshore structures and vessels. An accurate estimation of maximum wave height and prediction of freak wave occurrence frequency is important for marine safety and ocean developments. According to several studies on freak waves, the deep-water third-order nonlinearity (quasi-resonant four-wave interactions) can lead to a significant enhancement of freak wave occurrence from normality. However, it is not clear the behavior of offshore generated freak waves shoaling to shallow water regions. In general, a numerical simulation based on Boussinesq model has been frequently and widely used to estimate wave transformation in shallow water regions and has high-level performance in the design of coast and harbor structures in Japan. However, it is difficult to describe the freak wave occurrence from deep to shallow water regions by the Boussinesq model because it can express only up to the second-order nonlinear interactions. There is a gap of governing equation between deep and shallow water regions from the extreme wave modeling point of view. It is necessary to investigate the aftereffects of generated freak waves by the third-order nonlinear interactions in deep water regions and their propagation to shallow water regions using the Boussinesq model. In this study, the model experiments in a wave tank and numerical simulations based on the Boussinesq model were conducted to estimate the freak wave occurrence from deep to shallow water regions. In the model experiments, the maximum wave height increases with an increase in kurtosis by the third-order nonlinear interactions in deep water regions. The dependence of kurtosis on freak wave occurrence weakens by the second-order nonlinear interactions associated with wave shoaling if dimensionless water depth kph becomes shallower than 1.363, which kp
The general time fractional wave equation for a vibrating string
Energy Technology Data Exchange (ETDEWEB)
Sandev, Trifce [Radiation Safety Directorate, Blv. Partizanski odredi 143, PO Box 22, 1020 Skopje (Macedonia, The Former Yugoslav Republic of); Tomovski, Zivorad, E-mail: trifce.sandev@avis.gov.m, E-mail: tomovski@iunona.pmf.ukim.edu.m [Faculty of Natural Sciences and Mathematics, Institute of Mathematics, 1000 Skopje (Macedonia, The Former Yugoslav Republic of)
2010-02-05
The solution of a general time fractional wave equation for a vibrating string is obtained in terms of the Mittag-Leffler-type functions and complete set of eigenfunctions of the Sturm-Liouville problem. The time fractional derivative used is taken in the Caputo sense, and the method of separation of variables and the Laplace transform method are used to solve the equation. Some results for special cases of the initial and boundary conditions are obtained and it is shown that the corresponding solutions of the integer order equations are special cases of those of time fractional equations. The proposed general equation may be used for modeling different processes in complex or viscoelastic media, disordered materials, etc.
Directory of Open Access Journals (Sweden)
Weiguo Rui
2014-01-01
Full Text Available By using the integral bifurcation method together with factoring technique, we study a water wave model, a high-order nonlinear wave equation of KdV type under some newly solvable conditions. Based on our previous research works, some exact traveling wave solutions such as broken-soliton solutions, periodic wave solutions of blow-up type, smooth solitary wave solutions, and nonsmooth peakon solutions within more extensive parameter ranges are obtained. In particular, a series of smooth solitary wave solutions and nonsmooth peakon solutions are obtained. In order to show the properties of these exact solutions visually, we plot the graphs of some representative traveling wave solutions.
Comparison of fractional wave equations for power law attenuation in ultrasound and elastography.
Holm, Sverre; Näsholm, Sven Peter
2014-04-01
A set of wave equations with fractional loss operators in time and space are analyzed. The fractional Szabo equation, the power law wave equation and the causal fractional Laplacian wave equation are all found to be low-frequency approximations of the fractional Kelvin-Voigt wave equation and the more general fractional Zener wave equation. The latter two equations are based on fractional constitutive equations, whereas the former wave equations have been derived from the desire to model power law attenuation in applications like medical ultrasound. This has consequences for use in modeling and simulation, especially for applications that do not satisfy the low-frequency approximation, such as shear wave elastography. In such applications, the wave equations based on constitutive equations are the viable ones. Copyright © 2014 World Federation for Ultrasound in Medicine & Biology. Published by Elsevier Inc. All rights reserved.
Derivation of relativistic wave equation from the Poisson process
Indian Academy of Sciences (India)
A generalized linear photon wave equation in dispersive and homogeneous medium with dissipation is derived using the formulation of the Poisson process. This formulation provides a possible interpretation of the passage time of a photon moving in the medium, which never exceeds the speed of light in vacuum.
Blowing-up semilinear wave equation with exponential nonlinearity ...
Indian Academy of Sciences (India)
Indian Acad. Sci. (Math. Sci.) Vol. 123, No. 3, August 2013, pp. 365–372. c Indian Academy of Sciences. Blowing-up semilinear wave equation with exponential nonlinearity in two space dimensions. T SAANOUNI. Department of Mathematics, Faculty of Sciences of Tunis,. University of Tunis El Manar, El Manar 2092, Tunisia.
Invariant Solutions for a Class of Perturbed Nonlinear Wave Equations
Directory of Open Access Journals (Sweden)
Waheed A. Ahmed
2017-11-01
Full Text Available Approximate symmetries of a class of perturbed nonlinear wave equations are computed using two newly-developed methods. Invariant solutions associated with the approximate symmetries are constructed for both methods. Symmetries and solutions are compared through discussing the advantages and disadvantages of each method.
Exponential decay for solutions to semilinear damped wave equation
Gerbi, Stéphane
2011-10-01
This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Intro- ducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in [4].
Finite element solution of the Boussinesq wave equation | Akpobi ...
African Journals Online (AJOL)
In this work, we investigate a Boussinesq-type flow model for nonlinear dispersive waves by developing a computational model based on the finite element discretisation technique. Hermite interpolation functions were used to interpolate approximation elements. The system is modeled using a time dependent equation.
Solitary wave solutions to nonlinear evolution equations in ...
Indian Academy of Sciences (India)
Solitary wave solutions to nonlinear evolution equations in mathematical physics. ANWAR JA'AFAR MOHAMAD JAWAD1, M MIRZAZADEH2,∗ and. ANJAN BISWAS3,4. 1Computer Engineering Technique Department, Al-Rafidain University College, Baghdad, Iraq. 2Department of Engineering Sciences, Faculty of ...
Blowing-up semilinear wave equation with exponential nonlinearity ...
Indian Academy of Sciences (India)
We investigate the initial value problem for some semi-linear wave equation in two space dimensions with exponential nonlinearity growth. Author Affiliations. T Saanouni1. Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, El Manar 2092, Tunisia. Dates. Manuscript received: 26 April ...
Traveling wave solutions of the BBM-like equations
Energy Technology Data Exchange (ETDEWEB)
Kuru, S [Department of Physics, Faculty of Science, Ankara University 06100 Ankara (Turkey)], E-mail: kuru@science.ankara.edu.tr
2009-09-18
In this work, we apply the factorization technique to the Benjamin-Bona-Mahony-like equations, B(m, n), in order to get traveling wave solutions. We will focus on some special cases for which m {ne} n, and we will obtain these solutions in terms of the special forms of Weierstrass functions.
Derivation of relativistic wave equation from the Poisson process
Indian Academy of Sciences (India)
Abstract. A Poisson process is one of the fundamental descriptions for relativistic particles: both fermions and bosons. A generalized linear photon wave equation in dispersive and homogeneous medium with dissipation is derived using the formulation of the Poisson process. This formulation provides a possible ...
Comparison of artificial absorbing boundaries for acoustic wave equation modelling
Gao, Yingjie; Song, Hanjie; Zhang, Jinhai; Yao, Zhenxing
2017-12-01
Absorbing boundary conditions are necessary in numerical simulation for reducing the artificial reflections from model boundaries. In this paper, we overview the most important and typical absorbing boundary conditions developed throughout history. We first derive the wave equations of similar methods in unified forms; then, we compare their absorbing performance via theoretical analyses and numerical experiments. The Higdon boundary condition is shown to be the best one among the three main absorbing boundary conditions that are based on a one-way wave equation. The Clayton and Engquist boundary is a special case of the Higdon boundary but has difficulty in dealing with the corner points in implementaion. The Reynolds boundary does not have this problem but its absorbing performance is the poorest among these three methods. The sponge boundary has difficulties in determining the optimal parameters in advance and too many layers are required to achieve a good enough absorbing performance. The hybrid absorbing boundary condition (hybrid ABC) has a better absorbing performance than the Higdon boundary does; however, it is still less efficient for absorbing nearly grazing waves since it is based on the one-way wave equation. In contrast, the perfectly matched layer (PML) can perform much better using a few layers. For example, the 10-layer PML would perform well for absorbing most reflected waves except the nearly grazing incident waves. The 20-layer PML is suggested for most practical applications. For nearly grazing incident waves, convolutional PML shows superiority over the PML when the source is close to the boundary for large-scale models. The Higdon boundary and hybrid ABC are preferred when the computational cost is high and high-level absorbing performance is not required, such as migration and migration velocity analyses, since they are not as sensitive to the amplitude errors as the full waveform inversion.
Reinterpretation of Matter-Wave Interference Experiments Based on the Local-Ether Wave Equation
Su, Ching-Chuan
2002-01-01
Based on the local-ether wave equation for free particle, the dispersion of matter wave is examined. From the dispersion relation, the angular frequency and wavelength of matter wave are derived. These formulas look like the postulates of de Broglie in conjunction with the Lorentz mass-variation law. However, the fundamental difference is that for terrestrial particles their speeds are referred specifically to a geocentric inertial frame and hence incorporate the speed due to earth's rotation...
Water waves generated by impulsively moving obstacle
Makarenko, Nikolay; Kostikov, Vasily
2017-04-01
There are several mechanisms of tsunami-type wave formation such as piston displacement of the ocean floor due to a submarine earthquake, landslides, etc. We consider simplified mathematical formulation which involves non-stationary Euler equations of infinitely deep ideal fluid with submerged compact wave-maker. We apply semi-analytical method [1] based on the reduction of fully nonlinear water wave problem to the integral-differential system for the wave elevation together with normal and tangential fluid velocities at the free surface. Recently, small-time asymptotic solutions were constructed by this method for submerged piston modeled by thin elliptic cylinder which starts with constant acceleration from rest [2,3]. By that, the leading-order solution terms describe several regimes of non-stationary free surface flow such as formation of inertial fluid layer, splash jets and diverging waves over the obstacle. Now we construct asymptotic solution taking into account higher-order nonlinear terms in the case of submerged circular cylinder. The role of non-linearity in the formation mechanism of surface waves is clarified in comparison with linear approximations. This work was supported by RFBR (grant No 15-01-03942). References [1] Makarenko N.I. Nonlinear interaction of submerged cylinder with free surface, JOMAE Trans. ASME, 2003, 125(1), 75-78. [2] Makarenko N.I., Kostikov V.K. Unsteady motion of an elliptic cylinder under a free surface, J. Appl. Mech. Techn. Phys., 2013, 54(3), 367-376. [3] Makarenko N.I., Kostikov V.K. Non-linear water waves generated by impulsive motion of submerged obstacle, NHESS, 2014, 14(4), 751-756.
Real time wave measurements and wave hindcasting in deep waters
Digital Repository Service at National Institute of Oceanography (India)
Anand, N.M.; Mandal, S.; SanilKumar, V.; Nayak, B.U.
Deep water waves off Karwar (lat. 14~'45.1'N, long. 73~'34.8'E) at 75 m water depth pertaining to peak monsoon period have been measured using a Datawell waverider buoy. Measured wave data show that the significant wave height (Hs) predominantly...
Simon, Bruno; Seez, William; Abid, Malek; Kharif, Christian; Touboul, Julien
2017-04-01
During the last ten years, the topic of water waves interacting with sheared current has drawn a lot of attention, since the interaction of water waves with vorticity was recently found to be significant when modeling the propagation of water waves. In this framework, the configuration involving constantly sheared current (indeed a constant vorticity) is of special interst, since the equations remain tractable. In this framework, it is demonstrated that the flow related to water waves can be described by means of potential theory, since the source term in the vorticity equation is proportionnal to the curvature of the current profile (Nwogu, 2009). In the mean time, the community often wonders if this argument is valid, since the existence of a perfectly linearly sheared current is purely theoretical, and the presence of the vorticity within the wave field can be external (through wave generation mechanisms, for instance). Thus, this work is dedicated to investigate the magnitude of the vorticity related to the wave field, in conditions similar to this analytical case of constant vorticity. This approach is based on the comparison of experimental data, and three models. The first model is linear, supposing a constantly seared current and water waves described by potential theory. The second is fully nonlinear, but still supposing that water waves are potential, and finally, the third model is fully nonlinear, but solves the Euler equations, allowing the existence of vorticity related to the waves. The confrontation of these three approaches with the experimental data will allow to quantify the wave-related vorticity within the total flow, and analyze its importance as a function of nonlinearity and vorticity magnitude. ACKNOWLEDGEMENTS The DGA (Direction Générale de l'Armement, France) is acknowledged for its financial support through the ANR grant N°ANR-13-ASTR-0007.
The wave equation for stiff strings and piano tuning
Gràcia, Xavier
2016-01-01
We study the wave equation for a string with stiffness. We solve the equation and provide a uniqueness theorem with suitable boundary conditions. For a pinned string we compute the spectrum, which is slightly inharmonic. Therefore, the widespread scale of 12 equal divisions of the just octave is not the best choice to tune instruments like the piano. Basing in the theory of dissonance, we provide a way to tune the piano in order to improve its consonance. A good solution is obtained by tuning a note and its fifth by minimizing their beats.
Quaternion wave equations in curved space-time
Edmonds, J. D., Jr.
1974-01-01
The quaternion formulation of relativistic quantum theory is extended to include curvilinear coordinates and curved space-time in order to provide a framework for a unified quantum/gravity theory. Six basic quaternion fields are identified in curved space-time, the four-vector basis quaternions are identified, and the necessary covariant derivatives are obtained. Invariant field equations are derived, and a general invertable coordinate transformation is developed. The results yield a way of writing quaternion wave equations in curvilinear coordinates and curved space-time as well as a natural framework for solving the problem of second quantization for gravity.
Stochastic regulator theory for a class of abstract wave equations
Balakrishnan, A. V.
1991-01-01
A class of steady-state stochastic regulator problems for abstract wave equations in a Hilbert space - of relevance to the problem of feedback control of large space structures using co-located controls/sensors - is studied. Both the control operator, as well as the observation operator, are finite-dimensional. As a result, the usual condition of exponential stabilizability invoked for existence of solutions to the steady-state Riccati equations is not valid. Fortunately, for the problems considered it turns out that strong stabilizability suffices. In particular, a closed form expression is obtained for the minimal (asymptotic) performance criterion as the control effort is allowed to grow without bound.
Derivation of asymptotic two-dimensional time-dependent equations for ocean wave propagation
Lannes, David
2007-01-01
A general method for the derivation of asymptotic nonlinear shallow water and deep water models is presented. Starting from a general dimensionless version of the water-wave equations, we reduce the problem to a system of two equations on the surface elevation and the velocity potential at the free surface. These equations involve a Dirichlet-Neumann operator and we show that all the asymptotic models can be recovered by a simple asymptotic expansion of this operator, in function of the shallowness parameter (shallow water limit) or the steepness parameter (deep water limit). Based on this method, a new two-dimensional fully dispersive model for small wave steepness is also derived, which extends to uneven bottom the approach developed by Matsuno \\cite{matsuno3} and Choi \\cite{choi}. This model is still valid in shallow water but with less precision than what can be achieved with Green-Naghdi model, when fully nonlinear waves are considered. The combination, or the coupling, of the new fully dispersive equati...
Multiobjective optimal control of the linear wave equation
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Hassan Zarei
2014-12-01
Full Text Available In this paper, we propose a method for the solution of a multiobjective optimal control problem (MOOCP in a linear distributed-parameter system governed by a wave equation. An explicit solution for the wave equation is derived and the control problem of this distributed-parameter system is reduced to an approximate multiobjective programming problem. The fuzzy goals are incorporated for objectives and the equilibrium problem in terms of maximization of the degree of attainment for the aggregated fuzzy goals is considered. The solution of the equilibrium optimization problem is a Pareto optimal solution with the best satisfaction performance which is achieved by using a metaheuristic algorithm such as the simulated annealing (SA together with the simplex method of linear programming (LP problems. An illustrative numerical example is presented to indicate the efficiency of the proposed method and the capability of the SA in finding optimal solution compared with two popular metaheurestics.
Control Operator for the Two-Dimensional Energized Wave Equation
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Sunday Augustus REJU
2006-07-01
Full Text Available This paper studies the analytical model for the construction of the two-dimensional Energized wave equation. The control operator is given in term of space and time t independent variables. The integral quadratic objective cost functional is subject to the constraint of two-dimensional Energized diffusion, Heat and a source. The operator that shall be obtained extends the Conjugate Gradient method (ECGM as developed by Hestenes et al (1952, [1]. The new operator enables the computation of the penalty cost, optimal controls and state trajectories of the two-dimensional energized wave equation when apply to the Conjugate Gradient methods in (Waziri & Reju, LEJPT & LJS, Issues 9, 2006, [2-4] to appear in this series.
Nonlinear wave dynamics in self-consistent water channels
Pelinovsky, Efim; Didenkulova, Ira; Shurgalina, Ekaterina; Aseeva, Nataly
2017-12-01
We study long-wave dynamics in a self-consistent water channel of variable cross-section, taking into account the effects of weak nonlinearity and dispersion. The self-consistency of the water channel is considered within the linear shallow water theory, which implies that the channel depth and width are interrelated, so the wave propagates in such a channel without inner reflection from the bottom even if the water depth changes significantly. In the case of small-amplitude weakly dispersive waves, the reflection from the bottom is also small, which allows the use of a unidirectional approximation. A modified equation for Riemann waves is derived for the nondispersive case. The wave-breaking criterion (gradient catastrophe) for self-consistent channels is defined. If both weak nonlinearity and dispersion are accounted for, the variable-coefficient Korteweg–de Vries (KdV) equation for waves in self-consistent channels is derived. Note that this is the first time that a KdV equation has been derived for waves in strongly inhomogeneous media. Soliton transformation in a channel with an abrupt change in depth is also studied.
Turbulent wind waves on a water current
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M. V. Zavolgensky
2008-05-01
Full Text Available An analytical model of water waves generated by the wind over the water surface is presented. A simple modeling method of wind waves is described based on waves lengths diagram, azimuthal hodograph of waves velocities and others. Properties of the generated waves are described. The wave length and wave velocity are obtained as functions on azimuth of wave propagation and growth rate. Motionless waves dynamically trapped into the general picture of three dimensional waves are described. The gravitation force does not enter the three dimensional of turbulent wind waves. That is why these waves have turbulent and not gravitational nature. The Langmuir stripes are naturally modeled and existence of the rogue waves is theoretically proved.
Assessing Tsunami Vulnerabilities of Geographies with Shallow Water Equations
Aras, Rifat; Shen, Yuzhong
2012-01-01
Tsunami preparedness is crucial for saving human lives in case of disasters that involve massive water movement. In this work, we develop a framework for visual assessment of tsunami preparedness of geographies. Shallow water equations (also called Saint Venant equations) are a set of hyperbolic partial differential equations that are derived by depth-integrating the Navier-Stokes equations and provide a great abstraction of water masses that have lower depths compared to their free surface area. Our specific contribution in this study is to use Microsoft's XNA Game Studio to import underwater and shore line geographies, create different tsunami scenarios, and visualize the propagation of the waves and their impact on the shore line geography. Most importantly, we utilized the computational power of graphical processing units (GPUs) as HLSL based shader files and delegated all of the heavy computations to the GPU. Finally, we also conducted a validation study, in which we have tested our model against a controlled shallow water experiment. We believe that such a framework with an easy to use interface that is based on readily available software libraries, which are widely available and easily distributable, would encourage not only researchers, but also educators to showcase ideas.
A boundary value problem for the wave equation
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Nezam Iraniparast
1999-01-01
Full Text Available Traditionally, boundary value problems have been studied for elliptic differential equations. The mathematical systems described in these cases turn out to be “well posed”. However, it is also important, both mathematically and physically, to investigate the question of boundary value problems for hyperbolic partial differential equations. In this regard, prescribing data along characteristics as formulated by Kalmenov [5] is of special interest. The most recent works in this area have resulted in a number of interesting discoveries [3, 4, 5, 7, 8]. Our aim here is to extend some of these results to a more general domain which includes the characteristics of the underlying wave equation as a part of its boundary.
Solving Potential Scattering Equations without Partial Wave Decomposition
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Caia, George; Pascalutsa, Vladimir; Wright, Louis E
2004-03-01
Considering two-body integral equations we show how they can be dimensionally reduced by integrating exactly over the azimuthal angle of the intermediate momentum. Numerical solution of the resulting equation is feasible without employing a partial-wave expansion. We illustrate this procedure for the Bethe-Salpeter equation for pion-nucleon scattering and give explicit details for the one-nucleon-exchange term in the potential. Finally, we show how this method can be applied to pion photoproduction from the nucleon with {pi}N rescattering being treated so as to maintain unitarity to first order in the electromagnetic coupling. The procedure for removing the azimuthal angle dependence becomes increasingly complex as the spin of the particles involved increases.
Wave-equation Migration Velocity Analysis Using Plane-wave Common Image Gathers
Guo, Bowen
2017-06-01
Wave-equation migration velocity analysis (WEMVA) based on subsurface-offset, angle domain or time-lag common image gathers (CIGs) requires significant computational and memory resources because it computes higher dimensional migration images in the extended image domain. To mitigate this problem, a WEMVA method using plane-wave CIGs is presented. Plane-wave CIGs reduce the computational cost and memory storage because they are directly calculated from prestack plane-wave migration, and the number of plane waves is often much smaller than the number of shots. In the case of an inaccurate migration velocity, the moveout of plane-wave CIGs is automatically picked by a semblance analysis method, which is then linked to the migration velocity update by a connective function. Numerical tests on two synthetic datasets and a field dataset validate the efficiency and effectiveness of this method.
Parsimonious wave-equation travel-time inversion for refraction waves
Fu, Lei
2017-02-14
We present a parsimonious wave-equation travel-time inversion technique for refraction waves. A dense virtual refraction dataset can be generated from just two reciprocal shot gathers for the sources at the endpoints of the survey line, with N geophones evenly deployed along the line. These two reciprocal shots contain approximately 2N refraction travel times, which can be spawned into O(N2) refraction travel times by an interferometric transformation. Then, these virtual refraction travel times are used with a source wavelet to create N virtual refraction shot gathers, which are the input data for wave-equation travel-time inversion. Numerical results show that the parsimonious wave-equation travel-time tomogram has about the same accuracy as the tomogram computed by standard wave-equation travel-time inversion. The most significant benefit is that a reciprocal survey is far less time consuming than the standard refraction survey where a source is excited at each geophone location.
Wang, T.
2017-05-26
Elastic full waveform inversion (EFWI) provides high-resolution parameter estimation of the subsurface but requires good initial guess of the true model. The traveltime inversion only minimizes traveltime misfits which are more sensitive and linearly related to the low-wavenumber model perturbation. Therefore, building initial P and S wave velocity models for EFWI by using elastic wave-equation reflections traveltime inversion (WERTI) would be effective and robust, especially for the deeper part. In order to distinguish the reflection travletimes of P or S-waves in elastic media, we decompose the surface multicomponent data into vector P- and S-wave seismogram. We utilize the dynamic image warping to extract the reflected P- or S-wave traveltimes. The P-wave velocity are first inverted using P-wave traveltime followed by the S-wave velocity inversion with S-wave traveltime, during which the wave mode decomposition is applied to the gradients calculation. Synthetic example on the Sigbee2A model proves the validity of our method for recovering the long wavelength components of the model.
Cui, Lei; Tong, Fei-Fei; Shi, Feng
2011-09-01
Researches on breaking-induced currents by waves are summarized firstly in this paper. Then, a combined numerical model in orthogonal curvilinear coordinates is presented to simulate wave-induced current in areas with curved boundary or irregular coastline. The proposed wave-induced current model includes a nearshore current module established through orthogonal curvilinear transformation form of shallow water equations and a wave module based on the curvilinear parabolic approximation wave equation. The wave module actually serves as the driving force to provide the current module with required radiation stresses. The Crank-Nicolson finite difference scheme and the alternating directions implicit method are used to solve the wave and current module, respectively. The established surf zone currents model is validated by two numerical experiments about longshore currents and rip currents in basins with rip channel and breakwater. The numerical results are compared with the measured data and published numerical results.
Lecture Notes for the Course in Water Wave Mechanics
DEFF Research Database (Denmark)
Andersen, Thomas Lykke; Frigaard, Peter; Burcharth, Hans F.
The present notes are written for the course in water wave mechanics given on the 7th semester of the education in civil engineering at Aalborg University. The prerequisites for the course are the course in fluid dynamics also given on the 7th semester and some basic mathematical and physical...... knowledge. The course is at the same time an introduction to the course in coastal hydraulics on the 8th semester. The notes cover the first four lectures of the course: • Definitions. Governing equations and boundary conditions. • Derivation of velocity potential for linear waves. Dispersion relationship...... Particle velocities and accelerations. • Particle paths, pressure variation, deep and shallow water waves, wave energy and group velocity. • Shoaling, refraction, diffraction and wave breaking. The last part of the course is on analysis of irregular waves and was included in the first two editions...
A new iterative solver for the time-harmonic wave equation
Riyanti, C.D.; Erlangga, Y.A.; Plessix, R.E.; Mulder, W.A.; Vuik, C.; Oosterlee, C.
2006-01-01
The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can
TRAVELING WAVE SOLUTIONS OF SOME FRACTIONAL DIFFERENTIAL EQUATIONS
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SERIFE MUGE EGE
2016-07-01
Full Text Available The modified Kudryashov method is powerful, efficient and can be used as an alternative to establish new solutions of different type of fractional differential equations applied in mathematical physics. In this article, we’ve constructed new traveling wave solutions including symmetrical Fibonacci function solutions, hyperbolic function solutions and rational solutions of the space-time fractional Cahn Hillihard equation D_t^α u − γD_x^α u − 6u(D_x^α u^2 − (3u^2 − 1D_x^α (D_x^α u + D_x^α(D_x^α(D_x^α(D_x^α u = 0 and the space-time fractional symmetric regularized long wave (SRLW equation D_t^α(D_t^α u + D_x^α(D_x^α u + uD_t^α(D_x^α u + D_x^α u D_t^α u + D_t^α(D_t^α(D_x^α(D_x^α u = 0 via modified Kudryashov method. In addition, some of the solutions are described in the figures with the help of Mathematica.
Nonlinear diffraction of water waves by offshore stuctures
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Matiur Rahman
1986-01-01
Full Text Available This paper is concerned with a variational formulation of a nonaxisymmetric water wave problem. The full set of equations of motion for the problem in cylindrical polar coordinates is derived. This is followed by a review of the current knowledge on analytical theories and numerical treatments of nonlinear diffraction of water waves by offshore cylindrical structures. A brief discussion is made on water waves incident on a circular harbor with a narrow gap. Special emphasis is given to the resonance phenomenon associated with this problem. A new theoretical analysis is also presented to estimate the wave forces on large conical structures. Second-order (nonlinear effects are included in the calculation of the wave forces on the conical structures. A list of important references is also given.
Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru; Sushila
2018-02-01
In this work, we aim to present a new fractional extension of regularized long-wave equation. The regularized long-wave equation is a very important mathematical model in physical sciences, which unfolds the nature of shallow water waves and ion acoustic plasma waves. The existence and uniqueness of the solution of the regularized long-wave equation associated with Atangana-Baleanu fractional derivative having Mittag-Leffler type kernel is verified by implementing the fixed-point theorem. The numerical results are derived with the help of an iterative algorithm. In order to show the effects of various parameters and variables on the displacement, the numerical results are presented in graphical and tabular form.
Exact solitary wave and quasi-periodic wave solutions of the KdV-Sawada-Kotera-Ramani equation
National Research Council Canada - National Science Library
Zhang, Lijun; Khalique, Chaudry Masood
2015-01-01
In this paper we derive new exact solitary wave solutions and quasi-periodic traveling wave solutions of the KdV-Sawada-Kotera-Ramani equation by using a method which we introduce here for the first time...
Metamaterials, from electromagnetic waves to water waves, bending waves and beyond
Dupont, G.
2015-08-04
We will review our recent work on metamaterials for different types of waves. Transposition of transform optics to water waves and bending waves on plates will be considered with potential applications of cloaking to water waves protection and anti-vibrating systems.
Wave equation based microseismic source location and velocity inversion
Zheng, Yikang; Wang, Yibo; Chang, Xu
2016-12-01
The microseismic event locations and velocity information can be used to infer the stress field and guide hydraulic fracturing process, as well as to image the subsurface structures. How to get accurate microseismic event locations and velocity model is the principal problem in reservoir monitoring. For most location methods, the velocity model has significant relation with the accuracy of the location results. The velocity obtained from log data is usually too rough to be used for location directly. It is necessary to discuss how to combine the location and velocity inversion. Among the main techniques for locating microseismic events, time reversal imaging (TRI) based on wave equation avoids traveltime picking and offers high-resolution locations. Frequency dependent wave equation traveltime inversion (FWT) is an inversion method that can invert velocity model with source uncertainty at certain frequency band. Thus we combine TRI with FWT to produce improved event locations and velocity model. In the proposed approach, the location and model information are interactively used and updated. Through the proposed workflow, the inverted model is better resolved and the event locations are more accurate. We test this method on synthetic borehole data and filed data of a hydraulic fracturing experiment. The results verify the effectiveness of the method and prove it has potential for real-time microseismic monitoring.
Zecca, Antonio
2017-03-01
The arbitrary spin field equations that are not separable, contrarily to what happens in the Robertson-Walker and Schwarzschild metrics, are studied in a general comoving spherically symmetric metric. They result to be separable by variable separation in a class of metrics governing the Lemâitre Tolman Bondi cosmological models whose physical radius has a special factorized parametric representation. The result is proved by induction by explicitly considering the spin 1, 3/2, 2 case and then the higher spin values. The procedure is based on the Newman-Penrose formalism, which takes into account the strong analogy with the Robertson-Walker metric case. The existence of a nontrivial Weyl spinor requires a symmetrization of one of the spinor wave equations for spin values greater than 1.
Water waves generated by underwater explosion
Mehaute, Bernard Le
1996-01-01
This is the first book on explosion-generated water waves. It presents the theoretical foundations and experimental results of the generation and propagation of impulsively generated waves resulting from underwater explosions. Many of the theories and concepts presented herein are applicable to other types of water waves, in particular, tsunamis and waves generated by the fall of a meteorite. Linear and nonlinear theories, as well as experimental calibrations, are presented for cases of deep and shallow water explosions. Propagation of transient waves on dissipative, nonuniform bathymetries to
Well-posedness of semilinear stochastic wave equations with Hölder continuous coefficients
Masiero, Federica; Priola, Enrico
2017-08-01
We prove that semilinear stochastic abstract wave equations, including wave and plate equations, are well-posed in the strong sense with an α-Hölder continuous drift coefficient, if α ∈ (2 / 3 , 1). The uniqueness may fail for the corresponding deterministic PDE and well-posedness is restored by adding an external random forcing of white noise type. This shows a kind of regularization by noise for the semilinear wave equation. To prove the result we introduce an approach based on backward stochastic differential equations. We also establish regularizing properties of the transition semigroup associated to the stochastic wave equation by using control theoretic results.
Wave-equation dispersion inversion of surface waves recorded on irregular topography
Li, Jing
2017-08-17
Significant topographic variations will strongly influence the amplitudes and phases of propagating surface waves. Such effects should be taken into account, otherwise the S-velocity model inverted from the Rayleigh dispersion curves will contain significant inaccuracies. We now show that the recently developed wave-equation dispersion inversion (WD) method naturally takes into account the effects of topography to give accurate S-velocity tomograms. Application of topographic WD to demonstrates that WD can accurately invert dispersion curves from seismic data recorded over variable topography. We also apply this method to field data recorded on the crest of mountainous terrain and find with higher resolution than the standard WD tomogram.
On an Acoustic Wave Equation Arising in Non-Equilibrium Gasdynamics. Classroom Notes
Chandran, Pallath
2004-01-01
The sixth-order wave equation governing the propagation of one-dimensional acoustic waves in a viscous, heat conducting gaseous medium subject to relaxation effects has been considered. It has been reduced to a system of lower order equations corresponding to the finite speeds occurring in the equation, following a method due to Whitham. The lower…
Wave-equation migration in generalized coordinate systems
Shragge, Jeffrey
Wave-equation migration using one-way wavefield extrapolation operators is commonly used in industry to generate images of complex geologic structure from 3D seismic data. By design, most conventional wave-equation approaches restrict propagation to downward continuation, where wavefields are recursively extrapolated to depth on Cartesian meshes. In practice, this approach is limited in high-angle accuracy and is restricted to down-going waves, which precludes the use of some steep dip and all turning wave components important for imaging targets in such areas as steep salt body flanks. This thesis discusses a strategy for improving wavefield extrapolation based on extending wavefield propagation to generalized coordinate system geometries that are more conformal to the wavefield propagation direction and permit imaging with turning waves. Wavefield propagation in non-Cartesian coordinates requires properly specifying the Laplacian operator in the governing Helmholtz equation. By employing differential geometry theory, I demonstrate how generalized a Riemannian wavefield extrapolation (RWE) procedure can be developed for any 3D non-orthogonal coordinate system, including those constructed by smoothing ray-based coordinate meshes formed from a suite of traced rays. I present 2D and 3D generalized RWE propagation examples illustrating the improved steep-dip propagation afforded by the coordinate transformation. One consequence of using non-Cartesian coordinates, though, is that the corresponding 3D extrapolation operators have up to 10 non-stationary coefficients, which can lead to imposing (and limiting) computer memory constraints for realistic 3D applications. To circumvent this difficulty, I apply the generalized RWE theory to analytic coordinate systems, rather than numerically generated meshes. Analytic coordinates offer the advantage of having straightforward analytic dispersion relationships and easy-to-implement extrapolation operators that add little
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Mostafa M.A. Khater
Full Text Available In this article and for the first time, we introduce and describe Khater method which is a new technique for solving nonlinear partial differential equations (PDEs.. We apply this method for each of the following models Bogoyavlenskii equation, couple Boiti-Leon-Pempinelli system and Time-fractional Cahn-Allen equation. Khater method is very powerful, Effective, felicitous and fabulous method to get exact and solitary wave solution of (PDEs.. Not only just like that but it considers too one of the general methods for solving that kind of equations since it involves some methods as we will see in our discuss of the results. We make a comparison between the results of this new method and another method. Keywords: Bogoyavlenskii equations system, Couple Boiti-Leon-Pempinelli equations system, Time-fractional Cahn-Allen equation, Khater method, Traveling wave solutions, Solitary wave solutions
Parker, A.
1995-07-01
In this second of two articles (designated I and II), the bilinear transformation method is used to obtain stationary periodic solutions of the partially integrable regularized long-wave (RLW) equation. These solutions are expressed in terms of Riemann theta functions, and this approach leads to a new and compact expression for the important dispersion relation. The periodic solution (or cnoidal wave) can be represented as an infinite sum of sech2 ``solitary waves'': this remarkable property may be interpreted in the context of a nonlinear superposition principle. The RLW cnoidal wave approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes, respectively. Analytic approximations and error estimates are given which shed light on the character of the cnoidal wave in the different parameter regimes. Similar results are presented in brief for the related RLW Boussinesq (RLWB) equation.
Numerical treatment of interfaces for second-order wave equations
Cécere, Mariana; Reula, Oscar
2011-01-01
In this article we develop a numerical scheme to deal with interfaces between touching numerical grids when solving the second-order wave equation. In the spirit of the Simultaneous Approximation Term (SAT) schemes introduced in \\cite{Carpenter1999341}, information is passed among grids using the values of the fields only at the contact points between them (actually, in our case, just the values of the field corresponding to the time derivative of the field). The scheme seems to be as accurate as the space and time discretizations used for the corresponding derivatives. The semi-discrete approximation preserves the norm and uses standard finite-difference operators satisfying summation by parts. For the time integrator we use a semi-implicit IMEX Runge-Kutta method. This is crucial for, otherwise, the methods will be impractical given the severe restrictions its stiff parts would put on totally explicit integrators.
Time evolution of the wave equation using rapid expansion method
Pestana, Reynam C.
2010-07-01
Forward modeling of seismic data and reverse time migration are based on the time evolution of wavefields. For the case of spatially varying velocity, we have worked on two approaches to evaluate the time evolution of seismic wavefields. An exact solution for the constant-velocity acoustic wave equation can be used to simulate the pressure response at any time. For a spatially varying velocity, a one-step method can be developed where no intermediate time responses are required. Using this approach, we have solved for the pressure response at intermediate times and have developed a recursive solution. The solution has a very high degree of accuracy and can be reduced to various finite-difference time-derivative methods, depending on the approximations used. Although the two approaches are closely related, each has advantages, depending on the problem being solved. © 2010 Society of Exploration Geophysicists.
Lipschitz Metrics for a Class of Nonlinear Wave Equations
Bressan, Alberto; Chen, Geng
2017-12-01
The nonlinear wave equation {u_{tt}-c(u)(c(u)u_x)_x=0} determines a flow of conservative solutions taking values in the space {H^1(R)}. However, this flow is not continuous with respect to the natural H 1 distance. The aim of this paper is to construct a new metric which renders the flow uniformly Lipschitz continuous on bounded subsets of {H^1(R)}. For this purpose, H 1 is given the structure of a Finsler manifold, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, one can carefully estimate how the weighted length grows in time. By the generic regularity result proved in [7], these piecewise regular paths are dense and can be used to construct a geodesic distance with the desired Lipschitz property.
Characteristics of phase-averaged equations for modulated wave groups
Klopman, G.; Petit, H.A.H.; Battjes, J.A.
2000-01-01
The project concerns the influence of long waves on coastal morphology. The modelling of the combined motion of the long waves and short waves in the horizontal plane is done by phase-averaging over the short wave motion and using intra-wave modelling for the long waves, see e.g. Roelvink (1993).
Limiting Behavior of Travelling Waves for the Modified Degasperis-Procesi Equation
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Jiuli Yin
2014-01-01
Full Text Available Using an improved qualitative method which combines characteristics of several methods, we classify all travelling wave solutions of the modified Degasperis-Procesi equation in specified regions of the parametric space. Besides some popular exotic solutions including peaked waves, and looped and cusped waves, this equation also admits some very particular waves, such as fractal-like waves, double stumpons, double kinked waves, and butterfly-like waves. The last three types of solutions have not been reported in the literature. Furthermore, we give the limiting behavior of all periodic solutions as the parameters trend to some special values.
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H. Ullah
2015-01-01
Full Text Available The two-dimensional nonlinear wave equations are considered. Solution to the problem is approximated by using optimal homotopy asymptotic method (OHAM. The residual and convergence of the proposed method to nonlinear wave equation are presented through graphs. The resultant analytic series solution of the two-dimensional nonlinear wave equation shows the effectiveness of the proposed method. The comparison of results has been made with the existing results available in the literature.
Energy Technology Data Exchange (ETDEWEB)
Uesaka, S. [Kyoto University, Kyoto (Japan). Faculty of Engineering; Watanabe, T.; Sassa, K. [Kyoto University, Kyoto (Japan)
1997-05-27
Algorithm is constructed and a program developed for a full-wave inversion (FWI) method utilizing the elastic wave equation in seismic exploration. The FWI method is a method for obtaining a physical property distribution using the whole observed waveforms as the data. It is capable of high resolution which is several times smaller than the wavelength since it can handle such phenomena as wave reflection and dispersion. The method for determining the P-wave velocity structure by use of the acoustic wave equation does not provide information about the S-wave velocity since it does not consider S-waves or converted waves. In an analysis using the elastic wave equation, on the other hand, not only P-wave data but also S-wave data can be utilized. In this report, under such circumstances, an inverse analysis algorithm is constructed on the basis of the elastic wave equation, and a basic program is developed. On the basis of the methods of Mora and of Luo and Schuster, the correction factors for P-wave and S-wave velocities are formulated directly from the elastic wave equation. Computations are performed and the effects of the hypocenter frequency and vibration transmission direction are examined. 6 refs., 8 figs.
Wave-equation Q tomography and least-squares migration
Dutta, Gaurav
2016-03-01
This thesis designs new methods for Q tomography and Q-compensated prestack depth migration when the recorded seismic data suffer from strong attenuation. A motivation of this work is that the presence of gas clouds or mud channels in overburden structures leads to the distortion of amplitudes and phases in seismic waves propagating inside the earth. If the attenuation parameter Q is very strong, i.e., Q<30, ignoring the anelastic effects in imaging can lead to dimming of migration amplitudes and loss of resolution. This, in turn, adversely affects the ability to accurately predict reservoir properties below such layers. To mitigate this problem, I first develop an anelastic least-squares reverse time migration (Q-LSRTM) technique. I reformulate the conventional acoustic least-squares migration problem as a viscoacoustic linearized inversion problem. Using linearized viscoacoustic modeling and adjoint operators during the least-squares iterations, I show with numerical tests that Q-LSRTM can compensate for the amplitude loss and produce images with better balanced amplitudes than conventional migration. To estimate the background Q model that can be used for any Q-compensating migration algorithm, I then develop a wave-equation based optimization method that inverts for the subsurface Q distribution by minimizing a skeletonized misfit function ε. Here, ε is the sum of the squared differences between the observed and the predicted peak/centroid-frequency shifts of the early-arrivals. Through numerical tests on synthetic and field data, I show that noticeable improvements in the migration image quality can be obtained from Q models inverted using wave-equation Q tomography. A key feature of skeletonized inversion is that it is much less likely to get stuck in a local minimum than a standard waveform inversion method. Finally, I develop a preconditioning technique for least-squares migration using a directional Gabor-based preconditioning approach for isotropic
Nurijanyan, S.; van der Vegt, Jacobus J.W.; Bokhove, Onno
2013-01-01
A discontinuous Galerkin finite element method (DGFEM) has been developed and tested for the linear, three-dimensional, rotating incompressible Euler equations. These equations admit complicated wave solutions, which poses numerical challenges. These challenges concern: (i) discretisation of a
Aly R. Seadawy; Dianchen Lu; Mostafa M.A. Khater
2017-01-01
In this paper, we utilize the exp(−φ(ξ))-expansion method to find exact and solitary wave solutions of the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony nonlinear evolution equation. The generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony nonlinear evolution equation describes the model for the propagation of long waves that mingle with nonlinear and dissipative impact. This model is used in the analysis of the surface waves of long wavelength in hydro magnetic waves in cold plasma, liq...
Destrade, M.
2010-12-08
We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then, we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and Khokhlov- Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid mechanics. © 2010 The Royal Society.
CFD Analysis of Water Solitary Wave Reflection
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K. Smida
2011-12-01
Full Text Available A new numerical wave generation method is used to investigate the head-on collision of two solitary waves. The reflection at vertical wall of a solitary wave is also presented. The originality of this model, based on the Navier-Stokes equations, is the specification of an internal inlet velocity, defined as a source line within the computational domain for the generation of these non linear waves. This model was successfully implemented in the PHOENICS (Parabolic Hyperbolic Or Elliptic Numerical Integration Code Series code. The collision of two counter-propagating solitary waves is similar to the interaction of a soliton with a vertical wall. This wave generation method allows the saving of considerable time for this collision process since the counter-propagating wave is generated directly without reflection at vertical wall. For the collision of two solitary waves, numerical results show that the run-up phenomenon can be well explained, the solution of the maximum wave run-up is almost equal to experimental measurement. The simulated wave profiles during the collision are in good agreement with experimental results. For the reflection at vertical wall, the spatial profiles of the wave at fixed instants show that this problem is equivalent to the collision process.
Initial-value problem for the Gardner equation applied to nonlinear internal waves
Rouvinskaya, Ekaterina; Kurkina, Oxana; Kurkin, Andrey; Talipova, Tatiana; Pelinovsky, Efim
2017-04-01
The Gardner equation is a fundamental mathematical model for the description of weakly nonlinear weakly dispersive internal waves, when cubic nonlinearity cannot be neglected. Within this model coefficients of quadratic and cubic nonlinearity can both be positive as well as negative, depending on background conditions of the medium, where waves propagate (sea water density stratification, shear flow profile) [Rouvinskaya et al., 2014, Kurkina et al., 2011, 2015]. For the investigation of weakly dispersive behavior in the framework of nondimensional Gardner equation with fixed (positive) sign of quadratic nonlinearity and positive or negative cubic nonlinearity {eq1} partial η/partial t+6η( {1± η} )partial η/partial x+partial ^3η/partial x^3=0, } the series of numerical experiments of initial-value problem was carried out for evolution of a bell-shaped impulse of negative polarity (opposite to the sign of quadratic nonlinear coefficient): {eq2} η(x,t=0)=-asech2 ( {x/x0 } ), for which amplitude a and width x0 was varied. Similar initial-value problem was considered in the paper [Trillo et al., 2016] for the Korteweg - de Vries equation. For the Gardner equation with different signs of cubic nonlinearity the initial-value problem for piece-wise constant initial condition was considered in detail in [Grimshaw et al., 2002, 2010]. It is widely known, for example, [Pelinovsky et al., 2007], that the Gardner equation (1) with negative cubic nonlinearity has a family of classic solitary wave solutions with only positive polarity,and with limiting amplitude equal to 1. Therefore evolution of impulses (2) of negative polarity (whose amplitudes a were varied from 0.1 to 3, and widths at the level of a/2 were equal to triple width of solitons with the same amplitude for a 1) was going on a universal scenario with the generation of nonlinear Airy wave. For the Gardner equation (1) with the positive cubic nonlinearity coefficient there exist two one-parametric families of
Capturing the flow beneath water waves.
Nachbin, A; Ribeiro-Junior, R
2018-01-28
Recently, the authors presented two numerical studies for capturing the flow structure beneath water waves (Nachbin and Ribeiro-Junior 2014 Disc. Cont. Dyn. Syst. A 34 , 3135-3153 (doi:10.3934/dcds.2014.34.3135); Ribeiro-Junior et al. 2017 J. Fluid Mech. 812 , 792-814 (doi:10.1017/jfm.2016.820)). Closed orbits for irrotational waves with an opposing current and stagnation points for rotational waves were some of the issues addressed. This paper summarizes the numerical strategies adopted for capturing the flow beneath irrotational and rotational water waves. It also presents new preliminary results for particle trajectories, due to irrotational waves, in the presence of a bottom topography.This article is part of the theme issue 'Nonlinear water waves'. © 2017 The Author(s).
Sub- and superluminal kink-like waves in the kinetic limit of Maxwell-Bloch equations
Energy Technology Data Exchange (ETDEWEB)
Janowicz, Maciej [Instytut Fizyki Polskiej Akademii Nauk, Aleja Lotnikow 32/46, 02-668 Warszawa (Poland); Holthaus, Martin, E-mail: mjanow@ifpan.edu.pl [Institut fuer Physik, Carl von Ossietzky Universitaet, D-26111 Oldenburg (Germany)
2011-01-14
Running-wave solutions to three systems of partial differential equations describing wave propagation in atomic media in the kinetic limit have been obtained. Those systems include approximations to (i) standard two-level Maxwell-Bloch equations; (ii) equations describing processes with saturated absorption in three-level systems and (iii) equations describing processes with reversed saturation in four-level systems. It has been shown that in all three cases kink-like solitary waves can emerge if the dynamical equation for the intensity includes a linear contribution to the Lambert-Beer law. Those solitary waves can propagate with either sub- or superluminal velocity of the edge of the kink, and in a direction which can be either the same as or opposite to that of the carrier wave. In addition, simple qualitative information about the behaviour of waves near the wavefronts has been obtained.
Rouvinskaya, Ekaterina; Kurkin, Andrey; Kurkina, Oxana
2017-04-01
Intensive internal gravity waves influence bottom topography in the coastal zone. They induce substantial flows in the bottom layer that are essential for the formation of suspension and for the sediment transport. It is necessary to develop a mathematical model to predict the state of the seabed near the coastline to assess and ensure safety during the building and operation of the hydraulic engineering constructions. There are many models which are used to predict the impact of storm waves on the sediment transport processes. Such models for the impact of the tsunami waves are also actively developing. In recent years, the influence of intense internal waves on the sedimentation processes is also of a special interest. In this study we adapt one of such models, that is based on the advection-diffusion equation and allows to study processes of resuspension under the influence of internal gravity waves in the coastal zone, for solving the specific practical problems. During the numerical simulation precomputed velocity values are substituted in the advection - diffusion equation for sediment concentration at each time step and each node of the computational grid. Velocity values are obtained by the simulation of the internal waves' dynamics by using the IGW Research software package for numerical integration of fully nonlinear two-dimensional (vertical plane) system of equations of hydrodynamics of inviscid incompressible stratified fluid in the Boussinesq approximation bearing in mind the impact of barotropic tide. It is necessary to set the initial velocity and density distribution in the computational domain, bottom topography, as well as the value of the Coriolis parameter and, if necessary, the parameters of the tidal wave to carry out numerical calculations in the software package IGW Research. To initialize the background conditions of the numerical model we used data records obtained in the summer in the southern part of the shelf zone of Sakhalin Island
Some Further Results on Traveling Wave Solutions for the ZK-BBM( Equations
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Shaoyong Li
2013-01-01
Full Text Available We investigate the traveling wave solutions for the ZK-BBM( equations by using bifurcation method of dynamical systems. Firstly, for ZK-BBM(2, 2 equation, we obtain peakon wave, periodic peakon wave, and smooth periodic wave solutions and point out that the peakon wave is the limit form of the periodic peakon wave. Secondly, for ZK-BBM(3, 2 equation, we obtain some elliptic function solutions which include periodic blow-up and periodic wave. Furthermore, from the limit forms of the elliptic function solutions, we obtain some trigonometric and hyperbolic function solutions which include periodic blow-up, blow-up, and smooth solitary wave. We also show that our work extends some previous results.
Li, Jing
2017-12-22
A robust imaging technology is reviewed that provide subsurface information in challenging environments: wave-equation dispersion inversion (WD) of surface waves for the shear velocity model. We demonstrate the benefits and liabilities of the method with synthetic seismograms and field data. The benefits of WD are that 1) there is no layered medium assumption, as there is in conventional inversion of dispersion curves, so that the 2D or 3D S-velocity model can be reliably obtained with seismic surveys over rugged topography, and 2) WD mostly avoids getting stuck in local minima. The synthetic and field data examples demonstrate that WD can accurately reconstruct the S-wave velocity distributions in laterally heterogeneous media if the dispersion curves can be identified and picked. The WD method is easily extended to anisotropic media and the inversion of dispersion curves associated with Love wave. The liability is that is almost as expensive as FWI and only recovers the Vs distribution to a depth no deeper than about 1/2~1/3 wavelength.
On the wave equations with memory in noncylindrical domains
Directory of Open Access Journals (Sweden)
Mauro de Lima Santos
2007-10-01
Full Text Available In this paper we prove the exponential and polynomial decays rates in the case $n > 2$, as time approaches infinity of regular solutions of the wave equations with memory $$ u_{tt}-Delta u+int^{t}_{0}g(t-sDelta u(sds=0 quad mbox{in } widehat{Q} $$ where $widehat{Q}$ is a non cylindrical domains of $mathbb{R}^{n+1}$, $(nge1$. We show that the dissipation produced by memory effect is strong enough to produce exponential decay of solution provided the relaxation function $g$ also decays exponentially. When the relaxation function decay polynomially, we show that the solution decays polynomially with the same rate. For this we introduced a new multiplier that makes an important role in the obtaining of the exponential and polynomial decays of the energy of the system. Existence, uniqueness and regularity of solutions for any $n ge 1$ are investigated. The obtained result extends known results from cylindrical to non-cylindrical domains.
On the wave equation with semilinear porous acoustic boundary conditions
Graber, Philip Jameson
2012-05-01
The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function. © 2012 Elsevier Inc.
Diffusion phenomenon for linear dissipative wave equations in an exterior domain
Ikehata, Ryo
Under the general condition of the initial data, we will derive the crucial estimates which imply the diffusion phenomenon for the dissipative linear wave equations in an exterior domain. In order to derive the diffusion phenomenon for dissipative wave equations, the time integral method which was developed by Ikehata and Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.
Travelling wave solutions of the generalized Benjamin-Bona-Mahony equation
Energy Technology Data Exchange (ETDEWEB)
Estevez, P.G. [Departamento de Fisica Fundamental, Area de Fisica Teorica, Universidad de Salamanca, 37008 Salamanca (Spain); Kuru, S. [Departamento de Fisica Teorica, Atomica y Optica, Universidad de Valladolid, 47071 Valladolid (Spain); Department of Physics, Faculty of Science, Ankara University, 06100 Ankara (Turkey); Negro, J. [Departamento de Fisica Teorica, Atomica y Optica, Universidad de Valladolid, 47071 Valladolid (Spain)], E-mail: jnegro@fta.uva.es; Nieto, L.M. [Departamento de Fisica Teorica, Atomica y Optica, Universidad de Valladolid, 47071 Valladolid (Spain)
2009-05-30
A class of particular travelling wave solutions of the generalized Benjamin-Bona-Mahony equation is studied systematically using the factorization technique. Then, the general travelling wave solutions of Benjamin-Bona-Mahony equation, and of its modified version, are also recovered.
Rogue wave solutions of the nonlinear Schrödinger equation with ...
Indian Academy of Sciences (India)
solutions of the variable coefficient Schrödinger equation are also obtained. Two free functions of time t and several arbitrary parameters are involved to generate a large number of wave structures. Keywords. Nonlinear Schrödinger equation; exp-function method; breather soliton; rogue wave. PACS Nos 02.30.Jr; 05.45.
Radiation Boundary Conditions for the Two-Dimensional Wave Equation from a Variational Principle
Broeze, J.; Broeze, Jan; van Daalen, Edwin F.G.; van Daalen, E.F.G.
1992-01-01
A variational principle is used to derive a new radiation boundary condition for the two-dimensional wave equation. This boundary condition is obtained from an expression for the local energy flux velocity on the boundary in normal direction. The wellposedness of the wave equation with this boundary
Energy Technology Data Exchange (ETDEWEB)
Erlangga, Mokhammad Puput [Geophysical Engineering, Institut Teknologi Bandung, Ganesha Street no.10 Basic Science B Buliding fl.2-3 Bandung, 40132, West Java Indonesia puput.erlangga@gmail.com (Indonesia)
2015-04-16
Separation between signal and noise, incoherent or coherent, is important in seismic data processing. Although we have processed the seismic data, the coherent noise is still mixing with the primary signal. Multiple reflections are a kind of coherent noise. In this research, we processed seismic data to attenuate multiple reflections in the both synthetic and real seismic data of Mentawai. There are several methods to attenuate multiple reflection, one of them is Radon filter method that discriminates between primary reflection and multiple reflection in the τ-p domain based on move out difference between primary reflection and multiple reflection. However, in case where the move out difference is too small, the Radon filter method is not enough to attenuate the multiple reflections. The Radon filter also produces the artifacts on the gathers data. Except the Radon filter method, we also use the Wave Equation Multiple Elimination (WEMR) method to attenuate the long period multiple reflection. The WEMR method can attenuate the long period multiple reflection based on wave equation inversion. Refer to the inversion of wave equation and the magnitude of the seismic wave amplitude that observed on the free surface, we get the water bottom reflectivity which is used to eliminate the multiple reflections. The WEMR method does not depend on the move out difference to attenuate the long period multiple reflection. Therefore, the WEMR method can be applied to the seismic data which has small move out difference as the Mentawai seismic data. The small move out difference on the Mentawai seismic data is caused by the restrictiveness of far offset, which is only 705 meter. We compared the real free multiple stacking data after processing with Radon filter and WEMR process. The conclusion is the WEMR method can more attenuate the long period multiple reflection than the Radon filter method on the real (Mentawai) seismic data.
A boundary element model for diffraction of water waves on varying water depth
Energy Technology Data Exchange (ETDEWEB)
Poulin, Sanne
1997-12-31
In this thesis a boundary element model for calculating diffraction of water waves on varying water depth is presented. The varying water depth is approximated with a perturbed constant depth in the mild-slope wave equation. By doing this, the domain integral which is a result of the varying depth is no longer a function of the unknown wave potential but only a function of position and the constant depth wave potential. The number of unknowns is the resulting system of equations is thus reduced significantly. The integration procedures in the model are tested very thoroughly and it is found that a combination of analytical integration in the singular region and standard numerical integration outside works very well. The gradient of the wave potential is evaluated successfully using a hypersingular integral equation. Deviations from the analytical solution are only found on the boundary or very close to, but these deviations have no significant influence on the accuracy of the solution. The domain integral is evaluated using the dual reciprocity method. The results are compared with a direct integration of the integral, and the accuracy is quite satisfactory. The problem with irregular frequencies is taken care of by the CBIEM (or CHIEF-method) together with a singular value decomposition technique. This method is simple to implement and works very well. The model is verified using Homma`s island as a test case. The test cases are limited to shallow water since the analytical solution is only valid in this region. Several depth ratios are examined, and it is found that the accuracy of the model increases with increasing wave period and decreasing depth ratio. Short waves, e.g. wind generated waves, can allow depth variations up to approximately 2 before the error exceeds 10%, while long waves can allow larger depth ratios. It is concluded that the perturbation idea is highly usable. A study of (partially) absorbing boundary conditions is also conducted. (EG)
Solitary and cnoidal wave scattering by a submerged horizontal plate in shallow water
Directory of Open Access Journals (Sweden)
Masoud Hayatdavoodi
2017-06-01
Full Text Available Solitary and cnoidal wave transformation over a submerged, fixed, horizontal rigid plate is studied by use of the nonlinear, shallow-water Level I Green-Naghdi (GN equations. Reflection and transmission coefficients are defined for cnoidal and solitary waves to quantify the nonlinear wave scattering. Results of the GN equations are compared with the laboratory experiments and other theoretical solutions for linear and nonlinear waves in intermediate and deep waters. The GN equations are then used to study the nonlinear wave scattering by a plate in shallow water. It is shown that in deep and intermediate depths, the wave-scattering varies nonlinearly by both the wavelength over the plate length ratio, and the submergence depth. In shallow water, however, and for long-waves, only the submergence depth appear to play a significant role on wave scattering. It is possible to define the plate submergence depth and length such that certain wave conditions are optimized above, below, or downwave of the plate for different applications. A submerged plate in shallow water can be used as a means to attenuate energy, such as in wave breakers, or used for energy focusing, and in wave energy devices.
Turbulence beneath finite amplitude water waves
Energy Technology Data Exchange (ETDEWEB)
Beya, J.F. [Universidad de Valparaiso, Escuela de Ingenieria Civil Oceanica, Facultad de Ingenieria, Valparaiso (Chile); The University of New South Wales, Water Research Laboratory, School of Civil and Environmental Engineering, Sydney, NSW (Australia); Peirson, W.L. [The University of New South Wales, Water Research Laboratory, School of Civil and Environmental Engineering, Sydney, NSW (Australia); Banner, M.L. [The University of New South Wales, School of Mathematics and Statistics, Sydney, NSW (Australia)
2012-05-15
Babanin and Haus (J Phys Oceanogr 39:2675-2679, 2009) recently presented evidence of near-surface turbulence generated below steep non-breaking deep-water waves. They proposed a threshold wave parameter a {sup 2}{omega}/{nu} = 3,000 for the spontaneous occurrence of turbulence beneath surface waves. This is in contrast to conventional understanding that irrotational wave theories provide a good approximation of non-wind-forced wave behaviour as validated by classical experiments. Many laboratory wave experiments were carried out in the early 1960s (e.g. Wiegel 1964). In those experiments, no evidence of turbulence was reported, and steep waves behaved as predicted by the high-order irrotational wave theories within the accuracy of the theories and experimental techniques at the time. This contribution describes flow visualisation experiments for steep non-breaking waves using conventional dye techniques in the wave boundary layer extending above the wave trough level. The measurements showed no evidence of turbulent mixing up to a value of a {sup 2}{omega}/{nu} = 7,000 at which breaking commenced in these experiments. These present findings are in accord with the conventional understandings of wave behaviour. (orig.)
New soliton solutions of the system of equations for the ion sound and Langmuir waves
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Seyma Tuluce Demiray
2016-11-01
Full Text Available This study is based on new soliton solutions of the system of equations for the ion sound wave under the action of the ponderomotive force due to high-frequency field and for the Langmuir wave. The generalized Kudryashov method (GKM, which is one of the analytical methods, has been tackled for finding exact solutions of the system of equations for the ion sound wave and the Langmuir wave. By using this method, dark soliton solutions of this system of equations have been obtained. Also, by using Mathematica Release 9, some graphical simulations were designed to see the behavior of these solutions.
DG-FEM solution for nonlinear wave-structure interaction using Boussinesq-type equations
DEFF Research Database (Denmark)
Engsig-Karup, Allan Peter; Hesthaven, Jan; Bingham, Harry B.
2008-01-01
We present a high-order nodal Discontinuous Galerkin Finite Element Method (DG-FEM) solution based on a set of highly accurate Boussinesq-type equations for solving general water-wave problems in complex geometries. A nodal DG-FEM is used for the spatial discretization to solve the Boussinesq equ...... and absorbed in the interior of the computational domain using a flexible relaxation technique applied on the free surface variables....... waters within the breaking limit. To demonstrate the current applicability of the model both linear and mildly nonlinear test cases are considered in two horizontal dimensions where the water waves interact with bottom-mounted fully reflecting structures. It is established that, by simple symmetry...... considerations combined with a mirror principle, it is possible to impose weak slip boundary conditions for both structured and general curvilinear wall boundaries while maintaining the accuracy of the scheme. As is standard for current high-order Boussinesq-type models, arbitrary waves can be generated...
Johnson, Kennita A.; Vormohr, Hannah R.; Doinikov, Alexander A.; Bouakaz, Ayache; Shields, C. Wyatt; López, Gabriel P.; Dayton, Paul A.
2016-05-01
Acoustophoresis uses acoustic radiation force to remotely manipulate particles suspended in a host fluid for many scientific, technological, and medical applications, such as acoustic levitation, acoustic coagulation, contrast ultrasound imaging, ultrasound-assisted drug delivery, etc. To estimate the magnitude of acoustic radiation forces, equations derived for an inviscid host fluid are commonly used. However, there are theoretical predictions that, in the case of a traveling wave, viscous effects can dramatically change the magnitude of acoustic radiation forces, which make the equations obtained for an inviscid host fluid invalid for proper estimation of acoustic radiation forces. To date, experimental verification of these predictions has not been published. Experimental measurements of viscous effects on acoustic radiation forces in a traveling wave were conducted using a confocal optical and acoustic system and values were compared with available theories. Our results show that, even in a low-viscosity fluid such as water, the magnitude of acoustic radiation forces is increased manyfold by viscous effects in comparison with what follows from the equations derived for an inviscid fluid.
Approximate analytical time-domain Green's functions for the Caputo fractional wave equation.
Kelly, James F; McGough, Robert J
2016-08-01
The Caputo fractional wave equation [Geophys. J. R. Astron. Soc. 13, 529-539 (1967)] models power-law attenuation and dispersion for both viscoelastic and ultrasound wave propagation. The Caputo model can be derived from an underlying fractional constitutive equation and is causal. In this study, an approximate analytical time-domain Green's function is derived for the Caputo equation in three dimensions (3D) for power law exponents greater than one. The Green's function consists of a shifted and scaled maximally skewed stable distribution multiplied by a spherical spreading factor 1/(4πR). The approximate one dimensional (1D) and two dimensional (2D) Green's functions are also computed in terms of stable distributions. Finally, this Green's function is decomposed into a loss component and a diffraction component, revealing that the Caputo wave equation may be approximated by a coupled lossless wave equation and a fractional diffusion equation.
Mathematical aspects of extreme water waves
Karjanto, N.
2006-01-01
In this thesis we discuss mathematical aspects of extreme water wave generation in a hydrodynamic laboratory. The original problem comes from the Maritime Research Institute Netherlands (MARIN) to generate large amplitude and non-breaking waves to test ship and offshore construction. We choose the
Time scales and structures of wave interaction exemplified with water waves
Kartashova, Elena
2013-05-01
Presently two models for computing energy spectra in weakly nonlinear dispersive media are known: kinetic wave turbulence theory, using a statistical description of an energy cascade over a continuous spectrum (K-cascade), and the D-model, describing resonant clusters and energy cascades (D-cascade) in a deterministic way as interaction of distinct modes. In this letter we give an overview of these structures and their properties and a list of criteria about which model of energy cascade should be used in the analysis of a given experiment, using water waves as an example. Applying the time scale analysis to weakly nonlinear wave systems modeled by the focusing nonlinear Schödinger equation, we demonstrate that K-cascade and D-cascade are not competing processes but rather two processes taking place at different time scales, at different characteristic levels of nonlinearity and based on different physical mechanisms. Applying those criteria to data known from experiments with surface water waves we find that the energy cascades observed occur at short characteristic times compatible only with a D-cascade. The only pre-requisite for a D-cascade being a focusing nonlinear Schödinger equation, the same analysis may be applied to existing experiments with wave systems appearing in hydrodynamics, nonlinear optics, electrodynamics, plasma, convection theory, etc.
The (′/-Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation
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Hasibun Naher
2011-01-01
Full Text Available We construct the traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG equation by the (/-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the (/-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.
Higher-Order Hamiltonian Model for Unidirectional Water Waves
Bona, J. L.; Carvajal, X.; Panthee, M.; Scialom, M.
2017-10-01
Formally second-order correct, mathematical descriptions of long-crested water waves propagating mainly in one direction are derived. These equations are analogous to the first-order approximations of KdV- or BBM-type. The advantage of these more complex equations is that their solutions corresponding to physically relevant initial perturbations of the rest state may be accurate on a much longer timescale. The initial value problem for the class of equations that emerges from our derivation is then considered. A local well-posedness theory is straightforwardly established by a contraction mapping argument. A subclass of these equations possess a special Hamiltonian structure that implies the local theory can be continued indefinitely.
Energy decay of a variable-coefficient wave equation with nonlinear time-dependent localized damping
Directory of Open Access Journals (Sweden)
Jieqiong Wu
2015-09-01
Full Text Available We study the energy decay for the Cauchy problem of the wave equation with nonlinear time-dependent and space-dependent damping. The damping is localized in a bounded domain and near infinity, and the principal part of the wave equation has a variable-coefficient. We apply the multiplier method for variable-coefficient equations, and obtain an energy decay that depends on the property of the coefficient of the damping term.
Directory of Open Access Journals (Sweden)
Aly R. Seadawy
2017-06-01
Full Text Available In this paper, we utilize the exp(−φ(ξ-expansion method to find exact and solitary wave solutions of the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony nonlinear evolution equation. The generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony nonlinear evolution equation describes the model for the propagation of long waves that mingle with nonlinear and dissipative impact. This model is used in the analysis of the surface waves of long wavelength in hydro magnetic waves in cold plasma, liquids, acoustic waves in harmonic crystals and acoustic–gravity waves in compressible fluids. By using this method, seven different kinds of traveling wave solutions are successfully obtained for this model. The considered method and transformation techniques are efficient and consistent for solving nonlinear evolution equations and obtain exact solutions that are applied to the science and engineering fields.
Wave-Breaking Phenomena and Existence of Peakons for a Generalized Compressible Elastic-Rod Equation
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Xiaolian Ai
2014-01-01
Full Text Available Consideration in this paper is the Cauchy problem of a generalized hyperelastic-rod wave equation. We first derive a wave-breaking mechanism for strong solutions, which occurs in finite time for certain initial profiles. In addition, we determine the existence of some new peaked solitary wave solutions.
Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations
Xu, Y.; Shu, Chi-Wang
2005-01-01
In this paper, we develop, analyze and test local discontinuous Galerkin (DG) methods to solve two classes of two-dimensional nonlinear wave equations formulated by the Kadomtsev–Petviashvili (KP) equation and the Zakharov–Kuznetsov (ZK) equation. Our proposed scheme for the Kadomtsev–Petviashvili
Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves
DEFF Research Database (Denmark)
Eldeberky, Y.; Madsen, Per A.
1999-01-01
This paper presents a new and more accurate set of deterministic evolution equations for the propagation of fully dispersive, weakly nonlinear, irregular, multidirectional waves. The equations are derived directly from the Laplace equation with leading order nonlinearity in the surface boundary c...
WAVE-E: The WAter Vapour European-Explorer Mission
Jimenez-LLuva, David; Deiml, Michael; Pavesi, Sara
2017-04-01
In the last decade, stratosphere-troposphere coupling processes in the Upper Troposphere Lower Stratosphere (UTLS) have been increasingly recognized to severely impact surface climate and high-impact weather phenomena. Weakened stratospheric circumpolar jets have been linked to worldwide extreme temperature and high-precipitation events, while anomalously strong stratospheric jets can lead to an increase in surface winds and tropical cyclone intensity. Moreover, stratospheric water vapor has been identified as an important forcing for global decadal surface climate change. In the past years, operational weather forecast and climate models have adapted a high vertical resolution in the UTLS region in order to capture the dynamical processes occurring in this highly stratified region. However, there is an evident lack of available measurements in the UTLS region to consistently support these models and further improve process understanding. Consequently, both the IPCC fifth assessment report and the ESA-GEWEX report 'Earth Observation and Water Cycle Science Priorities' have identified an urgent need for long-term observations and improved process understanding in the UTLS region. To close this gap, the authors propose the 'WAter Vapour European - Explorer' (WAVE-E) space mission, whose primary goal is to monitor water vapor in the UTLS at 1 km vertical, 25 km horizontal and sub-daily temporal resolution. WAVE-E consists of three quasi-identical small ( 500 kg) satellites (WAVE-E 1-3) in a constellation of Sun-Synchronous Low Earth Orbits, each carrying a limb sounding and cross-track scanning mid-infrared passive spectrometer (824 cm-1 to 829 cm-1). The core of the instruments builds a monolithic, field-widened type of Michelson interferometer without any moving parts, rendering it rigid and fault tolerant. Synergistic use of WAVE-E and MetOp-NG operational satellites is identified, such that a data fusion algorithm could provide water vapour profiles from the
Pandey, Vikash; Holm, Sverre
2016-12-01
The characteristic time-dependent viscosity of the intergranular pore-fluid in Buckingham's grain-shearing (GS) model [Buckingham, J. Acoust. Soc. Am. 108, 2796-2815 (2000)] is identified as the property of rheopecty. The property corresponds to a rare type of a non-Newtonian fluid in rheology which has largely remained unexplored. The material impulse response function from the GS model is found to be similar to the power-law memory kernel which is inherent in the framework of fractional calculus. The compressional wave equation and the shear wave equation derived from the GS model are shown to take the form of the Kelvin-Voigt fractional-derivative wave equation and the fractional diffusion-wave equation, respectively. Therefore, an analogy is drawn between the dispersion relations obtained from the fractional framework and those from the GS model to establish the equivalence of the respective wave equations. Further, a physical interpretation of the characteristic fractional order present in the wave equations is inferred from the GS model. The overall goal is to show that fractional calculus is not just a mathematical framework which can be used to curve-fit the complex behavior of materials. Rather, it can also be derived from real physical processes as illustrated in this work by the example of GS.
Shock wave equation of state of powder material
Dijken, D.K.; Hosson, J.Th.M. De
1994-01-01
A model is proposed to predict the following quantities for powder materials compacted by shock waves: the pressure, the specific volume, the internal energy behind the shock wave, and the shock-wave velocity U-s. They are calculated as a function of flyerplate velocity u(p) and initial powder
Shallow water sound propagation with surface waves.
Tindle, Chris T; Deane, Grant B
2005-05-01
The theory of wavefront modeling in underwater acoustics is extended to allow rapid range dependence of the boundaries such as occurs in shallow water with surface waves. The theory allows for multiple reflections at surface and bottom as well as focusing and defocusing due to reflection from surface waves. The phase and amplitude of the field are calculated directly and used to model pulse propagation in the time domain. Pulse waveforms are obtained directly for all wavefront arrivals including both insonified and shadow regions near caustics. Calculated waveforms agree well with a reference solution and data obtained in a near-shore shallow water experiment with surface waves over a sloping bottom.
Zhang, Zhendong
2016-07-26
We present a surface-wave inversion method that inverts for the S-wave velocity from the Rayleigh wave dispersion curve using a difference approximation to the gradient of the misfit function. We call this wave equation inversion of skeletonized surface waves because the skeletonized dispersion curve for the fundamental-mode Rayleigh wave is inverted using finite-difference solutions to the multi-dimensional elastic wave equation. The best match between the predicted and observed dispersion curves provides the optimal S-wave velocity model. Our method can invert for lateral velocity variations and also can mitigate the local minimum problem in full waveform inversion with a reasonable computation cost for simple models. Results with synthetic and field data illustrate the benefits and limitations of this method. © 2016 Elsevier B.V.
Linking multiple relaxation, power-law attenuation, and fractional wave equations.
Näsholm, Sven Peter; Holm, Sverre
2011-11-01
The acoustic wave attenuation is described by an experimentally established frequency power law in a variety of complex media, e.g., biological tissue, polymers, rocks, and rubber. Recent papers present a variety of acoustical fractional derivative wave equations that have the ability to model power-law attenuation. On the other hand, a multiple relaxation model is widely recognized as a physically based description of the acoustic loss mechanisms as developed by Nachman et al. [J. Acoust. Soc. Am. 88, 1584-1595 (1990)]. Through assumption of a continuum of relaxation mechanisms, each with an effective compressibility described by a distribution related to the Mittag-Leffler function, this paper shows that the wave equation corresponding to the multiple relaxation approach is identical to a given fractional derivative wave equation. This work therefore provides a physically based motivation for use of fractional wave equations in acoustic modeling.
Radiative transfer of acoustic waves in continuous complex media: Beyond the Helmholtz equation
Baydoun, Ibrahim; Pierrat, Romain; Derode, Arnaud
2016-01-01
Heterogeneity can be accounted for by a random potential in the wave equation. For acoustic waves in a fluid with fluctuations of both density and compressibility (as well as for electromagnetic waves in a medium with fluctuation of both permittivity and permeability) the random potential entails a scalar and an operator contribution. For simplicity, the latter is usually overlooked in multiple scattering theory: whatever the type of waves, this simplification amounts to considering the Helmholtz equation with a sound speed $c$ depending on position $\\mathbf{r}$. In this work, a radiative transfer equation is derived from the wave equation, in order to study energy transport through a multiple scattering medium. In particular, the influence of the operator term on various transport parameters is studied, based on the diagrammatic approach of multiple scattering. Analytical results are obtained for fundamental quantities of transport theory such as the transport mean-free path $\\ell^*$, scattering phase functi...
Gusev, Vitalyi; Aleshin, Vladislav
2002-12-01
Nonlinear wave propagation in materials, where distribution function of mesoscopic mechanical elements has very different scales of variation along and normally to diagonal of Preisach-Mayergoyz space, is analyzed. An evolution equation for strain wave, which takes into account localization of element distribution near the diagonal and its slow variation along the diagonal, is proposed. The evolution equation provides opportunity to model propagation of elastic waves with strain amplitudes comparable to and even higher than characteristic scale of element localization near Preisach-Mayergoyz space diagonal. Analytical solutions of evolution equation predict nonmonotonous dependence of wave absorption on its amplitude in a particular regime. The regime of self-induced absorption for small-amplitude nonlinear waves is followed by the regime of self-induced transparency for high-amplitude waves. The developed theory might be useful in seismology, in high-pressure nonlinear acoustics, and in nonlinear acoustic diagnostics of damaged and fatigued materials.
Witten, Matthew
1983-01-01
Hyperbolic Partial Differential Equations, Volume 1: Population, Reactors, Tides and Waves: Theory and Applications covers three general areas of hyperbolic partial differential equation applications. These areas include problems related to the McKendrick/Von Foerster population equations, other hyperbolic form equations, and the numerical solution.This text is composed of 15 chapters and begins with surveys of age specific population interactions, populations models of diffusion, nonlinear age dependent population growth with harvesting, local and global stability for the nonlinear renewal eq
On the Generalized Maxwell Equations and Their Prediction of Electroscalar Wave
Directory of Open Access Journals (Sweden)
Arbab A. I.
2009-04-01
Full Text Available We have formulated the basic laws of electromagnetic theory in quaternion form. The formalism shows that Maxwell equations and Lorentz force are derivable from just one quaternion equation that only requires the Lorentz gauge. We proposed a quaternion form of the continuity equation from which we have derived the ordinary continuity equation. We introduce new transformations that produces a scalar wave and generalize the continuity equation to a set of three equations. These equations imply that both current and density are waves. Moreover, we have shown that the current can not cir- culate around a point emanating from it. Maxwell equations are invariant under these transformations. An electroscalar wave propagating with speed of light is derived upon requiring the invariance of the energy conservation equation under the new transforma- tions. The electroscalar wave function is found to be proportional to the electric field component along the charged particle motion. This scalar wave exists with or without considering the Lorentz gauge. We have shown that the electromagnetic fields travel with speed of light in the presence or absence of free charges.
Simulations and analysis of asteroid-generated tsunamis using the shallow water equations
Berger, M. J.; LeVeque, R. J.; Weiss, R.
2016-12-01
We discuss tsunami propagation for asteroid-generated air bursts and water impacts. We present simulations for a range of conditions using the GeoClaw simulation software. Examples include meteors that span 5 to 250 MT of kinetic energy, and use bathymetry from the U.S. coastline. We also study radially symmetric one-dimensional equations to better explore the nature and decay rate of waves generated by air burst pressure disturbances traveling at the speed of sound in air, which is much greater than the gravity wave speed of the tsunami generated. One-dimensional simulations along a transect of bathymetry are also used to explore the resolution needed for the full two-dimensional simulations, which are much more expensive even with the use of adaptive mesh refinement due to the short wave lengths of these tsunamis. For this same reason, shallow water equations may be inadequate and we also discuss dispersive effects.
Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map
Bellassoued, Mourad; Ferreira, David Dos Santos
2010-01-01
In this article we seek stability estimates in the inverse problem of determining the potential or the velocity in a wave equation in an anisotropic medium from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n\\geq 2$ that the knowledge of the Dirichlet-to-Neumann map for the wave equation uniquely determines the electric potential and we prove H\\"older-type stability in dete...
Wave power potential in Malaysian territorial waters
Asmida Mohd Nasir, Nor; Maulud, Khairul Nizam Abdul
2016-06-01
Up until today, Malaysia has used renewable energy technology such as biomass, solar and hydro energy for power generation and co-generation in palm oil industries and also for the generation of electricity, yet, we are still far behind other countries which have started to optimize waves for similar production. Wave power is a renewable energy (RE) transported by ocean waves. It is very eco-friendly and is easily reachable. This paper presents an assessment of wave power potential in Malaysian territorial waters including waters of Sabah and Sarawak. In this research, data from Malaysia Meteorology Department (MetMalaysia) is used and is supported by a satellite imaginary obtained from National Aeronautics and Space Administration (NASA) and Malaysia Remote Sensing Agency (ARSM) within the time range of the year 1992 until 2007. There were two types of analyses conducted which were mask analysis and comparative analysis. Mask analysis of a research area is the analysis conducted to filter restricted and sensitive areas. Meanwhile, comparative analysis is an analysis conducted to determine the most potential area for wave power generation. Four comparative analyses which have been carried out were wave power analysis, comparative analysis of wave energy power with the sea topography, hot-spot area analysis and comparative analysis of wave energy with the wind speed. These four analyses underwent clipping processes using Geographic Information System (GIS) to obtain the final result. At the end of this research, the most suitable area to develop a wave energy converter was found, which is in the waters of Terengganu and Sarawak. Besides that, it was concluded that the average potential energy that can be generated in Malaysian territorial waters is between 2.8kW/m to 8.6kW/m.
Shock wave equation of state of powder material
Dijken, D. K.; De Hosson, J. Th. M.
1994-01-01
A model is proposed to predict the following quantities for powder materials compacted by shock waves: the pressure, the specific volume, the internal energy behind the shock wave, and the shock-wave velocity Us. They are calculated as a function of flyerplate velocity up and initial powder specific volume V00. The model is tested on Cu, Al2024, and Fe. Calculated Us vs up curves agree well with experiments provided V00 is smaller than about two times the solid specific volume. The model can be used to predict shock-wave state points of powder or solid material with a lower or higher initial temperature than room temperature.
DEFF Research Database (Denmark)
Guo, Hairun; Zeng, Xianglong; Zhou, Binbin
2013-01-01
We interpret the purely spectral forward Maxwell equation with up to third-order induced polarizations for pulse propagation and interactions in quadratic nonlinear crystals. The interpreted equation, also named the nonlinear wave equation in the frequency domain, includes quadratic and cubic...... nonlinearities, delayed Raman effects, and anisotropic nonlinearities. The full potential of this wave equation is demonstrated by investigating simulations of solitons generated in the process of ultrafast cascaded second-harmonic generation. We show that a balance in the soliton delay can be achieved due...
Analysis of efficient preconditioned defect correction methods for nonlinear water waves
DEFF Research Database (Denmark)
Engsig-Karup, Allan Peter
2014-01-01
Robust computational procedures for the solution of non-hydrostatic, free surface, irrotational and inviscid free-surface water waves in three space dimensions can be based on iterative preconditioned defect correction (PDC) methods. Such methods can be made efficient and scalable to enable...... prediction of free-surface wave transformation and accurate wave kinematics in both deep and shallow waters in large marine areas or for predicting the outcome of experiments in large numerical wave tanks. We revisit the classical governing equations are fully nonlinear and dispersive potential flow...... equations. We present new detailed fundamental analysis using finite-amplitude wave solutions for iterative solvers. We demonstrate that the PDC method in combination with a high-order discretization method enables efficient and scalable solution of the linear system of equations arising in potential flow...
Speed ot travelling waves in reaction-diffusion equations
Energy Technology Data Exchange (ETDEWEB)
Benguria, R.D.; Depassier, M.C. [Facultad de Fisica, Pontificia Universidad Catolica de Chile, Avda. Vicuna Mackenna 4860, Santiago (Chile); Mendez, V. [Facultat de Ciencies de la Salut, Universidad Internacional de Catalunya, Gomera s/n 08190 Sant Cugat del Valles, Barcelona (Spain)
2002-07-01
Reaction diffusion equations arise in several problems of population dynamics, flame propagation and others. In one dimensional cases the systems may evolve into travelling fronts. Here we concentrate on a reaction diffusion equation which arises as a simple model for chemotaxis and present results for the speed of the travelling fronts. (Author)
From Galilean-invariant to relativistic wave equations
Elizalde, E. (Emili), 1950-; Lobo Gutiérrez, José Alberto
1980-01-01
Through an imaginary change of coordinates in the Galilei algebra in 4 space dimensions and making use of an original idea of Dirac and Lvy-Leblond, we are able to obtain the relativistic equations of Dirac and of Bargmann and Wigner starting with the (Galilean-invariant) Schrdinger equation.
Comments on ``The Depth-Dependent Current and Wave Interaction Equations: A Revision''
Bennis, Anne-Claire; Ardhuin, Fabrice
2011-10-01
Equations for the wave-averaged three-dimensional momentum equations have been published in this journal. It appears that these equations are not consistent with the known depth-integrated momentum balance, especially over a sloping bottom. These equations should thus be considered with caution as they can produce erroneous flows, in particular outside of the surf zone. It is suggested that the inconsistency in the equations may arise from the different averaging operators applied to the different terms of the momentum equation. It is concluded that other forms of the momentum equations, expressed in terms of the quasi-Eulerian velocity, are better suited for three dimensional modelling of wave-current interactions.
Perturbation series for the double cnoidal wave of the Korteweg-de Vries equation
Boyd, John P.
1984-12-01
By means of the theorems proved earlier by the author, the problem of the double cnoidal wave of the Korteweg-de Vries equation is reduced to four algebraic equations in four unknowns. Two of the unknowns are the nonlinear phase speeds c1 and c2. Another is a physically irrelevant integration constant. The fourth unknown is the off-diagonal element of the symmetric, 2×2 theta matrix, which in turn gives the explicit coefficients of the Riemann theta function. The double cnoidal wave u(x,t) is then obtained by taking the second x-derivative of the logarithm of the theta function. Two separate forms of these four nonlinear ``residual'' equations are given. One is obtained from the Fourier series of the theta function and is useful for small wave amplitude. The other is based on the Gaussian series of the theta function and is highly efficient in the large amplitude regime where the double cnoidal wave is the sum of two solitary waves. Both sets of residual equations can be solved via perturbation theory and results are given to fourth order in the Fourier case and second order in the Gaussian case. The Gaussian-based perturbation series has the remarkable property that it converges more and more rapidly as the wave amplitude increases; the zeroth-order solution is the familiar double solitary wave. Numerical comparisons show that the two complementary perturbation series give accurate results in all the important regions of parameter space. (The ``unimportant'' regions are those in which the double cnoidal wave is an ordinary cnoidal wave subject to a very weak perturbation.) This is turn implies that even for moderate wave amplitude where the nonlinear interactions are not weak, and yet the solitary wave peaks are not well separated, at least to the eye, it is still qualitatively legitimate to describe the double cnoidal wave as either the sum of two sine waves or of two solitary waves of different heights.
Spinor-electron wave guided modes in coupled quantum wells structures by solving the Dirac equation
Energy Technology Data Exchange (ETDEWEB)
Linares, Jesus [Area de Optica, Departamento de Fisica Aplicada, Facultade de Fisica, Escola Universitaria de Optica e Optometria, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia (Spain)], E-mail: suso.linares.beiras@usc.es; Nistal, Maria C. [Area de Optica, Departamento de Fisica Aplicada, Facultade de Fisica, Escola Universitaria de Optica e Optometria, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia (Spain)
2009-05-04
A quantum analysis based on the Dirac equation of the propagation of spinor-electron waves in coupled quantum wells, or equivalently coupled electron waveguides, is presented. The complete optical wave equations for Spin-Up (SU) and Spin-Down (SD) spinor-electron waves in these electron guides couplers are derived from the Dirac equation. The relativistic amplitudes and dispersion equations of the spinor-electron wave-guided modes in a planar quantum coupler formed by two coupled quantum wells, or equivalently by two coupled slab electron waveguides, are exactly derived. The main outcomes related to the spinor modal structure, such as the breaking of the non-relativistic degenerate spin states, the appearance of phase shifts associated with the spin polarization and so on, are shown.
Critical string wave equations and the QCD (U(N{sub c})) string. (Some comments)
Energy Technology Data Exchange (ETDEWEB)
Botelho, Luiz C.L. [Universidade Federal Fluminense (UFF), Niteroi, RJ (Brazil). Inst. de Matematica. Dept. de Matematica Aplicada], e-mail: botelho.luiz@superig.com.br
2009-07-01
We present a simple proof that self-avoiding fermionic strings solutions solve formally (in a Quantum Mechanical Framework) the QCD(U(N{sub c})) loop wave equation written in terms of random loops. (author)
Traveling Wave Solutions of ZK-BBM Equation Sine-Cosine Method
Directory of Open Access Journals (Sweden)
Sadaf Bibi
2014-03-01
Full Text Available Travelling wave solutions are obtained by using a relatively new technique which is called sine-cosine method for ZK-BBM equations. Solution procedure and obtained results re-confirm the efficiency of the proposed scheme.
Small amplitude periodic solutions in time for one-dimensional nonlinear wave equations
Liu, Zhenjie
2017-09-01
This paper is devoted to the construction of solutions for one-dimensional wave equations with Dirichlet or Neumann boundary conditions by means of a Nash-Moser iteration scheme, for a large set of frequencies.
Jiang, Lijian
2010-08-01
In this paper, we discuss a numerical multiscale approach for solving wave equations with heterogeneous coefficients. Our interest comes from geophysics applications and we assume that there is no scale separation with respect to spatial variables. To obtain the solution of these multiscale problems on a coarse grid, we compute global fields such that the solution smoothly depends on these fields. We present a Galerkin multiscale finite element method using the global information and provide a convergence analysis when applied to solve the wave equations. We investigate the relation between the smoothness of the global fields and convergence rates of the global Galerkin multiscale finite element method for the wave equations. Numerical examples demonstrate that the use of global information renders better accuracy for wave equations with heterogeneous coefficients than the local multiscale finite element method. © 2010 IMACS.
Four ways to justify temporal memory operators in the lossy wave equation
Holm, Sverre
2015-01-01
Attenuation of ultrasound often follows near power laws which cannot be modeled with conventional viscous or relaxation wave equations. The same is often the case for shear wave propagation in tissue also. More general temporal memory operators in the wave equation can describe such behavior. They can be justified in four ways: 1) Power laws for attenuation with exponents other than two correspond to the use of convolution operators with a temporal memory kernel which is a power law in time. 2) The corresponding constitutive equation is also a convolution, often with a temporal power law function. 3) It is also equivalent to an infinite set of relaxation processes which can be formulated via the complex compressibility. 4) The constitutive equation can also be expressed as an infinite sum of higher order derivatives. An extension to longitudinal waves in a nonlinear medium is also provided.
Energy Technology Data Exchange (ETDEWEB)
Barut, A.O. (Colorado Univ., Boulder (USA). Dept. of Physics); Oezaltin, O.; Uenal, N. (Dicle Univ., Diyarbakir (Turkey). Dept. of Physics)
1985-01-01
The Heisenberg equations for the Dirac electron in an external electromagnetic plane wave have been solved exactly in terms of incomplete ..gamma..-functions. As a special case the solution for a crossed constant electric and magnetic field is given.
Yang, Zhijian; Liu, Zhiming
2017-03-01
The paper investigates the well-posedness and the longtime dynamics of the quasilinear wave equations with structural damping and supercritical nonlinearities: {{u}tt}- Δ u+{{≤ft(- Δ \\right)}α}{{u}t}-\
Zhao, Xiaofeng; McGough, Robert J.
2016-01-01
The attenuation of ultrasound propagating in human tissue follows a power law with respect to frequency that is modeled by several different causal and noncausal fractional partial differential equations. To demonstrate some of the similarities and differences that are observed in three related time-fractional partial differential equations, time-domain Green's functions are calculated numerically for the power law wave equation, the Szabo wave equation, and for the Caputo wave equation. These Green's functions are evaluated for water with a power law exponent of y = 2, breast with a power law exponent of y = 1.5, and liver with a power law exponent of y = 1.139. Simulation results show that the noncausal features of the numerically calculated time-domain response are only evident very close to the source and that these causal and noncausal time-domain Green's functions converge to the same result away from the source. When noncausal time-domain Green's functions are convolved with a short pulse, no evidence of noncausal behavior remains in the time-domain, which suggests that these causal and noncausal time-fractional models are equally effective for these numerical calculations. PMID:27250193
Pointwise estimates for solutions to a system of nonlinear damped wave equations
Directory of Open Access Journals (Sweden)
Wenjun Wang
2013-11-01
Full Text Available In this article, we consider the existence of global solutions and pointwise estimates for the Cauchy problem of a nonlinear damped wave equation. We obtain the existence by using the approach introduced by Li and Chen in [7] and some estimates of the solution. The proofs of the estimates are based on a detailed analysis of the Green function of the linear damped wave equations. Also, we show the L^p convergence rate of the solution.
A nine-point finite difference scheme for one-dimensional wave equation
Szyszka, Barbara
2017-07-01
The paper is devoted to an implicit finite difference method (FDM) for solving initial-boundary value problems (IBVP) for one-dimensional wave equation. The second-order derivatives in the wave equation have been approximated at the four intermediate points, as a consequence, an implicit nine-point difference scheme has been obtained. Von Neumann stability analysis has been conducted and we have demonstrated, that the presented difference scheme is unconditionally stable.
The influence of damping and source terms on solutions of nonlinear wave equations
Directory of Open Access Journals (Sweden)
Mohammad A. Rammaha
2007-11-01
Full Text Available We discuss in this paper some recent development in the study of nonlinear wave equations. In particular, we focus on those results that deal with wave equations that feature two competing forces.One force is a damping term and the other is a strong source. Our central interest here is to analyze the influence of these forces on the long-time behavior of solutions.
Hamilton, Brian; Bilbao, Stefan
2013-01-01
Finite difference schemes for the 2-D wave equation operating on hexagonal grids and the accompanyingnumerical dispersion properties have received little attention in comparison to schemes operating on rectilinear grids. This paper considers the hexagonal tiling of the wavenumber plane in order to show that thehexagonal grid is a more natural choice to emulate the isotropy of the Laplacian operator and the wave equation. Performance of the 7-point scheme on a hexagonal grid is better than pre...
Alvarez, Roberto; van Hecke, Martin; van Saarloos, Wim
1996-01-01
In many pattern forming systems that exhibit traveling waves, sources and sinks occur which separate patches of oppositely traveling waves. We show that simple qualitative features of their dynamics can be compared to predictions from coupled amplitude equations. In heated wire convection experiments, we find a discrepancy between the observed multiplicity of sources and theoretical predictions. The expression for the observed motion of sinks is incompatible with any amplitude equation descri...
Existence and Stability of Steady Waves for the Hasegawa-Mima Equation
Directory of Open Access Journals (Sweden)
Guo Boling
2009-01-01
Full Text Available Abstract By introducing a compactness lemma and considering a constrained variational problem, we obtain a set of steady waves for Hasegawa-Mima equation, which describes the motion of drift waves in plasma. Moreover, we prove that is a stable set for the initial value problem of the equation, in the sense that a solution which starts near will remain near it for all time.
Said-Houari, Belkacem
2012-03-01
In this paper, we consider a viscoelastic wave equation with an absorbing term and space-time dependent damping term. Based on the weighted energy method, and by assuming that the kernel decaying exponentially, we obtain the L2 decay rates of the solutions. More precisely, we show that the decay rates are the same as those obtained in Lin et al. (2010) [15] for the semilinear wave equation with absorption term. © 2011 Elsevier Inc.
Wapenaar, Kees
2017-06-01
A unified scalar wave equation is formulated, which covers three-dimensional (3D) acoustic waves, 2D horizontally-polarised shear waves, 2D transverse-electric EM waves, 2D transverse-magnetic EM waves, 3D quantum-mechanical waves and 2D flexural waves. The homogeneous Green's function of this wave equation is a combination of the causal Green's function and its time-reversal, such that their singularities at the source position cancel each other. A classical representation expresses this homogeneous Green's function as a closed boundary integral. This representation finds applications in holographic imaging, time-reversed wave propagation and Green's function retrieval by cross correlation. The main drawback of the classical representation in those applications is that it requires access to a closed boundary around the medium of interest, whereas in many practical situations the medium can be accessed from one side only. Therefore, a single-sided representation is derived for the homogeneous Green's function of the unified scalar wave equation. Like the classical representation, this single-sided representation fully accounts for multiple scattering. The single-sided representation has the same applications as the classical representation, but unlike the classical representation it is applicable in situations where the medium of interest is accessible from one side only.
Wave Breaking Phenomenon for DGH Equation with Strong Dissipation
Directory of Open Access Journals (Sweden)
Zhengguang Guo
2014-01-01
Full Text Available The present work is mainly concerned with the Dullin-Gottwald-Holm (DGH equation with strong dissipative term. We establish some sufficient conditions to guarantee finite time blow-up of strong solutions.
Evolution of wave and tide over vegetation region in nearshore waters
Zhang, Mingliang; Zhang, Hongxing; Zhao, Kaibin; Tang, Jun; Qin, Huifa
2017-08-01
Coastal wetlands are an important ecosystem in nearshore regions, where complex flow characteristics occur because of the interactions among tides, waves, and plants, especially in the discontinuous flow of the intertidal zone. In order to simulate the wave and wave-induced current in coastal waters, in this study, an explicit depth-averaged hydrodynamic (HD) model has been dynamically coupled with a wave spectral model (CMS-Wave) by sharing the tide and wave data. The hydrodynamic model is based on the finite volume method; the intercell flux is computed using the Harten-Lax-van Leer (HLL) approximate Riemann solver for computing the dry-to-wet interface; the drag force of vegetation is modeled as the sink terms in the momentum equations. An empirical wave energy dissipation term with plant effect has been derived from the wave action balance equation to account for the resistance induced by aquatic vegetation in the CMS-Wave model. The results of the coupling model have been verified using the measured data for the case with wave-tide-vegetation interactions. The results show that the wave height decreases significantly along the wave propagation direction in the presence of vegetation. In the rip channel system, the oblique waves drive a meandering longshore current; it moves from left to right past the cusps with oscillations. In the vegetated region, the wave height is greatly attenuated due to the presence of vegetation, and the radiation stresses are noticeably changed as compared to the region without vegetation. Further, vegetation can affect the spatial distribution of mean velocity in a rip channel system. In the co-exiting environment of tides, waves, and vegetation, the locations of wave breaking and wave-induced radiation stress also vary with the water level of flooding or ebb tide in wetland water, which can also affect the development and evolution of wave-induced current.
Directory of Open Access Journals (Sweden)
M.G. Hafez
2015-06-01
Full Text Available The modeling of wave propagation in microstructured materials should be able to account for various scales of microstructure. Based on the proposed new exponential expansion method, we obtained the multiple explicit and exact traveling wave solutions of the strain wave equation for describing different types of wave propagation in microstructured solids. The solutions obtained in this paper include the solitary wave solutions of topological kink, singular kink, non-topological bell type solutions, solitons, compacton, cuspon, periodic solutions, and solitary wave solutions of rational functions. It is shown that the new exponential method, with the help of symbolic computation, provides an effective and straightforward mathematical tool for solving nonlinear evolution equations arising in mathematical physics and engineering.
Rogue wave solutions of the nonlinear Schrödinger equation with ...
Indian Academy of Sciences (India)
In this paper, a unified formula of a series of rogue wave solutions for the standard (1+1)-dimensional nonlinear Schrödinger equation is obtained through exp-function method. Further, by means of an appropriate transformation and previously obtained solutions, rogue wave solutions of the variable coefficient Schrödinger ...
Fundamental solutions of the wave equation in Robertson-Walker spaces
Yagdjian, Karen; Galstian, Anahit
2008-10-01
In this article we construct the fundamental solutions for the wave equation in the Robertson-Walker spaces arising in the de Sitter model of the universe. We then use these fundamental solutions to represent solutions of the Cauchy problem for the equation with and without a source term.
A similarity relation of the coupled equations for RF waves in a tokamak
Lee, Jungpyo; Smithe, David; Jaeger, Erwin; Berry, Lee; Harvey, R. W.; Bonoli, Paul
2017-10-01
The propagation and damping of RF waves in plasmas are modeled kinetically by solving the coupled equations between Maxwell's equation and Fokker-Planck equation. When the plasmas are magnetized, the wave dielectric tensor strongly depends on the background magnetic field, which can be calculated using Grad-Shafranov equation in a toroidally symmetric geometry. We found a similarity in the solutions of the coupled equations above, which keep the several dimensionless parameters constant. By changing plasma density and pressure, machine geometry (major radius), and RF wave frequency and power according to the similarity rule, there exists a set of solutions that show the consistent change in the background magnetic fields in the Grad-Shafranov equation, the electric and magnetic fields in the Maxwell's equation, and the distribution function of the Fokker-Planck equation. By investigating the numerical errors of the solutions from the expected results by the similarity rule, we verify the coupled numerical code for the RF waves in a tokamak (e.g. TORIC or AORSA/CQL3D/ECOM). This work was supported by US DoE Contract No. DE-FC02-01ER54648 under a Scientific Discovery through Advanced Computing Initiative.
Global Existence of Solutions to the Fowler Equation in a Neighbourhood of Travelling-Waves
Directory of Open Access Journals (Sweden)
Afaf Bouharguane
2011-01-01
Full Text Available We investigate a fractional diffusion/anti-diffusion equation proposed by Andrew C. Fowler to describe the dynamics of sand dunes sheared by a fluid flow. In this paper, we prove the global-in-time well-posedness in the neighbourhood of travelling-waves solutions of the Fowler equation.
Nurijanyan, S.; van der Vegt, Jacobus J.W.; Bokhove, Onno
2011-01-01
A discontinuous Galerkin nite element method (DGFEM) has been developed and tested for linear, three-dimensional, rotating incompressible Euler equations. These equations admit complicated wave solutions. The numerical challenges concern: (i) discretisation of a divergence-free velocity eld; (ii)
Energy Technology Data Exchange (ETDEWEB)
Fierros Palacios, Angel [Instituto de Investigaciones Electricas, Temixco, Morelos (Mexico)
2001-02-01
In this work the complete set of differential field equations which describes the dynamic state of a continuos conducting media which flow in presence of a perturbed magnetic field is obtained. Then, the thermic equation of state, the wave equation and the conservation law of energy for the Alfven MHD waves are obtained. [Spanish] Es este trabajo se obtiene el conjunto completo de ecuaciones diferenciales de campo que describen el estado dinamico de un medio continuo conductor que se mueve en presencia de un campo magnetico externo perturbado. Asi, se obtiene la ecuacion termica de estado, la ecuacion de onda y la ley de la conservacion de la energia para las ondas de Alfven de la MHD.
Exact Traveling Wave Solutions for Wick-Type Stochastic Schamel KdV Equation
Directory of Open Access Journals (Sweden)
Hossam A. Ghany
2014-01-01
Full Text Available F-expansion method is proposed to seek exact solutions of nonlinear partial differential equations. By means of Hermite transform, inverse Hermite transform, and white noise analysis, the variable coefficients and Wick-type stochastic Schamel KdV equations are completely described. Abundant exact traveling wave solutions for variable coefficients Schamel KdV equations are given. These solutions include exact stochastic Jacobi elliptic functions, trigonometric functions, and hyperbolic functions solutions.
Global existence and decay of solutions of a nonlinear system of wave equations
Said-Houari, Belkacem
2012-03-01
This work is concerned with a system of two wave equations with nonlinear damping and source terms acting in both equations. Under some restrictions on the nonlinearity of the damping and the source terms, we show that our problem has a unique local solution. Also, we prove that, for some restrictions on the initial data, the rate of decay of the total energy is exponential or polynomial depending on the exponents of the damping terms in both equations.
A Possible Generalization of Acoustic Wave Equation Using the Concept of Perturbed Derivative Order
Directory of Open Access Journals (Sweden)
Abdon Atangana
2013-01-01
Full Text Available The standard version of acoustic wave equation is modified using the concept of the generalized Riemann-Liouville fractional order derivative. Some properties of the generalized Riemann-Liouville fractional derivative approximation are presented. Some theorems are generalized. The modified equation is approximately solved by using the variational iteration method and the Green function technique. The numerical simulation of solution of the modified equation gives a better prediction than the standard one.
Conservation Laws and Traveling Wave Solutions of a Generalized Nonlinear ZK-BBM Equation
Directory of Open Access Journals (Sweden)
Khadijo Rashid Adem
2014-01-01
Full Text Available We study a generalized two-dimensional nonlinear Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM equation, which is in fact Benjamin-Bona-Mahony equation formulated in the ZK sense. Conservation laws for this equation are constructed by using the new conservation theorem due to Ibragimov and the multiplier method. Furthermore, traveling wave solutions are obtained by employing the (G'/G-expansion method.
Traveling Wave Solutions of Reaction-Diffusion Equations Arising in Atherosclerosis Models
Directory of Open Access Journals (Sweden)
Narcisa Apreutesei
2014-05-01
Full Text Available In this short review article, two atherosclerosis models are presented, one as a scalar equation and the other one as a system of two equations. They are given in terms of reaction-diffusion equations in an infinite strip with nonlinear boundary conditions. The existence of traveling wave solutions is studied for these models. The monostable and bistable cases are introduced and analyzed.
Dutta, Gaurav
2013-08-20
Attenuation leads to distortion of amplitude and phase of seismic waves propagating inside the earth. Conventional acoustic and least-squares reverse time migration do not account for this distortion which leads to defocusing of migration images in highly attenuative geological environments. To account for this distortion, we propose to use the visco-acoustic wave equation for least-squares reverse time migration. Numerical tests on synthetic data show that least-squares reverse time migration with the visco-acoustic wave equation corrects for this distortion and produces images with better balanced amplitudes compared to the conventional approach. © 2013 SEG.
Some Exact Results for the Schroedinger Wave Equation with a Time Dependent Potential
Campbell, Joel
2009-01-01
The time dependent Schroedinger equation with a time dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wave function at the origin, one may derive the wave function everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the p otential lead to conservation of the normalization of the probability density.
Stochastic wave-function unravelling of the generalized Lindblad equation
Semin, V.; Semina, I.; Petruccione, F.
2017-12-01
We investigate generalized non-Markovian stochastic Schrödinger equations (SSEs), driven by a multidimensional counting process and multidimensional Brownian motion introduced by A. Barchielli and C. Pellegrini [J. Math. Phys. 51, 112104 (2010), 10.1063/1.3514539]. We show that these SSEs can be translated in a nonlinear form, which can be efficiently simulated. The simulation is illustrated by the model of a two-level system in a structured bath, and the results of the simulations are compared with the exact solution of the generalized master equation.
Polynomial expansions for solution of wave equation in quantum calculus
Directory of Open Access Journals (Sweden)
Akram Nemri
2010-12-01
Full Text Available In this paper, using the q^2 -Laplace transform early introduced by Abdi [1], we study q-Wave polynomials related with the q-difference operator ∆q,x . We show in particular that they are linked to the q-little Jacobi polynomials p_n (x; α, β | q^2 .
A causal and fractional all-frequency wave equation for lossy media.
Holm, Sverre; Näsholm, Sven Peter
2011-10-01
This work presents a lossy partial differential acoustic wave equation including fractional derivative terms. It is derived from first principles of physics (mass and momentum conservation) and an equation of state given by the fractional Zener stress-strain constitutive relation. For a derivative order α in the fractional Zener relation, the resulting absorption α(k) obeys frequency power-laws as α(k) ∝ ω(1+α) in a low-frequency regime, α(k) ∝ ω(1-α/2) in an intermediate-frequency regime, and α(k) ∝ ω(1-α) in a high-frequency regime. The value α=1 corresponds to the case of a single relaxation process. The wave equation is causal for all frequencies. In addition the sound speed does not diverge as the frequency approaches infinity. This is an improvement over a previously published wave equation building on the fractional Kelvin-Voigt constitutive relation. © 2011 Acoustical Society of America
Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue
2018-01-01
This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0 . Furthermore, we prove the global existence and uniqueness of C^{α ,β } -solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1 -space. The exponential convergence rate is also derived.
Solitary wave solutions to nonlinear evolution equations in ...
Indian Academy of Sciences (India)
This paper obtains solitons as well as other solutions to a few nonlinear evolution equations that appear in various areas of mathematical physics. The two analytical integrators that are applied to extract solutions are tan–cot method and functional variable approaches. The soliton solutions can be used in the further study of ...
Spectral characteristics of high shallow water waves
Digital Repository Service at National Institute of Oceanography (India)
SanilKumar, V.; AshokKumar, K.
–Moskowitz and Kitaigordskii et al. (1975) for the south-west coast of India. Ochi and Hubble (1976) found that JONSWAP spectrum provided good approx- imation to the data for the uni-model spectra with mean JONSWAP parameters of g (peak enhancement para- meter) ¼ 2.2 and a.... 4.1–4.55. Young, I.R., 1992. The determination of spectral parameters from significant wave height and peak period. Ocean Engineering 19 (5), 497–508. Young, I.R., Verhagen, L.A., 1996. The growth of fetch limited waves in water of finite depth: Part...
Galilean exotic planar supersymmetries and nonrelativistic supersymmetric wave equations
Energy Technology Data Exchange (ETDEWEB)
Lukierski, J. [Institute for Theoretical Physics, University of Wroclaw, pl. Maxa Borna 9, 50-205 Wroclaw (Poland)]. E-mail: lukier@ift.uni.wroc.pl; Prochnicka, I. [Institute for Theoretical Physics, University of Wroclaw, pl. Maxa Borna 9, 50-205 Wroclaw (Poland)]. E-mail: ipro@ift.uni.wroc.pl; Stichel, P.C. [An der Krebskuhle 21, D-33619 Bielefeld (Germany)]. E-mail: peter@physik.uni-bielefeld.de; Zakrzewski, W.J. [Department of Mathematical Sciences, University of Durham, Durham DH1 3LE (United Kingdom)]. E-mail: w.j.zakrzewski@durham.ac.uk
2006-08-10
We describe the general class of N-extended D=(2+1) Galilean supersymmetries obtained, respectively, from the N-extended D=3 Poincare superalgebras with maximal sets of central charges. We confirm the consistency of supersymmetry with the presenc the 'exotic' second central charge {theta}. We show further how to introduce a N=2 Galilean superfield equation describing nonrelativistic spin 0 and spin 12 free particles.
Hu, Wenjing
2017-08-01
This paper uses Fourier’s triple integral transform method to simplify the calculation of the non-homogeneous wave equations of the time-varying electromagnetic field. By adding several special definite conditions to the wave equation, it becomes a mathematical problem of definite condition. Then by using Fourier’s triple integral transform method, this three-dimension non-homogeneous partial differential wave equation is changed into an ordinary differential equation. Through the solution to this ordinary differential equation, the expression of the relationship between the time-varying scalar potential and electromagnetic wave excitation source is developed precisely. This method simplifies the solving process effectively.
Coupled equations of electromagnetic waves in nonlinear metamaterial waveguides.
Azari, Mina; Hatami, Mohsen; Meygoli, Vahid; Yousefi, Elham
2016-11-01
Over the past decades, scientists have presented ways to manipulate the macroscopic properties of a material at levels unachieved before, and called them metamaterials. This research can be considered an important step forward in electromagnetics and optics. In this study, higher-order nonlinear coupled equations in a special kind of metamaterial waveguides (a planar waveguide with metamaterial core) will be derived from both electric and magnetic components of the transverse electric mode of electromagnetic pulse propagation. On the other hand, achieving the refractive index in this research is worthwhile. It is also shown that the coupled equations are not symmetric with respect to the electric and magnetic fields, unlike these kinds of equations in fiber optics and dielectric waveguides. Simulations on the propagation of a fundamental soliton pulse in a nonlinear metamaterial waveguide near the resonance frequency (a little lower than the magnetic resonant frequency) are performed to study its behavior. These pulses are recommended to practice in optical communications in controlled switching by external voltage, even in low power.
Orbital stability of periodic traveling-wave solutions for the log-KdV equation
Natali, Fábio; Pastor, Ademir; Cristófani, Fabrício
2017-09-01
In this paper we establish the orbital stability of periodic waves related to the logarithmic Korteweg-de Vries equation. Our motivation is inspired in the recent work [3], in which the authors established the well-posedness and the linear stability of Gaussian solitary waves. By using the approach put forward recently in [20] to construct a smooth branch of periodic waves as well as to get the spectral properties of the associated linearized operator, we apply the abstract theories in [13] and [25] to deduce the orbital stability of the periodic traveling waves in the energy space.
Nonlinear water waves: introduction and overview
Constantin, A.
2017-12-01
For more than two centuries progress in the study of water waves proved to be interdependent with innovative and deep developments in theoretical and experimental directions of investigation. In recent years, considerable progress has been achieved towards the understanding of waves of large amplitude. Within this setting one cannot rely on linear theory as nonlinearity becomes an essential feature. Various analytic methods have been developed and adapted to come to terms with the challenges encountered in settings where approximations (such as those provided by linear or weakly nonlinear theory) are ineffective. Without relying on simpler models, progress becomes contingent upon the discovery of structural properties, the exploitation of which requires a combination of creative ideas and state-of-the-art technical tools. The successful quest for structure often reveals unexpected patterns and confers aesthetic value on some of these studies. The topics covered in this issue are both multi-disciplinary and interdisciplinary: there is a strong interplay between mathematical analysis, numerical computation and experimental/field data, interacting with each other via mutual stimulation and feedback. This theme issue reflects some of the new important developments that were discussed during the programme `Nonlinear water waves' that took place at the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) from 31st July to 25th August 2017. A cross-section of the experts in the study of water waves who participated in the programme authored the collected papers. These papers illustrate the diversity, intensity and interconnectivity of the current research activity in this area. They offer new insight, present emerging theoretical methodologies and computational approaches, and describe sophisticated experimental results. This article is part of the theme issue 'Nonlinear water waves'.
Nonlinear water waves: introduction and overview.
Constantin, A
2018-01-28
For more than two centuries progress in the study of water waves proved to be interdependent with innovative and deep developments in theoretical and experimental directions of investigation. In recent years, considerable progress has been achieved towards the understanding of waves of large amplitude. Within this setting one cannot rely on linear theory as nonlinearity becomes an essential feature. Various analytic methods have been developed and adapted to come to terms with the challenges encountered in settings where approximations (such as those provided by linear or weakly nonlinear theory) are ineffective. Without relying on simpler models, progress becomes contingent upon the discovery of structural properties, the exploitation of which requires a combination of creative ideas and state-of-the-art technical tools. The successful quest for structure often reveals unexpected patterns and confers aesthetic value on some of these studies. The topics covered in this issue are both multi-disciplinary and interdisciplinary: there is a strong interplay between mathematical analysis, numerical computation and experimental/field data, interacting with each other via mutual stimulation and feedback. This theme issue reflects some of the new important developments that were discussed during the programme 'Nonlinear water waves' that took place at the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) from 31st July to 25th August 2017. A cross-section of the experts in the study of water waves who participated in the programme authored the collected papers. These papers illustrate the diversity, intensity and interconnectivity of the current research activity in this area. They offer new insight, present emerging theoretical methodologies and computational approaches, and describe sophisticated experimental results.This article is part of the theme issue 'Nonlinear water waves'. © 2017 The Author(s).
On shallow water waves in a medium with time-dependent
Directory of Open Access Journals (Sweden)
Hamdy I. Abdel-Gawad
2015-07-01
Full Text Available In this paper, we studied the progression of shallow water waves relevant to the variable coefficient Korteweg–de Vries (vcKdV equation. We investigated two kinds of cases: when the dispersion and nonlinearity coefficients are proportional, and when they are not linearly dependent. In the first case, it was shown that the progressive waves have some geometric structures as in the case of KdV equation with constant coefficients but the waves travel with time dependent speed. In the second case, the wave structure is maintained when the nonlinearity balances the dispersion. Otherwise, water waves collapse. The objectives of the study are to find a wide class of exact solutions by using the extended unified method and to present a new algorithm for treating the coupled nonlinear PDE’s.
Conditional Second Order Short-crested Water Waves Applied to Extreme Wave Episodes
DEFF Research Database (Denmark)
Jensen, Jørgen Juncher
2005-01-01
A derivation of the mean second order short-crested wave pattern and associated wave kinematics, conditional on a given magnitude of the wave crest, is presented. The analysis is based on the second order Sharma and Dean finite water wave theory. A comparison with a measured extreme wave profile...
Exact Travelling Wave Solutions of two Important Nonlinear Partial Differential Equations
Kim, Hyunsoo; Bae, Jae-Hyeong; Sakthivel, Rathinasamy
2014-04-01
Coupled nonlinear partial differential equations describing the spatio-temporal dynamics of predator-prey systems and nonlinear telegraph equations have been widely applied in many real world problems. So, finding exact solutions of such equations is very helpful in the theories and numerical studies. In this paper, the Kudryashov method is implemented to obtain exact travelling wave solutions of such physical models. Further, graphic illustrations in two and three dimensional plots of some of the obtained solutions are also given to predict their behaviour. The results reveal that the Kudryashov method is very simple, reliable, and effective, and can be used for finding exact solution of many other nonlinear evolution equations.
Energy Technology Data Exchange (ETDEWEB)
Watanabe, T.; Sassa, K. [Kyoto University, Kyoto (Japan); Uesaka, S. [Kyoto University, Kyoto (Japan). Faculty of Engineering
1996-10-01
The effect of initial models on full-wave inversion (FWI) analysis based on acoustic wave-equation was studied for elastic wave tomography of underground structures. At present, travel time inversion using initial motion travel time is generally used, and inverse analysis is conducted using the concept `ray,` assuming very high wave frequency. Although this method can derive stable solutions relatively unaffected by initial model, it uses only the data of initial motion travel time. FWI calculates theoretical waveform at each receiver using all of observed waveforms as data by wave equation modeling where 2-D underground structure is calculated by difference calculus under the assumption that wave propagation is described by wave equation of P wave. Although it is a weak point that FWI is easily affected by noises in an initial model and data, it is featured by high resolution of solutions. This method offers very excellent convergence as a proper initial model is used, resulting in sufficient performance, however, it is strongly affected by initial model. 2 refs., 7 figs., 1 tab.
Solvability of the Initial Value Problem to the Isobe-Kakinuma Model for Water Waves
Nemoto, Ryo; Iguchi, Tatsuo
2017-09-01
We consider the initial value problem to the Isobe-Kakinuma model for water waves and the structure of the model. The Isobe-Kakinuma model is the Euler-Lagrange equations for an approximate Lagrangian which is derived from Luke's Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The Isobe-Kakinuma model is a system of second order partial differential equations and is classified into a system of nonlinear dispersive equations. Since the hypersurface t=0 is characteristic for the Isobe-Kakinuma model, the initial data have to be restricted in an infinite dimensional manifold for the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh-Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces. We also discuss the linear dispersion relation to the model.
Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity
Directory of Open Access Journals (Sweden)
Claudio Cremaschini
2017-07-01
Full Text Available Key aspects of the manifestly-covariant theory of quantum gravity (Cremaschini and Tessarotto 2015–2017 are investigated. These refer, first, to the establishment of the four-scalar, manifestly-covariant evolution quantum wave equation, denoted as covariant quantum gravity (CQG wave equation, which advances the quantum state ψ associated with a prescribed background space-time. In this paper, the CQG-wave equation is proved to follow at once by means of a Hamilton–Jacobi quantization of the classical variational tensor field g ≡ g μ ν and its conjugate momentum, referred to as (canonical g-quantization. The same equation is also shown to be variational and to follow from a synchronous variational principle identified here with the quantum Hamilton variational principle. The corresponding quantum hydrodynamic equations are then obtained upon introducing the Madelung representation for ψ , which provides an equivalent statistical interpretation of the CQG-wave equation. Finally, the quantum state ψ is proven to fulfill generalized Heisenberg inequalities, relating the statistical measurement errors of quantum observables. These are shown to be represented in terms of the standard deviations of the metric tensor g ≡ g μ ν and its quantum conjugate momentum operator.
Shiryaeva, E V
2014-01-01
In paper [S.I. Senashov, A. Yakhno. 2012. SIGMA. Vol.8. 071] the variant of the hodograph method based on the conservation laws for two hyperbolic quasilinear equations of the first order is described. Using these results we propose a method which allows to reduce the Cauchy problem for the two quasilinear PDE's to the Cauchy problem for ODE's. The proposed method is actually some similar method of characteristics for a system of two hyperbolic quasilinear equations. The method can be used effectively in all cases, when the linear hyperbolic equation in partial derivatives of the second order with variable coefficients, resulting from the application of the hodograph method, has an explicit expression for the Riemann-Green function. One of the method's features is the possibility to construct a multi-valued solutions. In this paper we present examples of method application for solving the classical shallow water equations.
Spectral power density of the random excitation for the photoacoustic wave equation
Directory of Open Access Journals (Sweden)
Hakan Erkol
2014-09-01
Full Text Available The superposition of the Green's function and its time reversal can be extracted from the photoacoustic point sources applying the representation theorems of the convolution and correlation type. It is shown that photoacoustic pressure waves at locations of random point sources can be calculated with the solution of the photoacoustic wave equation and utilization of the continuity and the discontinuity conditions of the pressure waves in the frequency domain although the pressure waves cannot be measured at these locations directly. Therefore, with the calculated pressure waves at the positions of the sources, the spectral power density can be obtained for any system consisting of two random point sources. The methodology presented here can also be generalized to any finite number of point like sources. The physical application of this study includes the utilization of the cross-correlation of photoacoustic waves to extract functional information associated with the flow dynamics inside the tissue.
Visco-acoustic wave-equation traveltime inversion and its sensitivity to attenuation errors
Yu, Han
2018-02-23
A visco-acoustic wave-equation traveltime inversion method is presented that inverts for the shallow subsurface velocity distribution. Similar to the classical wave equation traveltime inversion, this method finds the velocity model that minimizes the squared sum of the traveltime residuals. Even though, wave-equation traveltime inversion can partly avoid the cycle skipping problem, a good initial velocity model is required for the inversion to converge to a reasonable tomogram with different attenuation profiles. When Q model is far away from the real model, the final tomogram is very sensitive to the starting velocity model. Nevertheless, a minor or moderate perturbation of the Q model from the true one does not strongly affect the inversion if the low wavenumber information of the initial velocity model is mostly correct. These claims are validated with numerical tests on both the synthetic and field data sets.
On radiating solitary waves in bi-layers with delamination and coupled Ostrovsky equations.
Khusnutdinova, K R; Tranter, M R
2017-01-01
We study the scattering of a long longitudinal radiating bulk strain solitary wave in the delaminated area of a two-layered elastic structure with soft ("imperfect") bonding between the layers within the scope of the coupled Boussinesq equations. The direct numerical modelling of this and similar problems is challenging and has natural limitations. We develop a semi-analytical approach, based on the use of several matched asymptotic multiple-scale expansions and averaging with respect to the fast space variable, leading to the coupled Ostrovsky equations in bonded regions and uncoupled Korteweg-de Vries equations in the delaminated region. We show that the semi-analytical approach agrees well with direct numerical simulations and use it to study the nonlinear dynamics and scattering of the radiating solitary wave in a wide range of bi-layers with delamination. The results indicate that radiating solitary waves could help us to control the integrity of layered structures with imperfect interfaces.
Theory of a ring laser. [electromagnetic field and wave equations
Menegozzi, L. N.; Lamb, W. E., Jr.
1973-01-01
Development of a systematic formulation of the theory of a ring laser which is based on first principles and uses a well-known model for laser operation. A simple physical derivation of the electromagnetic field equations for a noninertial reference frame in uniform rotation is presented, and an attempt is made to clarify the nature of the Fox-Li modes for an open polygonal resonator. The polarization of the active medium is obtained by using a Fourier-series method which permits the formulation of a strong-signal theory, and solutions are given in terms of continued fractions. It is shown that when such a continued fraction is expanded to third order in the fields, the familiar small-signal ring-laser theory is obtained.
Traveling Waves for Nonlinear Schrödinger Equations with Nonzero Conditions at Infinity
Chiron, David; Mariş, Mihai
2017-10-01
We prove the existence of nontrivial finite energy traveling waves for a large class of nonlinear Schrödinger equations with nonzero conditions at infinity (includindg the Gross-Pitaevskii and the so-called "cubic-quintic" equations) in space dimension { N ≥ 2}. We show that minimization of the energy at fixed momentum can be used whenever the associated nonlinear potential is nonnegative and it gives a set of orbitally stable traveling waves, while minimization of the action at constant kinetic energy can be used in all cases. We also explore the relationship between the families of traveling waves obtained by different methods and we prove a sharp nonexistence result for traveling waves with small energy.
Angel, Jordan B.; Banks, Jeffrey W.; Henshaw, William D.
2018-01-01
High-order accurate upwind approximations for the wave equation in second-order form on overlapping grids are developed. Although upwind schemes are well established for first-order hyperbolic systems, it was only recently shown by Banks and Henshaw [1] how upwinding could be incorporated into the second-order form of the wave equation. This new upwind approach is extended here to solve the time-domain Maxwell's equations in second-order form; schemes of arbitrary order of accuracy are formulated for general curvilinear grids. Taylor time-stepping is used to develop single-step space-time schemes, and the upwind dissipation is incorporated by embedding the exact solution of a local Riemann problem into the discretization. Second-order and fourth-order accurate schemes are implemented for problems in two and three space dimensions, and overlapping grids are used to treat complex geometry and problems with multiple materials. Stability analysis of the upwind-scheme on overlapping grids is performed using normal mode theory. The stability analysis and computations confirm that the upwind scheme remains stable on overlapping grids, including the difficult case of thin boundary grids when the traditional non-dissipative scheme becomes unstable. The accuracy properties of the scheme are carefully evaluated on a series of classical scattering problems for both perfect conductors and dielectric materials in two and three space dimensions. The upwind scheme is shown to be robust and provide high-order accuracy.
Excitation of surface electromagnetic waves on water.
Singh, A K; Goben, C A; Davarpanah, M; Boone, J L
1978-11-01
Excitation of surface electromagnetic waves (SEW) on water was studied using optical coupling techniques at microwave frequencies. Excitation of SEW was also achieved using direct horn antenna coupling. The transmitted SEW power was increased by adding acid and salt to water. The horn antenna gave the maximum excitation efficiency 70%. It was increased to 75% by collimating the electromagnetic beam in the vertical direction. Excitation efficiency for the prism (0 degrees pitch angle) and grating couplers were 15.2% and 10.5% respectively. By changing the prism coupler pitch angle to +36 degrees , its excitation efficiency was increased to 82%.
On the new soliton and optical wave structures to some nonlinear evolution equations
Bulut, Hasan; Sulaiman, Tukur Abdulkadir; Baskonus, Haci Mehmet
2017-11-01
In this study, with the aid of the Wolfram Mathematica software, the powerful sine-Gordon expansion method is utilized to search for the solutions to some important nonlinear mathematical models arising in nonlinear sciences, namely, the (2 + 1) -dimensional Zakharov-Kuznetsov modified equal width equation, the cubic Boussinesq equation and the modified regularized long wave equation. We successfully obtain some new soliton, singular soliton, singular periodic waves and kink-type solutions with complex hyperbolic structures to these equations. We also present the two- and three-dimensional shapes of all the solutions obtained in this study. We further give the physical meaning of all the obtained solutions. We compare our results with the existing results in the literature.
Shock formation in small-data solutions to 3D quasilinear wave equations
Speck, Jared
2016-01-01
In 1848 James Challis showed that smooth solutions to the compressible Euler equations can become multivalued, thus signifying the onset of a shock singularity. Today it is known that, for many hyperbolic systems, such singularities often develop. However, most shock-formation results have been proved only in one spatial dimension. Serge Alinhac's groundbreaking work on wave equations in the late 1990s was the first to treat more than one spatial dimension. In 2007, for the compressible Euler equations in vorticity-free regions, Demetrios Christodoulou remarkably sharpened Alinhac's results and gave a complete description of shock formation. In this monograph, Christodoulou's framework is extended to two classes of wave equations in three spatial dimensions. It is shown that if the nonlinear terms fail to satisfy the null condition, then for small data, shocks are the only possible singularities that can develop. Moreover, the author exhibits an open set of small data whose solutions form a shock, and he prov...
Hydromagnetic waves in a compressed-dipole field via field-aligned Klein–Gordon equations
Directory of Open Access Journals (Sweden)
J. Zheng
2016-05-01
Full Text Available Hydromagnetic waves, especially those of frequencies in the range of a few millihertz to a few hertz observed in the Earth's magnetosphere, are categorized as ultra low-frequency (ULF waves or pulsations. They have been extensively studied due to their importance in the interaction with radiation belt particles and in probing the structures of the magnetosphere. We developed an approach to examining the toroidal standing Aflvén waves in a background magnetic field by recasting the wave equation into a Klein–Gordon (KG form along individual field lines. The eigenvalue solutions to the system are characteristic of a propagation type when the corresponding eigenfrequency is greater than a critical frequency and a decaying type otherwise. We apply the approach to a compressed-dipole magnetic field model of the inner magnetosphere and obtain the spatial profiles of relevant parameters and the spatial wave forms of harmonic oscillations. We further extend the approach to poloidal-mode standing Alfvén waves along field lines. In particular, we present a quantitative comparison with a recent spacecraft observation of a poloidal standing Alfvén wave in the Earth's magnetosphere. Our analysis based on the KG equation yields consistent results which agree with the spacecraft measurements of the wave period and the amplitude ratio between the magnetic field and electric field perturbations.
Directory of Open Access Journals (Sweden)
Victor Onomza WAZIRI
2006-07-01
Full Text Available The paper computes the optimal control and state of the two-dimensional Energized wave equation using the Extended Conjugate gradient Method (ECGM. This piece of work has to do with all the vital computational elements as derived in the implementation of the ECGM algorithm on the two-dimensional Energized Wave equation in (Waziri, 1 and (Waziri & Reju, LEJPT & LJS, Issues 9, 2006, [7-9]. With these recalls, program codes were derived which gave various numerical optimal controls and states. These optimal controls and states were considered at various points in a plane surface.
Exact solutions of time fractional heat-like and wave-like equations with variable coefficients
Directory of Open Access Journals (Sweden)
Zhang Sheng
2016-01-01
Full Text Available In this paper, a variable-coefficient time fractional heat-like and wave-like equation with initial and boundary conditions is solved by the use of variable separation method and the properties of Mittag-Leffler function. As a result, exact solutions are obtained, from which some known special solutions are recovered. It is shown that the variable separation method can also be used to solve some others time fractional heat-like and wave-like equation in science and engineering.
Baumeiste, K. J.
1983-01-01
A time-dependent finite difference formulation to the inhomogeneous wave equation is derived for plane wave propagation with harmonic noise sources. The difference equation and boundary conditions are developed along with the techniques to simulate the Dirac delta function associated with a concentrated noise source. Example calculations are presented for the Green's function and distributed noise sources. For the example considered, the desired Fourier transformed acoustic pressures are determined from the transient pressures by use of a ramping function and an integration technique, both of which eliminates the nonharmonic pressure associated with the initial transient.
Baumeister, K. J.
1983-01-01
A time-dependent finite difference formulation to the inhomogeneous wave equation is derived for plane wave propagation with harmonic noise sources. The difference equation and boundary conditions are developed along with the techniques to simulate the Dirac delta function associated with a concentrated noise source. Example calculations are presented for the Green's function and distributed noise sources. For the example considered, the desired Fourier transformed acoustic pressures are determined from the transient pressures by use of a ramping function and an integration technique, both of which eliminates the nonharmonic pressure associated with the initial transient.
NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION.
Liu, F; Meerschaert, M M; McGough, R J; Zhuang, P; Liu, Q
2013-03-01
In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.
Grosse, Ralf
1990-01-01
Propagation of sound through the turbulent atmosphere is a statistical problem. The randomness of the refractive index field causes sound pressure fluctuations. Although no general theory to predict sound pressure statistics from given refractive index statistics exists, there are several approximate solutions to the problem. The most common approximation is the parabolic equation method. Results obtained by this method are restricted to small refractive index fluctuations and to small wave lengths. While the first condition is generally met in the atmosphere, it is desirable to overcome the second. A generalization of the parabolic equation method with respect to the small wave length restriction is presented.
Three-dimensional wave-induced current model equations and radiation stresses
Xia, Hua-yong
2017-08-01
After the approach by Mellor (2003, 2008), the present paper reports on a repeated effort to derive the equations for three-dimensional wave-induced current. Via the vertical momentum equation and a proper coordinate transformation, the phase-averaged wave dynamic pressure is well treated, and a continuous and depth-dependent radiation stress tensor, rather than the controversial delta Dirac function at the surface shown in Mellor (2008), is provided. Besides, a phase-averaged vertical momentum flux over a sloping bottom is introduced. All the inconsistencies in Mellor (2003, 2008), pointed out by Ardhuin et al. (2008) and Bennis and Ardhuin (2011), are overcome in the presently revised equations. In a test case with a sloping sea bed, as shown in Ardhuin et al. (2008), the wave-driving forces derived in the present equations are in good balance, and no spurious vertical circulation occurs outside the surf zone, indicating that Airy's wave theory and the approach of Mellor (2003, 2008) are applicable for the derivation of the wave-induced current model.
Tchinang Tchameu, J. D.; Togueu Motcheyo, A. B.; Tchawoua, C.
2016-09-01
The discrete multi-rogue waves (DMRW) as solution of the discrete nonlinear Schrödinger (DNLS) equation with saturable nonlinearities is studied numerically. These biological rogue waves represent the complex probability amplitude of finding an amide-I vibrational quantum at a site. We observe that the growth in the higher order saturable nonlinearity implies the formation of DMRW including an increase in the short-living DMRW and a decrease in amplitude of the long-living DMRW.
Some Wave Simulation Properties of the (2+1 Dimensional Breaking Soliton Equation
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Aksan Emine Nesligül
2017-01-01
Full Text Available In this paper, we apply an efficient method which is improved Bernoulli sub-equation function method (IBSEFM to (2+1 dimensional Breaking Soliton equation. It gives some new wave simulations like complex and exponential structures. We test whether all structures verify the (2+1 dimensional Breaking Soliton model. Then, we draw three and two dimensional plane by using Wolfram Mathematica 9.
A memory type boundary stabilization of a mildly damped wave equation
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Mokhtar Kirane
1999-01-01
Full Text Available We consider the wave equation with a mild internal dissipation. It is proved that any small dissipation inside the domain is sufficient to uniformly stabilize the solution of this equation by means of a nonlinear feedback of memory type acting on a part of the boundary. This is established without any restriction on the space dimension and without geometrical conditions on the domain or its boundary.
Mixing and large deviations for nonlinear wave equation with white noise
Martirosyan, Davit
2015-01-01
This thesis is devoted to the study of ergodicity and large deviations for the stochastic nonlinear wave (NLW) equation with smooth white noise in 3D. Under some standard growth and dissipativity assumptions on the nonlinearity, we show that the Markov process associated with the flow of NLW equation has a unique stationary measure that attracts the law of any solution with exponential rate. This result implies, in particular, the strong law of large numbers as well as the central limit theor...
Al Ali, Usamah S.; Bokhari, Ashfaque H.; Kara, A. H.; Zaman, F. D.
Nonlinear evolution equations represent some of the most fundamental processes in both physics as well engineering. Considering this, we analyze and classify the three dimensional wave equation with a power law nonlinearity in presence of damping and external force terms. In view of the significance of conservation laws in physics, a study of the invariance properties is presented and conservation laws are constructed and classified. An illustrative case of a symmetry reduction in one special case is presented.
Directory of Open Access Journals (Sweden)
Nilesh P. BARDE
2015-05-01
Full Text Available The concept of time dependent Schrödinger equation (TDSE illustrated in literature and even during class room teaching is mostly either complex or meant for advanced learners. This article is intended to enlighten the concept to the beginners in the field and further to improve knowledge about detailed steps for abstract mathematical formulation used which helps in understanding to derive TDSE using various tools and in more comprehensible manner. It is shown that TDSE may be derived using wave mechanics, time independent equation, classical & Hamilton-Jacobi’s equations. Similar attempts have been done earlier by some researchers. However, this article provides a comprehensive, lucid and well derived derivation, derived using various approaches, which would make this article unique.
Favrie, N.; Gavrilyuk, S.
2017-07-01
A new numerical method for solving the Serre-Green-Naghdi (SGN) equations describing dispersive waves on shallow water is proposed. From the mathematical point of view, the SGN equations are the Euler-Lagrange equations for a ‘master’ lagrangian submitted to a differential constraint which is the mass conservation law. One major numerical challenge in solving the SGN equations is the resolution of an elliptic problem at each time instant. This is the most time-consuming part of the numerical method. The idea is to replace the ‘master’ lagrangian by a one-parameter family of ‘augmented’ lagrangians, depending on a greater number of variables, for which the corresponding Euler-Lagrange equations are hyperbolic. In such an approach, the ‘master’ lagrangian is recovered by the augmented lagrangian in some limit (for example, when the corresponding parameter is large). The choice of such a family of augmented lagrangians is proposed and discussed. The corresponding hyperbolic system is numerically solved by a Godunov type method. Numerical solutions are compared with exact solutions to the SGN equations. It appears that the computational time in solving the hyperbolic system is much lower than in the case where the elliptic operator is inverted. The new method is applied, in particular, to the study of ‘Favre waves’ representing non-stationary undular bores produced after reflection of the fluid flow with a free surface at an immobile wall.
Multi-wave solutions of the space–time fractional Burgers and Sharma–Tasso–Olver equations
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Emad A.-B. Abdel-Salam
2016-03-01
Full Text Available Based on the improved generalized exp-function method, the space–time fractional Burgers and Sharma–Tasso–Olver equations were studied. The single-wave, double-wave, three-wave and four-wave solution discussed. With the best of our knowledge, some of the results are obtained for the first time. The improved generalized exp-function method can be applied to other fractional differential equations.
Chirped self-similar waves for quadratic-cubic nonlinear Schrödinger equation
Pal, Ritu; Loomba, Shally; Kumar, C. N.
2017-12-01
We have constructed analytical self-similar wave solutions for quadratic-cubic Nonlinear Schrödinger equation (QC-NLSE) by means of similarity transformation method. Then, we have investigated the role of chirping on these self-similar waves as they propagate through the tapered graded index waveguide. We have revealed that the chirping leads to interesting features and allows us to control the propagation of self-similar waves. This has been demonstrated for two cases (i) periodically distributed system and (ii) constant choice of system parameters. We expect our results to be useful in designing high performance optical devices.
Ayub, Kamran; Khan, M. Yaqub; Mahmood-Ul-Hassan, Qazi; Ahmad, Jamshad
2017-09-01
Nonlinear mathematical problems and their solutions attain much attention in solitary waves. In soliton theory, an efficient tool to attain various types of soliton solutions is the \\exp (-φ (ζ ))-expansion technique. This article is devoted to find exact travelling wave solutions of Drinfeld-Sokolov equation via a reliable mathematical technique. By using the proposed technique, we attain soliton wave solution of various types. It is observed that the technique under discussion is user friendly with minimum computational work, and can be extended for physical problems of different nature in mathematical physics.
Unified Theory of Wave-Particle Duality and the Schr\\"odinger Equations
Gilson, Greyson
2011-01-01
Individual quantum objects display coexisting wave properties and particle properties. A wave is ordinarily associated with spatial extension while a particle is ordinarily associated with a point-like locality. Coexistence of spatial extension and a point-like locality as properties of a single entity seems paradoxical. The apparent paradox is resolved by the unified theory of wave-particle duality developed in this paper. Using this theory, a straightforward derivation of the Schr\\"odinger equations (time-independent and time-dependent) is presented where previously no such derivation was considered to be possible.
Utilization of Double-Water-Chamber Seawall type for Wave Energy Extraction and Wave Dissipation
Husain, Firman
2016-01-01
Variation type and model of wave energy converter have been applied in many countries around the world in order to harvest the ocean wave power. A number of other devices were developing and testing by researchers in the experimental scale. In the present paper investigates double-water-chamber seawall performance for wave energy converter. The main body of water chamber seawall is like OWC structure. Savonius water turbine and guide vanes used to extract a wave power instead of air turbine a...
Ermstål, Johan
2012-01-01
Two nonlinear dispersive wave equations arising in elasto-plastic flow have been investigated for self-adjointness. For these equations their symmetries are calculated and conservation laws are constructed using two different methods: an old method based on Noether´s Theorem and a new one developed by Prof. Nail Ibragimov. The new method works for a larger number of equations than the old one. It is complementing the old one in the way that it gives some conservation laws that otherwise would...
Modified wave operators for nonlinear Schrodinger equations in one and two dimensions
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Nakao Hayashi
2004-04-01
Full Text Available We study the asymptotic behavior of solutions, in particular the scattering theory, for the nonlinear Schr"{o}dinger equations with cubic and quadratic nonlinearities in one or two space dimensions. The nonlinearities are summation of gauge invariant term and non-gauge invariant terms. The scattering problem of these equations belongs to the long range case. We prove the existence of the modified wave operators to those equations for small final data. Our result is an improvement of the previous work [13
High-order rogue wave solutions of the classical massive Thirring model equations
Guo, Lijuan; Wang, Lihong; Cheng, Yi; He, Jingsong
2017-11-01
The nth-order solutions of the classical massive Thirring model (MTM) equations are derived by using the n-fold Darboux transformation. These solutions are expressed by the ratios of the two determinants consisted of 2n eigenfunctions under the reduction conditions. Using this method, rogue waves are constructed explicitly up to the third-order. Three patterns, i.e., fundamental, triangular and circular patterns, of the rogue waves are discussed. The parameter μ in the MTM model plays the role of the mass in the relativistic field theory while in optics it is related to the medium periodic constant, which also results in a significant rotation and a remarkable lengthening of the first-order rogue wave. These results provide new opportunities to observe rouge waves by using a combination of electromagnetically induced transparency and the Bragg scattering four-wave mixing because of large amplitudes.
Development of predictive equations for total body water using the ...
African Journals Online (AJOL)
Objectives: The study aimed to derive predictive equations for total body water determinations with bioelectrical impedance and anthropometric measurements in a population of asymptomatic human immunodeficiency virus (HIV) -positive Zulu women. Design: Cross-sectional data from within an ongoing prospective study ...
Directory of Open Access Journals (Sweden)
Pedro Beirão
2015-09-01
Full Text Available The energy that can be captured from the sea waves and converted into electricity should be seen as a contribution to decrease the excessive dependency and growing demand of fossil fuels. Devices suitable to harness this kind of renewable energy source and convert it into electricity—wave energy converters (WECs—are not yet commercially competitive. There are several types of WECs, with different designs and working principles. One possible classification is their distance to the shoreline and thus their depth. Near-shore devices are one of them since they are typically deployed at intermediate water depth (IWD. The selection of the WEC deployment site should be a balance between several parameters; water depth is one of them. Another way of classifying WECs is grouping them by their geometry, size and orientation. Considering a near-shore WEC belonging to the floating point category, this paper is focused on the numerical study about the differences arising in the power captured from the sea waves when the typical deep water (DW assumption is compared with the more realistic IWD consideration. Actually, the production of electricity will depend, among other issues, on the depth of the deployment site. The development of a dynamic model including specific equations for the usual DW assumption as well as for IWD is also described. Derived equations were used to build a time domain simulator (TDS. Numerical results were obtained by means of simulations performed using the TDS. The objective is to simulate the dynamic behavior of the WEC due to the action of sea waves and to characterize the wave power variations according with the depth of the deployment site.
Chin-Joe-Kong, M.J.S.; Mulder, W.A.; van Veldhuizen, M.
1999-01-01
The higher-order finite-element scheme with mass lumping for triangles and tetrahedra is an efficient method for solving the wave equation. A number of lower-order elements have already been found. Here the search for elements of higher order is continued. Elements are constructed in a systematic
Global existence of solutions for semilinear damped wave equation in 2-D exterior domain
Ikehata, Ryo
We consider a mixed problem of a damped wave equation utt-Δ u+ ut=| u| p in the two dimensional exterior domain case. Small global in time solutions can be constructed in the case when the power p on the nonlinear term | u| p satisfies p ∗=2Japon. 55 (2002) 33) plays an effective role.
Non-monotonic Travelling Wave Fronts in a System of Fractional Flow Equations from Porous Media
Zegeling, P.A.; Hönig, O.; Doster, F.; Hilfer, R.
2016-01-01
Motivated by observations of saturation overshoot, this article investigates generic classes of smooth travelling wave solutions of a system of two coupled nonlinear parabolic partial differential equations resulting from a flux function of high symmetry. All boundary resp. limit value problems of
On spurious reflections, nonuniform grids and finite difference discretizations of wave equations
J.E. Frank (Jason); S. Reich
2004-01-01
textabstractThis paper addresses nonphysical reflections encountered in the discretization of wave equations on nonuniform grids. Such nonphysical solutions are commonly attributed to spurious modes in the numerical dispersion relation. We provide an example of a discretization in which a
Decay estimates for fractional wave equations on H-type groups
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Manli Song
2016-10-01
Full Text Available Abstract The aim of this paper is to establish the decay estimate for the fractional wave equation semigroup on H-type groups given by e i t Δ α $e^{it\\Delta^{\\alpha}}$ , 0 < α < 1 $0<\\alpha<1$ . Combining the dispersive estimate and a standard duality argument, we also derive the corresponding Strichartz inequalities.
Magnetic virial identities and applications to blow-up for Schroedinger and wave equations
Energy Technology Data Exchange (ETDEWEB)
Garcia, Andoni, E-mail: andoni.garcia@ehu.es [Departamento de Matematicas, Universidad del Pais Vasco, Apartado 644, 48080 Bilbao (Spain)
2012-01-13
We prove blow-up results for the solution of the initial-value problem with negative energy of the focusing mass-critical and supercritical nonlinear Schroedinger and the focusing energy-subcritical nonlinear wave equations with electromagnetic potential. (paper)
Blow-Up of Solutions for a Class of Sixth Order Nonlinear Strongly Damped Wave Equation
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Huafei Di
2014-01-01
Full Text Available We consider the blow-up phenomenon of sixth order nonlinear strongly damped wave equation. By using the concavity method, we prove a finite time blow-up result under assumptions on the nonlinear term and the initial data.
van Oers, A.M.; Maas, L.R.M.; Bokhove, O.
2017-01-01
The linear equations governing internal gravity waves in a stratified ideal fluid possess a Hamiltonian structure. A discontinuous Galerkin finite element method has been developed in which this Hamiltonian structure is discretized, resulting in conservation of discrete analogs of phase space and
Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion.
Leszczynski, Henryk; Wrzosek, Monika
2017-02-01
We consider nonlinear stochastic wave equations driven by one-dimensional white noise with respect to time. The existence of solutions is proved by means of Picard iterations. Next we apply Newton's method. Moreover, a second-order convergence in a probabilistic sense is demonstrated.
Coarse-Graining Can Beat the Rotating Wave Approximation in Quantum Markovian Master Equations
DEFF Research Database (Denmark)
Majenz, Christian; Albash, Tameem; Breuer, Heinz-Peter
2013-01-01
We present a first-principles derivation of the Markovian semi-group master equation without invoking the rotating wave approximation (RWA). Instead we use a time coarse-graining approach which leaves us with a free timescale parameter, which we can optimize. Comparing this approach to the standard...
Traveling wave solutions of the nonlinear Schrödinger equation
Akbari-Moghanjoughi, M.
2017-10-01
In this paper, we investigate the traveling soliton and the periodic wave solutions of the nonlinear Schrödinger equation (NLSE) with generalized nonlinear functionality. We also explore the underlying close connection between the well-known KdV equation and the NLSE. It is remarked that both one-dimensional KdV and NLSE models share the same pseudoenergy spectrum. We also derive the traveling wave solutions for two cases of weakly nonlinear mathematical models, namely, the Helmholtz and the Duffing oscillators' potentials. It is found that these models only allow gray-type NLSE solitary propagations. It is also found that the pseudofrequency ratio for the Helmholtz potential between the nonlinear periodic carrier and the modulated sinusoidal waves is always in the range 0.5 ≤ Ω/ω ≤ 0.537285 regardless of the potential parameter values. The values of Ω/ω = {0.5, 0.537285} correspond to the cnoidal waves modulus of m = {0, 1} for soliton and sinusoidal limits and m = 0.5, respectively. Moreover, the current NLSE model is extended to fully NLSE (FNLSE) situation for Sagdeev oscillator pseudopotential which can be derived using a closed set of hydrodynamic fluid equations with a fully integrable Hamiltonian system. The generalized quasi-three-dimensional traveling wave solution is also derived. The current simple hydrodynamic plasma model may also be generalized to two dimensions and other complex situations including different charged species and cases with magnetic or gravitational field effects.
Anomalous wave structure in magnetized materials described by non-convex equations of state
Energy Technology Data Exchange (ETDEWEB)
Serna, Susana, E-mail: serna@mat.uab.es [Departament de Matematiques, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona (Spain); Marquina, Antonio, E-mail: marquina@uv.es [Departamento de Matematicas, Universidad de Valencia, 46100 Burjassot, Valencia (Spain)
2014-01-15
We analyze the anomalous wave structure appearing in flow dynamics under the influence of magnetic field in materials described by non-ideal equations of state. We consider the system of magnetohydrodynamics equations closed by a general equation of state (EOS) and propose a complete spectral decomposition of the fluxes that allows us to derive an expression of the nonlinearity factor as the mathematical tool to determine the nature of the wave phenomena. We prove that the possible formation of non-classical wave structure is determined by both the thermodynamic properties of the material and the magnetic field as well as its possible rotation. We demonstrate that phase transitions induced by material properties do not necessarily imply the loss of genuine nonlinearity of the wavefields as is the case in classical hydrodynamics. The analytical expression of the nonlinearity factor allows us to determine the specific amount of magnetic field necessary to prevent formation of complex structure induced by phase transition in the material. We illustrate our analytical approach by considering two non-convex EOS that exhibit phase transitions and anomalous behavior in the evolution. We present numerical experiments validating the analysis performed through a set of one-dimensional Riemann problems. In the examples we show how to determine the appropriate amount of magnetic field in the initial conditions of the Riemann problem to transform a thermodynamic composite wave into a simple nonlinear wave.
Evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations
Zhang, Linghai
2017-10-01
The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ut =uxx + α [ βH (u - θ) - u ] - w, wt = ε (u - γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ], under different conditions on the model constants. To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0 1; the existence and instability of two standing wave fronts if 2 (1 + αγ) θ = αβγ and 0 0 is a positive constant, if 0 mathematical neuroscience.
An energy absorbing far-field boundary condition for the elastic wave equation
Energy Technology Data Exchange (ETDEWEB)
Petersson, N A; Sjogreen, B
2008-07-15
The authors present an energy absorbing non-reflecting boundary condition of Clayton-Engquist type for the elastic wave equation together with a discretization which is stable for any ratio of compressional to shear wave speed. They prove stability for a second order accurate finite-difference discretization of the elastic wave equation in three space dimensions together with a discretization of the proposed non-reflecting boundary condition. The stability proof is based on a discrete energy estimate and is valid for heterogeneous materials. The proof includes all six boundaries of the computational domain where special discretizations are needed at the edges and corners. The stability proof holds also when a free surface boundary condition is imposed on some sides of the computational domain.
DEFF Research Database (Denmark)
Mariegaard, Jesper Sandvig
We consider a control problem for the wave equation: Given the initial state, find a specific boundary condition, called a control, that steers the system to a desired final state. The Hilbert uniqueness method (HUM) is a mathematical method for the solution of such control problems. It builds....... As an example, we employ a HUM solution to an inverse source problem for the wave equation: Given boundary measurements for a wave problem with a separable source, find the spatial part of the source term. The reconstruction formula depends on a set of HUM eigenfunction controls; we suggest a discretization...... and show its convergence. We compare results obtained by L-FEM controls and DG-FEM controls. The reconstruction formula is seen to be quite sensitive to control inaccuracies which indeed favors DG-FEM over L-FEM....
George, David L.
2008-03-01
We present a class of augmented approximate Riemann solvers for the shallow water equations in the presence of a variable bottom surface. These belong to the class of simple approximate solvers that use a set of propagating jump discontinuities, or waves, to approximate the true Riemann solution. Typically, a simple solver for a system of m conservation laws uses m such discontinuities. We present a four wave solver for use with the the shallow water equations—a system of two equations in one dimension. The solver is based on a decomposition of an augmented solution vector—the depth, momentum as well as momentum flux and bottom surface. By decomposing these four variables into four waves the solver is endowed with several desirable properties simultaneously. This solver is well-balanced: it maintains a large class of steady states by the use of a properly defined steady state wave—a stationary jump discontinuity in the Riemann solution that acts as a source term. The form of this wave is introduced and described in detail. The solver also maintains depth non-negativity and extends naturally to Riemann problems with an initial dry state. These are important properties for applications with steady states and inundation, such as tsunami and flood modeling. Implementing the solver with LeVeque's wave propagation algorithm [R.J. LeVeque, Wave propagation algorithms for multi-dimensional hyperbolic systems, J. Comput. Phys. 131 (1997) 327-335] is also described. Several numerical simulations are shown, including a test problem for tsunami modeling.
A goal-oriented reduced basis method for the wave equation in inverse analysis
Hoang, Khac Chi; Bordas, Stephane P A
2013-01-01
In this paper, we extend the reduced-basis methods developed earlier for wave equations to goal-oriented wave equations with affine parameter dependence. The essential new ingredient is the dual (or adjoint) problem and the use of its solution in a sampling procedure to pick up "goal-orientedly" parameter samples. First, we introduce the reduced-basis recipe --- Galerkin projection onto a space $Y_N$ spanned by the reduced basis functions which are constructed from the solutions of the governing partial differential equation at several selected points in parameter space. Second, we propose a new "goal-oriented" Proper Orthogonal Decomposition (POD)--Greedy sampling procedure to construct these associated basis functions. Third, based on the assumption of affine parameter dependence, we use the offline-online computational procedures developed earlier to split the computational procedure into offline and online stages. We verify the proposed computational procedure by applying it to a three-dimensional simulat...
A space-time spectral collocation algorithm for the variable order fractional wave equation.
Bhrawy, A H; Doha, E H; Alzaidy, J F; Abdelkawy, M A
2016-01-01
The variable order wave equation plays a major role in acoustics, electromagnetics, and fluid dynamics. In this paper, we consider the space-time variable order fractional wave equation with variable coefficients. We propose an effective numerical method for solving the aforementioned problem in a bounded domain. The shifted Jacobi polynomials are used as basis functions, and the variable-order fractional derivative is described in the Caputo sense. The proposed method is a combination of shifted Jacobi-Gauss-Lobatto collocation scheme for the spatial discretization and the shifted Jacobi-Gauss-Radau collocation scheme for temporal discretization. The aforementioned problem is then reduced to a problem consists of a system of easily solvable algebraic equations. Finally, numerical examples are presented to show the effectiveness of the proposed numerical method.
Application of perturbation theory to a P-wave eikonal equation in orthorhombic media
Stovas, Alexey
2016-10-12
The P-wave eikonal equation for orthorhombic (ORT) anisotropic media is a highly nonlinear partial differential equation requiring the solution of a sixth-order polynomial to obtain traveltimes, resulting in complex and time-consuming numerical solutions. To alleviate this complexity, we approximate the solution of this equation by applying a multiparametric perturbation approach. We also investigated the sensitivity of traveltime surfaces inORT mediawith respect to three anelliptic parameters. As a result, a simple and accurate P-wave traveltime approximation valid for ORT media was derived. Two different possible anelliptic parameterizations were compared. One of the parameterizations includes anelliptic parameters defined at zero offset: η1, η2, and ηxy. Another parameterization includes anelliptic parameters defined for all symmetry planes: η1, η2, and η3. The azimuthal behavior of sensitivity coefficients with different parameterizations was used to analyze the crosstalk between anelliptic parameters. © 2016 Society of Exploration Geophysicists.
An Inverse Source Problem for a One-dimensional Wave Equation: An Observer-Based Approach
Asiri, Sharefa M.
2013-05-25
Observers are well known in the theory of dynamical systems. They are used to estimate the states of a system from some measurements. However, recently observers have also been developed to estimate some unknowns for systems governed by Partial differential equations. Our aim is to design an observer to solve inverse source problem for a one dimensional wave equation. Firstly, the problem is discretized in both space and time and then an adaptive observer based on partial field measurements (i.e measurements taken form the solution of the wave equation) is applied to estimate both the states and the source. We see the effectiveness of this observer in both noise-free and noisy cases. In each case, numerical simulations are provided to illustrate the effectiveness of this approach. Finally, we compare the performance of the observer approach with Tikhonov regularization approach.
Dissipation-preserving spectral element method for damped seismic wave equations
Cai, Wenjun; Zhang, Huai; Wang, Yushun
2017-12-01
This article describes the extension of the conformal symplectic method to solve the damped acoustic wave equation and the elastic wave equations in the framework of the spectral element method. The conformal symplectic method is a variation of conventional symplectic methods to treat non-conservative time evolution problems, which has superior behaviors in long-time stability and dissipation preservation. To reveal the intrinsic dissipative properties of the model equations, we first reformulate the original systems in their equivalent conformal multi-symplectic structures and derive the corresponding conformal symplectic conservation laws. We thereafter separate each system into a conservative Hamiltonian system and a purely dissipative ordinary differential equation system. Based on the splitting methodology, we solve the two subsystems respectively. The dissipative one is cheaply solved by its analytic solution. While for the conservative system, we combine a fourth-order symplectic Nyström method in time and the spectral element method in space to cover the circumstances in realistic geological structures involving complex free-surface topography. The Strang composition method is adopted thereby to concatenate the corresponding two parts of solutions and generate the completed conformal symplectic method. A relative larger Courant number than that of the traditional Newmark scheme is found in the numerical experiments in conjunction with a spatial sampling of approximately 5 points per wavelength. A benchmark test for the damped acoustic wave equation validates the effectiveness of our proposed method in precisely capturing dissipation rate. The classical Lamb problem is used to demonstrate the ability of modeling Rayleigh wave in elastic wave propagation. More comprehensive numerical experiments are presented to investigate the long-time simulation, low dispersion and energy conservation properties of the conformal symplectic methods in both the attenuating
Acoustic wave and eikonal equations in a transformed metric space for various types of anisotropy.
Noack, Marcus M; Clark, Stuart
2017-03-01
Acoustic waves propagating in anisotropic media are important for various applications. Even though these wave phenomena do not generally occur in nature, they can be used to approximate wave motion in various physical settings. We propose a method to derive wave equations for anisotropic wave propagation by adjusting the dispersion relation according to a selected type of anisotropy and transforming it into another metric space. The proposed method allows for the derivation of acoustic wave and eikonal equations for various types of anisotropy, and generalizes anisotropy by interpreting it as a change of the metric instead of a change of velocity with direction. The presented method reduces the scope of acoustic anisotropy to a selection of a velocity or slowness surface and a tensor that describes the transformation into a new metric space. Experiments are shown for spatially dependent ellipsoidal anisotropy in homogeneous and inhomogeneous media and sandstone, which shows vertical transverse isotropy. The results demonstrate the stability and simplicity of the solution process for certain types of anisotropy and the equivalency of the solutions.
Directory of Open Access Journals (Sweden)
Md. Nur Alam
2014-03-01
Full Text Available The new approach of generalized (G′/G-expansion method is significant, powerful and straightforward mathematical tool for finding exact traveling wave solutions of nonlinear evolution equations (NLEEs arise in the field of engineering, applied mathematics and physics. Dispersive effects due to microstructure of materials combined with nonlinearities give rise to solitary waves. In this article, the new approach of generalized (G′/G-expansion method has been applied to construct general traveling wave solutions of the strain wave equation in microstructured solids. Abundant exact traveling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important role in engineering fields.
Ballast, A.
2004-01-01
To increase the capabilities of the computer calculations a computer code for fully nonlinear potential calculations with water waves and floating bodies had been developed earlier. It uses a boundary integral equation formulation, which is discretized to give a higher order panel method. In the
Shallow water equations: viscous solutions and inviscid limit
Chen, Gui-Qiang; Perepelitsa, Mikhail
2012-12-01
We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C 2 test-functions, are confined in a compact set in H -1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.
Amplitude equation for under water sand-ripples in one dimension
DEFF Research Database (Denmark)
Sand-ripples under oscillatory water flow form periodic patterns with wave lengths primarily controlled by the amplitude d of the water motion. We present an amplitude equation for sand-ripples in one spatial dimension which captures the formation of the ripples as well as secondary bifurcations...... observed when the amplitude $d$ is suddenly varied. The equation has the form h_t=- ε(h-mean(h))+((h_x)^2-1)h_(xx)- h_(xxxx)+ δ((h_x)^2)_(xx) which, due to the first term, is neither completely local (it has long-range coupling through the average height mean(h)) nor has local sand conservation. We argue...
A family of nonlinear Schrödinger equations admitting q-plane wave solutions
Nobre, F. D.; Plastino, A. R.
2017-08-01
Nonlinear Schrödinger equations with power-law nonlinearities have attracted considerable attention recently. Two previous proposals for these types of equations, corresponding respectively to the Gross-Pitaievsky equation and to the one associated with nonextensive statistical mechanics, are here unified into a single, parameterized family of nonlinear Schrödinger equations. Power-law nonlinear terms characterized by exponents depending on a real index q, typical of nonextensive statistical mechanics, are considered in such a way that the Gross-Pitaievsky equation is recovered in the limit q → 1. A classical field theory shows that, due to these nonlinearities, an extra field Φ (x → , t) (besides the usual one Ψ (x → , t)) must be introduced for consistency. The new field can be identified with Ψ* (x → , t) only when q → 1. For q ≠ 1 one has a pair of coupled nonlinear wave equations governing the joint evolution of the complex valued fields Ψ (x → , t) and Φ (x → , t). These equations reduce to the usual pair of complex-conjugate ones only in the q → 1 limit. Interestingly, the nonlinear equations obeyed by Ψ (x → , t) and Φ (x → , t) exhibit a common, soliton-like, traveling solution, which is expressible in terms of the q-exponential function that naturally emerges within nonextensive statistical mechanics.
The Cauchy Problem and Stability of Solitary-Wave Solutions for RLW-KP-Type Equations
Bona, Jerry L.; Liu, Yue; Tom, Michael M.
The Kadomtsev-Petviashvilli (KP) equation,( ut+ ux+ uux+ uxxx) x+ ɛuyy=0, (*) arises in various contexts where nonlinear dispersive waves propagate principally along the x-axis, but with weak dispersive effects being felt in the direction parallel to the y-axis perpendicular to the main direction of propagation. We propose and analyze here a class of evolution equations of the form ( ut+ ux+ upux+ Lut) x+ ɛuyy=0, (**) which provides an alternative to Eq.(*) in the same way the regularized long-wave equation is related to the classical Korteweg-de Vries (KdV) equation. The operator L is a pseudo-differential operator in the x-variable, p⩾1 is an integer and ɛ=±1. After discussing the underlying motivation for the class (**), a local well-posedness theory for the initial-value problem is developed. With assumptions on L and p that include conditions appertaining to models of interesting physical phenomenon, the solutions defined locally in time t are shown to be smoothly extendable to the entire time-axis. In the particularly interesting case where L=-∂ x2 and ɛ=-1, (*) possesses travelling-wave solutions u( x, y, t)=π c( x- ct, y) provided c>1 and 01 and for {4}/{3}(4 p)/(4+ p). The paper concludes with commentary on extensions of the present theory to more than two space dimensions.
A Structural Equation Modeling approach to water quality perceptions.
Levêque, Jonas G; Burns, Robert C
2017-07-15
Researches on water quality perceptions have used various techniques and models to explain relationships between specific variables. Surprisingly, Structural Equation Modeling (SEM) has received little attention in water quality perceptions studies, and reporting has been inconsistent among existing studies. One objective of this article is to provide readers with a methodological example for conducting and reporting SEM. Another objective is to build a model that explains the different relationships among the diverse factors highlighted by previous studies on water quality perceptions. Our study focuses on the factors influencing people's perceptions of water quality in the Appalachian region. As such, researchers have conducted a survey in a mid-sized city in northcentral West Virginia to assess residents' perceptions of water quality for drinking and recreational purposes. Specifically, we aimed to understand the relationships between perceived water quality, health risk perceptions, organoleptic perceptions, environmental concern, area satisfaction and perceptions of surface water quality. Our model provided a good fit that explained about 50% of the variance in health risk perceptions and 43% of the variance in organoleptic perceptions. Environmental concern, area satisfaction and perceived surface water quality are important factors in explaining these variances. Perceived water quality was dismissed in our analysis due to multicollinearity. Our study demonstrates that risk communication needs to be better addressed by local decision-makers and water managers. Copyright © 2017 Elsevier Ltd. All rights reserved.
On Dirac equations for linear magnetoacoustic waves propagating in an isothermal atmosphere
Alicki, R.; Musielak, E. Z.; Sikorski, J.; Makowiec, D.
1994-01-01
A new analytical approach to study linear magnetoacoustic waves propagating in an isothermal, stratified, and uniformly magnetized atmosphere is presented. The approach is based on Dirac equations, and the theory of Sturm-Liouville operators is used to investigate spectral properties of the obtained Dirac Hamiltonians. Two cases are considered: (1) the background magnetic field is vertical, and the waves are separated into purely magnetic (transverse) and purely acoustic (longitudinal) modes; and (2) the field is tilted with respect to the vertical direction and the magnetic and acoustic modes become coupled giving magnetoacoustic waves. For the first case, the Dirac Hamiltonian possesses either a discrete spectrum, which corresponds to standing magnetic waves, or a continuous spectrum, which can be clearly identified with freely propagating acoustic waves. For the second case, the quantum mechanical perturbation calculus is used to study coupling and energy exchange between the magnetic and acoustic components of magnetoacoustic waves. It is shown that this coupling may efficiently prevent trapping of magnetoacoustic waves instellar atmospheres.
Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain
Yang, Xiao-Jun; Machado, J. A. Tenreiro; Baleanu, Dumitru
The new Boussinesq-type model in a fractal domain is derived based on the formulation of the local fractional derivative. The novel traveling wave transform of the non-differentiable type is adopted to convert the local fractional Boussinesq equation into a nonlinear local fractional ODE. The exact traveling wave solution is also obtained with aid of the non-differentiable graph. The proposed method, involving the fractal special functions, is efficient for finding the exact solutions of the nonlinear PDEs in fractal domains.
Laser driven shock wave experiments for equation of state studies at megabar pressures
Pant, H C; Senecha, V K; Bandyopadhyay, S; Rai, V N; Khare, P; Bhat, R K; Gupta, N K; Godwal, B K
2002-01-01
We present the results from laser driven shock wave experiments for equation of state (EOS) studies of gold metal. An Nd:YAG laser chain (2 J, 1.06 mu m wavelength, 200 ps pulse FWHM) is used to generate shocks in planar Al foils and Al + Au layered targets. The EOS of gold in the pressure range of 9-13 Mbar is obtained using the impedance matching technique. The numerical simulations performed using the one-dimensional radiation hydrodynamic code support the experimental results. The present experimental data show remarkable agreement with the existing standard EOS models and with other experimental data obtained independently using laser driven shock wave experiments.
Gilbert, Kenneth E
2015-01-01
The original formulation of the Green's function parabolic equation (GFPE) can have numerical accuracy problems for large normalized surface impedances. To solve the accuracy problem, an improved form of the GFPE has been developed. The improved GFPE formulation is similar to the original formulation, but it has the surface-wave pole "subtracted." The improved GFPE is shown to be accurate for surface impedances varying over 2 orders of magnitude, with the largest having a magnitude exceeding 1000. Also, the improved formulation is slightly faster than the original formulation because the surface-wave component does not have to be computed separately.
ON FINITE DIFFERENCE SCHEMES FOR THE 3-D WAVE EQUATION USING NON-CARTESIAN GRIDS
B. Hamilton; S. Bilbao
2013-01-01
In this paper, we investigate ﬁnite difference schemes forthe 3-D wave equation using 27-point stencils on the cubiclattice, a 13-point stencil on the face-centered cubic (FCC)lattice, and a 9-point stencil on the body-centered cubic(BCC) lattice. The tiling of the wavenumber space for nonCartesian grids is considered in order to analyse numericaldispersion. Schemes are compared for computational efﬁ-ciency in terms of minimising numerical wave speed error.It is shown that the 13-point scheme...
van Oers, Alexander M.; Maas, Leo R. M.; Bokhove, Onno
2017-02-01
The linear equations governing internal gravity waves in a stratified ideal fluid possess a Hamiltonian structure. A discontinuous Galerkin finite element method has been developed in which this Hamiltonian structure is discretized, resulting in conservation of discrete analogs of phase space and energy. This required (i) the discretization of the Hamiltonian structure using alternating flux functions and symplectic time integration, (ii) the discretization of a divergence-free velocity field using Dirac's theory of constraints and (iii) the handling of large-scale computational demands due to the 3-dimensional nature of internal gravity waves and, in confined, symmetry-breaking fluid domains, possibly its narrow zones of attraction.
Self-similar shock wave solutions of the non-linear Maxwell equations
Barna, I F
2013-01-01
In our study we consider nonlinear, power-law field-dependent electrical permitivity and magnetic permeability and investigate the time-dependent Maxwell equations with the self-similar Ansatz. This is a first-order hyperbolic PDE system which can conserve non-continuous initial conditions describing electromagnetic shock-waves. Besides shock-waves other interesting solutions (e.g. with localized compact support) can be found with delicate physical properties. Such phenomena may happen in complex materials induced by the planned powerful Extreme Light Infrastructure(ELI) laser pulses.
An Adaptive Observer-Based Algorithm for Solving Inverse Source Problem for the Wave Equation
Asiri, Sharefa M.
2015-08-31
Observers are well known in control theory. Originally designed to estimate the hidden states of dynamical systems given some measurements, the observers scope has been recently extended to the estimation of some unknowns, for systems governed by partial differential equations. In this paper, observers are used to solve inverse source problem for a one-dimensional wave equation. An adaptive observer is designed to estimate the state and source components for a fully discretized system. The effectiveness of the algorithm is emphasized in noise-free and noisy cases and an insight on the impact of measurements’ size and location is provided.
Exact controllability for a wave equation with mixed boundary conditions in a non-cylindrical domain
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Lizhi Cui
2014-04-01
Full Text Available In this article we study the exact controllability of a one-dimensional wave equation with mixed boundary conditions in a non-cylindrical domain. The fixed endpoint has a Dirichlet-type boundary condition, while the moving end has a Neumann-type condition. When the speed of the moving endpoint is less than the characteristic speed, the exact controllability of this equation is established by Hilbert Uniqueness Method. Moreover, we shall give the explicit dependence of the controllability time on the speed of the moving endpoint.
A study of wave forces on an offshore platform by direct CFD and Morison equation
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Zhang D.
2015-01-01
The next step is the presentation of 3D multiphase RANS simulation of the wind-turbine platform in single-harmonic regular waves. Simulation results from full 3D simulation will be compared to the results from Morison’s equation. We are motivated by the challenges of a floating platform which has complex underwater geometry (e.g. tethered semi-submersible. In cases like this, our hypothesis is that Morison’s equation will result in inaccurate prediction of forces, due to the limitations of 2D coefficients of simple geometries, and that 3D multiphase RANS CFD will be required to generate reliable predictions of platform loads and motions.
Viscoacoustic wave-equation traveltime inversion with correct and incorrect attenuation profiles
Yu, Han
2017-08-17
A visco-acoustic wave-equation traveltime inversion method is presented that inverts for a shallow subsurface velocity distribution with correct and incorrect attenuation profiles. Similar to the classical wave equation traveltime inversion, this method applies the misfit functional that minimizes the first break differences between the observed and predicted data. Although, WT can partly avoid the cycle skipping problem, an initial velocity model approaches to the right or wrong velocity models under different setups of the attenuation profiles. However, with a Q model far away from the real model, the inverted tomogram is obviously different from the true velocity model while a small change of the Q model does not improve the inversion quality in a strong manner if low frequency information is not lost.
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Ibrahim K. Abu Seif
2015-01-01
Full Text Available In this paper a fast numerical algorithm to solve an integral equation model for wave propagation along a perfectly conducting two-dimensional terrain is suggested. It is applied to different actual terrain profiles and the results indicate very good agreement with published work. In addition, the proposed algorithm has achieved considerable saving in processing time. The formulation is extended to solve the propagation over lossy dielectric surfaces. A combined field integral equation (CFIE for wave propagation over dielectric terrain is solved efficiently by utilizing the method of moments with complex basis functions. The numerical results for different cases of dielectric surfaces are compared with the results of perfectly conducting surface evaluated by the IE conventional algorithm.
Manipulating Effective Gravity and Trapping Shallow Water Waves
Zareei, Ahmad; Alam, Mohammad-Reza
2017-11-01
A perfect manipulation of water waves in shallow water using transformation media methods usually requires changes in both water depth and gravitational acceleration as medium properties; however gravitational acceleration is always a physical constant. Reduced models and conformal transformations are used to keep the gravitational acceleration as a constant at the cost of performance and restriction of use. Here we present a novel method of changing effective gravitational acceleration using a visco-elastic bottom topography. This method of manipulating effective gravitational acceleration, beside changes in bottom topography, opens new applications toward controlling surface waves and enables perfect manipulation of water waves in a broad range of frequencies. Using the visco-elastic bottom topography, we present a GRIN-lens based wave-guide that traps water waves in a region along the axis of the lens. The presented method of manipulating effective gravitational acceleration can as well be applied to perfectly focus and rotate the waves for energy harvesting applications.
Numerical Simulations for the Space-Time Variable Order Nonlinear Fractional Wave Equation
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Nasser Hassan Sweilam
2013-01-01
Full Text Available The explicit finite-difference method for solving variable order fractional space-time wave equation with a nonlinear source term is considered. The concept of variable order fractional derivative is considered in the sense of Caputo. The stability analysis and the truncation error of the method are discussed. To demonstrate the effectiveness of the method, some numerical test examples are presented.
The use of adomian decomposition method for solving the regularized long-wave equation
Energy Technology Data Exchange (ETDEWEB)
El-Danaf, Talaat S. [Department of Mathematics, Faculty of Science, Menoufia University, Shiben El-Kom (Egypt); Ramadan, Mohamed A. [Department of Mathematics, Faculty of Science, Menoufia University, Shiben El-Kom (Egypt)] e-mail: mramadan@mailer.eun.eg; Abd Alaal, Faysal E.I. [Department of Mathematics, Faculty of Science, Menoufia University, Shiben El-Kom (Egypt)
2005-11-01
In this paper, an accurate method to obtain an approximate numerical solution for the nonlinear regularized long-wave (in short RLW) equation is considered. The theoretical analysis of the method is investigated. The performance and the accuracy of the algorithm are illustrated by solving two test examples of the problem. The obtained results are presented and compared with the analytical solutions. It is observed that only few terms of the series expansion are required to obtain approximate solutions with good accuracy.
Povstenko, Y.
2013-09-01
The axisymmetric time-fractional diffusion-wave equation with the Caputo derivative of the order 0 function and the values of its normal derivative at the boundary. The fundamental solutions to the Cauchy, source, and boundary problems are investigated. The Laplace transform with respect to time and finite Hankel transform with respect to the radial coordinate are used. The solutions are obtained in terms of Mittag-Leffler functions. The numerical results are illustrated graphically.
Solutions to Time-Fractional Diffusion-Wave Equation in Cylindrical Coordinates
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Povstenko YZ
2011-01-01
Full Text Available Nonaxisymmetric solutions to time-fractional diffusion-wave equation with a source term in cylindrical coordinates are obtained for an infinite medium. The solutions are found using the Laplace transform with respect to time , the Hankel transform with respect to the radial coordinate , the finite Fourier transform with respect to the angular coordinate , and the exponential Fourier transform with respect to the spatial coordinate . Numerical results are illustrated graphically.
Gerbi, Stéphane
2011-12-01
In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the KelvinVoigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained. © 2011 Elsevier Ltd. All rights reserved.
Variational Approach to the Orbital Stability of Standing Waves of the Gross-Pitaevskii Equation
Hadj Selem, Fouad
2014-08-26
This paper is concerned with the mathematical analysis of a masssubcritical nonlinear Schrödinger equation arising from fiber optic applications. We show the existence and symmetry of minimizers of the associated constrained variational problem. We also prove the orbital stability of such solutions referred to as standing waves and characterize the associated orbit. In the last section, we illustrate our results with few numerical simulations. © 2014 Springer Basel.
A functional integral approach to shock wave solutions of Euler equations with spherical symmetry
Yang, Tong
1995-08-01
For n×n systems of conservation laws in one dimension without source terms, the existence of global weak solutions was proved by Glimm [1]. Glimm constructed approximate solutions using a difference scheme by solving a class of Riemann problems. In this paper, we consider the Cauchy problem for the Euler equations in the spherically symmetric case when the initial data are small perturbations of the trivial solution, i.e., u≡0 and ρ≡ constant, where u is velocity and ρ is density. We show that this Cauchy problem can be reduced to an ideal nonlinear problem approximately. If we assume all the waves move at constant speeds in the ideal problem, by using Glimm's scheme and an integral approach to sum the contributions of the reflected waves that correspond to each path through the solution, we get uniform bounds on the L ∞ norm and total variational norm of the solutions for all time. The geometric effects of spherical symmetry leads to a non-integrable source term in the Euler equations. Correspondingly, we consider an infinite reflection problem and solve it by considering the cancellations between reflections of different orders in our ideal problem. Thus we view this as an analysis of the interaction effects at the quadratic level in a nonlinear model problem for the Euler equations. Although it is far more difficult to obtain estimates in the exact solutions of the Euler equations due to the problem of controlling the time at which the cancellations occur, we believe that this analysis of the wave behaviour will be the first step in solving the problem of existence of global weak solutions for the spherically symmetric Euler equations outside of fixed ball.
Hydromagnetic Waves in a Compressed Dipole Field via Field-Aligned Klein-Gordon Equations
Zheng, Jinlei; McKenzie, J F; Webb, G M
2014-01-01
Hydromagnetic waves, especially those of frequencies in the range of a few milli-Hz to a few Hz observed in the Earth's magnetosphere, are categorized as Ultra Low Frequency (ULF) waves or pulsations. They have been extensively studied due to their importance in the interaction with radiation belt particles and in probing the structures of the magnetosphere. We developed an approach in examining the toroidal standing Aflv\\'{e}n waves in a background magnetic field by recasting the wave equation into a Klein-Gordon (KG) form along individual field lines. The eigenvalue solutions to the system are characteristic of a propagation type when the corresponding eigen-frequency is greater than a cut-off frequency and an evanescent type otherwise. We apply the approach to a compressed dipole magnetic field model of the inner magnetosphere, and obtain the spatial profiles of relevant parameters and the spatial wave forms of harmonic oscillations. We further extend the approach to poloidal mode standing Alfv\\'{e}n waves...
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N. N. Romanova
1998-01-01
Full Text Available The dynamics of weakly nonlinear wave trains in unstable media is studied. This dynamics is investigated in the framework of a broad class of dynamical systems having a Hamiltonian structure. Two different types of instability are considered. The first one is the instability in a weakly supercritical media. The simplest example of instability of this type is the Kelvin-Helmholtz instability. The second one is the instability due to a weak linear coupling of modes of different nature. The simplest example of a geophysical system where the instability of this and only of this type takes place is the three-layer model of a stratified shear flow with a continuous velocity profile. For both types of instability we obtain nonlinear evolution equations describing the dynamics of wave trains having an unstable spectral interval of wavenumbers. The transformation to appropriate canonical variables turns out to be different for each case, and equations we obtained are different for the two types of instability we considered. Also obtained are evolution equations governing the dynamics of wave trains in weakly subcritical media and in media where modes are coupled in a stable way. Presented results do not depend on a specific physical nature of a medium and refer to a broad class of dynamical systems having the Hamiltonian structure of a special form.
Seadawy, Aly R.
2017-09-01
Nonlinear two-dimensional Kadomtsev-Petviashvili (KP) equation governs the behaviour of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions. By using the reductive perturbation method, the two-dimensional dust-acoustic solitary waves (DASWs) in unmagnetized cold plasma consisting of dust fluid, ions and electrons lead to a KP equation. We derived the solitary travelling wave solutions of the two-dimensional nonlinear KP equation by implementing sech-tanh, sinh-cosh, extended direct algebraic and fraction direct algebraic methods. We found the electrostatic field potential and electric field in the form travelling wave solutions for two-dimensional nonlinear KP equation. The solutions for the KP equation obtained by using these methods can be demonstrated precisely and efficiency. As an illustration, we used the readymade package of Mathematica program 10.1 to solve the original problem. These solutions are in good agreement with the analytical one.
Energy Technology Data Exchange (ETDEWEB)
Tchinang Tchameu, J.D., E-mail: jtchinang@gmail.com; Togueu Motcheyo, A.B., E-mail: abtogueu@yahoo.fr; Tchawoua, C., E-mail: ctchawa@yahoo.fr
2016-09-07
The discrete multi-rogue waves (DMRW) as solution of the discrete nonlinear Schrödinger (DNLS) equation with saturable nonlinearities is studied numerically. These biological rogue waves represent the complex probability amplitude of finding an amide-I vibrational quantum at a site. We observe that the growth in the higher order saturable nonlinearity implies the formation of DMRW including an increase in the short-living DMRW and a decrease in amplitude of the long-living DMRW. - Highlights: • Discrete Multi-Rogue Waves (DMRW), representing the localization of vibrational energy in protein chain, are found numerically. • The higher order saturable nonlinearity of DNLS promotes the increase in the short-living RW. • The higher order saturable nonlinearity of DNLS promotes the decrease in amplitude of the long-living RW.
Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations
Luk, Jonathan
2013-01-01
In this paper, we study the problem of the nonlinear interaction of impulsive gravitational waves for the Einstein vacuum equations. The problem is studied in the context of a characteristic initial value problem with data given on two null hypersurfaces and containing curvature delta singularities. We establish an existence and uniqueness result for the spacetime arising from such data and show that the resulting spacetime represents the interaction of two impulsive gravitational waves germinating from the initial singularities. In the spacetime, the curvature delta singularities propagate along 3-dimensional null hypersurfaces intersecting to the future of the data. To the past of the intersection, the spacetime can be thought of as containing two independent, non-interacting impulsive gravitational waves and the intersection represents the first instance of their nonlinear interaction. Our analysis extends to the region past their first interaction and shows that the spacetime still remains smooth away fro...
Solution of the nonrelativistic wave equation using the tridiagonal representation approach
Alhaidari, A. D.
2017-07-01
We choose a complete set of square integrable functions as a basis for the expansion of the wavefunction in configuration space such that the matrix representation of the nonrelativistic time-independent linear wave operator is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. The recursion relation is then solved exactly in terms of orthogonal polynomials in the energy. Some of these polynomials are not found in the mathematics literature. The asymptotics of these polynomials give the phase shift for the continuous energy scattering states and the spectrum for the discrete energy bound states. Depending on the space and boundary conditions, the basis functions are written in terms of either the Laguerre or Jacobi polynomials. The tridiagonal requirement limits the number of potential functions that yield exact solutions of the wave equation. Nonetheless, the class of exactly solvable problems in this approach is larger than the conventional class (see, for example, Table XII in the text). We also give very accurate results for cases where the wave operator matrix is not tridiagonal but its elements could be evaluated either exactly or numerically with high precision.
Angle-domain Migration Velocity Analysis using Wave-equation Reflection Traveltime Inversion
Zhang, Sanzong
2012-11-04
The main difficulty with an iterative waveform inversion is that it tends to get stuck in a local minima associated with the waveform misfit function. This is because the waveform misfit function is highly non-linear with respect to changes in the velocity model. To reduce this nonlinearity, we present a reflection traveltime tomography method based on the wave equation which enjoys a more quasi-linear relationship between the model and the data. A local crosscorrelation of the windowed downgoing direct wave and the upgoing reflection wave at the image point yields the lag time that maximizes the correlation. This lag time represents the reflection traveltime residual that is back-projected into the earth model to update the velocity in the same way as wave-equation transmission traveltime inversion. The residual movemout analysis in the angle-domain common image gathers provides a robust estimate of the depth residual which is converted to the reflection traveltime residual for the velocity inversion. We present numerical examples to demonstrate its efficiency in inverting seismic data for complex velocity model.
Shot- and angle-domain wave-equation traveltime inversion of reflection data: Theory
Zhang, Sanzong
2015-05-26
The main difficulty with iterative waveform inversion is that it tends to get stuck in local minima associated with the waveform misfit function. To mitigate this problem and avoid the need to fit amplitudes in the data, we have developed a wave-equation method that inverts the traveltimes of reflection events, and so it is less prone to the local minima problem. Instead of a waveform misfit function, the penalty function was a crosscorrelation of the downgoing direct wave and the upgoing reflection wave at the trial image point. The time lag, which maximized the crosscorrelation amplitude, represented the reflection-traveltime residual (RTR) that was back projected along the reflection wavepath to update the velocity. Shot- and angle-domain crosscorrelation functions were introduced to estimate the RTR by semblance analysis and scanning. In theory, only the traveltime information was inverted and there was no need to precisely fit the amplitudes or assume a high-frequency approximation. Results with synthetic data and field records revealed the benefits and limitations of wave-equation reflection traveltime inversion.
WAVE CHARACTERISTICS ANALYSIS OF PERAK WATERS SURABAYA
Zainul Hidayah
2009-01-01
This research was intended to understand the wave’s characteristics of the study area. Wave’s parameters that were observed in this research including wave period (T), wave length (L), wave height (H), wave velocity (C) and wave energy (E). Another objective of this study was also to produce a topographic map of the sea floor for the study area. Wave’s data of this study was gain from an electronic sensor called MAWS (Marine Automatic Wave Sensor). This sensor is located in a Naval Base of Su...
On analytic solutions of wave equations in regular coordinate systems on Schwarzschild background
Philipp, Dennis
2015-01-01
The propagation of (massless) scalar, electromagnetic and gravitational waves on fixed Schwarzschild background spacetime is described by the general time-dependent Regge-Wheeler equation. We transform this wave equation to usual Schwarzschild, Eddington-Finkelstein, Painleve-Gullstrand and Kruskal-Szekeres coordinates. In the first three cases, but not in the last one, it is possible to separate a harmonic time-dependence. Then the resulting radial equations belong to the class of confluent Heun equations, i.e., we can identify one irregular and two regular singularities. Using the generalized Riemann scheme we collect properties of all the singular points and construct analytic (local) solutions in terms of the standard confluent Heun function HeunC, Frobenius and asymptotic Thome series. We study the Eddington-Finkelstein case in detail and obtain a solution that is regular at the black hole horizon. This solution satisfies causal boundary conditions, i.e., it describes purely ingoing radiation at $r=2M$. ...
Boundary-value problems for wave equations with data on the whole boundary
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Makhmud A. Sadybekov
2016-10-01
Full Text Available In this article we propose a new formulation of boundary-value problem for a one-dimensional wave equation in a rectangular domain in which boundary conditions are given on the whole boundary. We prove the well-posedness of boundary-value problem in the classical and generalized senses. To substantiate the well-posedness of this problem it is necessary to have an effective representation of the general solution of the problem. In this direction we obtain a convenient representation of the general solution for the wave equation in a rectangular domain based on d'Alembert classical formula. The constructed general solution automatically satisfies the boundary conditions by a spatial variable. Further, by setting different boundary conditions according to temporary variable, we get some functional or functional-differential equations. Thus, the proof of the well-posedness of the formulated problem is reduced to question of the existence and uniqueness of solutions of the corresponding functional equations.
Wave equations and computational models for sonic boom propagation through a turbulent atmosphere
Pierce, Allan D.
1992-01-01
The improved simulation of sonic boom propagation through the real atmosphere requires greater understanding of how the transient acoustic pulses popularly termed sonic booms are affected by atmospheric turbulence. A nonlinear partial differential equation that can be used to simulate the effects of smaller-scale atmospheric turbulence on sonic boom waveforms is described. The equation is first order in the time derivative and involves an extension of geometrical acoustics to include diffraction phenomena. Various terms in the equation are explained in physical terms. Such terms include those representing convection at the wave speed, diffraction, molecular relaxation, classical dissipation, and nonlinear steepening. The atmospheric turbulence enters through an effective sound speed, which varies with all three spatial coordinates, and which is the sum of the local sound speed and the component of the turbulent flow velocity projected along a central ray that connects the aircraft trajectory with the listener.
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Wei Li
2014-01-01
Full Text Available Based on Jumarie’s modified Riemann-Liouville derivative, the fractional complex transformation is used to transform fractional differential equations to ordinary differential equations. Exact solutions including the hyperbolic functions, the trigonometric functions, and the rational functions for the space-time fractional bidirectional wave equations are obtained using the (G′/G-expansion method. The method provides a promising tool for solving nonlinear fractional differential equations.
Oil slicks on water surface: Breakup, coalescence, and droplet formation under breaking waves.
Nissanka, Indrajith D; Yapa, Poojitha D
2017-01-15
The ability to calculate the oil droplet size distribution (DSD) and its dynamic behavior in the water column is important in oil spill modeling. Breaking waves disperse oil from a surface slick into the water column as droplets of varying sizes. Oil droplets undergo further breakup and coalescence in the water column due to the turbulence. Available models simulate oil DSD based on empirical/equilibrium equations. However, the oil DSD evolution due to subsequent droplet breakup and coalescence in the water column can be best represented by a dynamic population model. This paper develops a phenomenological model to calculate the oil DSD in wave breaking conditions and ocean turbulence and is based on droplet breakup and coalescence. Its results are compared with data from laboratory experiments that include different oil types, different weathering times, and different breaking wave heights. The model comparisons showed a good agreement with experimental data. Copyright © 2016 Elsevier Ltd. All rights reserved.
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Hongwei Yang
2014-01-01
Full Text Available In the paper, by using multiple-scale method, the Benjamin-Ono-Burgers-MKdV (BO-B-MKdV equation is obtained which governs algebraic Rossby solitary waves in stratified fluids. This equation is first derived for Rossby waves. By analysis and calculation, some conservation laws are derived from the BO-B-MKdV equation without dissipation. The results show that the mass, momentum, energy, and velocity of the center of gravity of algebraic Rossby waves are conserved and the presence of a small dissipation destroys these conservations.
7 CFR 610.12 - Equations for predicting soil loss due to water erosion.
2010-01-01
... 7 Agriculture 6 2010-01-01 2010-01-01 false Equations for predicting soil loss due to water... ASSISTANCE Soil Erosion Prediction Equations § 610.12 Equations for predicting soil loss due to water erosion. (a) The equation for predicting soil loss due to erosion for both the USLE and the RUSLE is A = R × K...
Wazwaz, Abdul-Majid
2012-06-01
In this work, we explore a variety of solitary wave ansatze and periodic wave ansatze to some nonlinear equations. Three complex systems of nonlinear equations that appear in mathematical physics are investigated. We derive abundant soliton and periodic wave solutions for the coupled Higgs field equation, the Maccari system and the Hirota-Maccari system. The results obtained show that these three coupled equations exhibit the richness of explicit solutions: solitons, periodic and rational wave solutions.
Wave-induced extreme water levels in the Puerto Morelos fringing reef lagoon
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A. Torres-Freyermuth
2012-12-01
Full Text Available Wave-induced extreme water levels in the Puerto Morelos fringing reef lagoon are investigated by means of a phase-resolving non-hydrostatic wave model (SWASH. This model solves the nonlinear shallow water equations including non-hydrostatic pressure. The one-dimensional version of the model is implemented in order to investigate wave transformation in fringing reefs. Firstly, the numerical model is validated with (i laboratory experiments conducted on a physical model (Demirbilek et al., 2007and (ii field observations (Coronado et al., 2007. Numerical results show good agreement with both experimental and field data. The comparison against the physical model results, for energetic wave conditions, indicates that high- and low-frequency wave transformation is well reproduced. Moreover, extreme water-level conditions measured during the passage of Hurricane Ivan in Puerto Morelos are also estimated by the numerical tool. Subsequently, the model is implemented at different along-reef locations in Puerto Morelos. Extreme water levels, wave-induced setup, and infragravity wave energy are estimated inside the reef lagoon for different storm wave conditions (H_{s} >2 m. The numerical results revealed a strong correlation between the offshore sea-swell wave energy and the setup. In contrast, infragravity waves are shown to be the result of a more complex pattern which heavily relies on the reef geometry. Indeed, the southern end of the reef lagoon provides evidence of resonance excitation, suggesting that the reef barrier may act as either a natural flood protection morphological feature, or as an inundation hazard enhancer depending on the incident wave conditions.
Lecture Notes for the Course in Water Wave Mechanics
DEFF Research Database (Denmark)
Andersen, Thomas Lykke; Frigaard, Peter
The present notes are written for the course in water wave mechanics given on the 7th semester of the education in civil engineering at Aalborg University.......The present notes are written for the course in water wave mechanics given on the 7th semester of the education in civil engineering at Aalborg University....
Hydrodynamic coefficients for water-wave diffraction by spherical ...
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
The work presented here is the result of water-wave interaction with submerged spheres. Analytical expressions for various hydrodynamic coefficients and loads due to the diffraction of water waves by a submerged sphere are obtained. The exciting force components due to surge and heave motions are derived by solving ...
Analysis and Computation of Acoustic and Elastic Wave Equations in Random Media
Motamed, Mohammad
2014-01-06
We propose stochastic collocation methods for solving the second order acoustic and elastic wave equations in heterogeneous random media and subject to deterministic boundary and initial conditions [1, 4]. We assume that the medium consists of non-overlapping sub-domains with smooth interfaces. In each sub-domain, the materials coefficients are smooth and given or approximated by a finite number of random variable. One important example is wave propagation in multi-layered media with smooth interfaces. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems [2, 3], the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence is only algebraic. A fast spectral rate of convergence is still possible for some quantities of interest and for the wave solutions with particular types of data. We also show that the semi-discrete solution is analytic with respect to the random variables with the radius of analyticity proportional to the grid/mesh size h. We therefore obtain an exponential rate of convergence which deteriorates as the quantity h p gets smaller, with p representing the polynomial degree in the stochastic space. We have shown that analytical results and numerical examples are consistent and that the stochastic collocation method may be a valid alternative to the more traditional Monte Carlo method. Here we focus on the stochastic acoustic wave equation. Similar results are obtained for stochastic elastic equations.
Traveltime sensitivity kernels for wave equation tomography using the unwrapped phase
Djebbi, Ramzi
2014-02-18
Wave equation tomography attempts to improve on traveltime tomography, by better adhering to the requirements of our finite-frequency data. Conventional wave equation tomography, based on the first-order Born approximation followed by cross-correlation traveltime lag measurement, or on the Rytov approximation for the phase, yields the popular hollow banana sensitivity kernel indicating that the measured traveltime at a point is insensitive to perturbations along the ray theoretical path at certain finite frequencies. Using the instantaneous traveltime, which is able to unwrap the phase of the signal, instead of the cross-correlation lag, we derive new finite-frequency traveltime sensitivity kernels. The kernel reflects more the model-data dependency, we typically encounter in full waveform inversion. This result confirms that the hollow banana shape is borne of the cross-correlation lag measurement, which exposes the Born approximations weakness in representing transmitted waves. The instantaneous traveltime can thus mitigate the additional component of nonlinearity introduced by the hollow banana sensitivity kernels in finite-frequency traveltime tomography. The instantaneous traveltime simply represents the unwrapped phase of Rytov approximation, and thus is a good alternative to Born and Rytov to compute the misfit function for wave equation tomography. We show the limitations of the cross-correlation associated with Born approximation for traveltime lag measurement when the source signatures of the measured and modelled data are different. The instantaneous traveltime is proven to be less sensitive to the distortions in the data signature. The unwrapped phase full banana shape of the sensitivity kernels shows smoother update compared to the banana–doughnut kernels. The measurement of the traveltime delay caused by a small spherical anomaly, embedded into a 3-D homogeneous model, supports the full banana sensitivity assertion for the unwrapped phase.
Energy Technology Data Exchange (ETDEWEB)
Huang, Lianjie [Los Alamos National Laboratory; Simonetti, Francesco [IMPERIAL COLLEGE LONDON; Huthwaite, Peter [IMPERIAL COLLEGE LONDON; Rosenberg, Robert [UNM; Williamson, Michael [UNM
2010-01-01
Ultrasound image resolution and quality need to be significantly improved for breast microcalcification detection. Super-resolution imaging with the factorization method has recently been developed as a promising tool to break through the resolution limit of conventional imaging. In addition, wave-equation reflection imaging has become an effective method to reduce image speckles by properly handling ultrasound scattering/diffraction from breast heterogeneities during image reconstruction. We explore the capabilities of a novel super-resolution ultrasound imaging method and a wave-equation reflection imaging scheme for detecting breast microcalcifications. Super-resolution imaging uses the singular value decomposition and a factorization scheme to achieve an image resolution that is not possible for conventional ultrasound imaging. Wave-equation reflection imaging employs a solution to the acoustic-wave equation in heterogeneous media to backpropagate ultrasound scattering/diffraction waves to scatters and form images of heterogeneities. We construct numerical breast phantoms using in vivo breast images, and use a finite-difference wave-equation scheme to generate ultrasound data scattered from inclusions that mimic microcalcifications. We demonstrate that microcalcifications can be detected at full spatial resolution using the super-resolution ultrasound imaging and wave-equation reflection imaging methods.
Localised Nonlinear Waves in the Three-Component Coupled Hirota Equations
Xu, Tao; Chen, Yong
2017-10-01
We construct the Lax pair and Darboux transformation for the three-component coupled Hirota equations including higher-order effects such as third-order dispersion, self-steepening, and stimulated Raman scattering. A special vector solution of the Lax pair with 4×4 matrices for the three-component Hirota system is elaborately generated, based on this vector solution, various types of mixed higher-order localised waves are derived through the generalised Darboux transformation. Instead of considering various arrangements of the three potential functions q1, q2, and q3, here, the same combination is considered as the same type solution. The first- and second-order localised waves are mainly discussed in six mixed types: (1) the hybrid solutions degenerate to the rational ones and three components are all rogue waves; (2) two components are hybrid solutions between rogue wave (RW) and breather (RW+breather), and one component is interactional solution between RW and dark soliton (RW+dark soliton); (3) two components are RW+dark soliton, and one component is RW+bright soliton; (4) two components are RW+breather, and one component is RW+bright soliton; (5) two components are RW+dark soliton, and one component is RW+bright soliton; (6) three components are all RW+breather. Moreover, these nonlinear localised waves merge with each other by increasing the absolute values of two free parameters α, β. These results further uncover some striking dynamic structures in the multicomponent coupled system.
Born reflection kernel analysis and wave-equation reflection traveltime inversion in elastic media
Wang, Tengfei
2017-08-17
Elastic reflection waveform inversion (ERWI) utilize the reflections to update the low and intermediate wavenumbers in the deeper part of model. However, ERWI suffers from the cycle-skipping problem due to the objective function of waveform residual. Since traveltime information relates to the background model more linearly, we use the traveltime residuals as objective function to update background velocity model using wave equation reflected traveltime inversion (WERTI). The reflection kernel analysis shows that mode decomposition can suppress the artifacts in gradient calculation. We design a two-step inversion strategy, in which PP reflections are firstly used to invert P wave velocity (Vp), followed by S wave velocity (Vs) inversion with PS reflections. P/S separation of multi-component seismograms and spatial wave mode decomposition can reduce the nonlinearity of inversion effectively by selecting suitable P or S wave subsets for hierarchical inversion. Numerical example of Sigsbee2A model validates the effectiveness of the algorithms and strategies for elastic WERTI (E-WERTI).
Luneburg modified lens for surface water waves
Pichard, Helene; Maurel, Agnes; Petitjeans, Phillipe; Martin, Paul; Pagneux, Vincent
2015-11-01
It is well known that when the waves pass across an elevated bathymetry, refraction often results in amplification of waves behind it. In this sense, focusing of liquid surface waves can be used to enhance the harvest efficiency of ocean power. An ocean wave focusing lens concentrates waves on a certain focal point by transforming straight crest lens of incident waves into circular ones just like an optical lens. These devices have attracted ocean engineers and are promising because they enable the effective utilization of wave energy, the remaining challenge being to increase the harvest efficiency of the lens. In this work, in order to improve well known focusing of surface liquid waves by lens, the propagation of liquid surface waves through a Luneburg modified lens is investigated. The traditional Luneburg lens is a rotationally symmetric lens with a spatially varying refractive-index profile that focuses an incident plane wave on the rim of the lens. The modified Luneburg lens allows to choose the position of the focal point, which can lie inside or outside the lens. This new degree of freedom leads to enhanced focusing and tunable focusing. The focusing of linear surface waves through this lens is investigated and is shown to be more efficient than classical profile lenses.
Directory of Open Access Journals (Sweden)
Ahmed Elhanafi
2016-12-01
Full Text Available In this paper, hydrodynamic wave loads on an offshore stationary–floating oscillating water column (OWC are investigated via a 2D and 3D computational fluid dynamics (CFD modeling based on the RANS equations and the VOF surface capturing scheme. The CFD model is validated against previous experiments for nonlinear regular wave interactions with a surface-piercing stationary barge. Following the validation stage, the numerical model is modified to consider the pneumatic damping effect, and an extensive campaign of numerical tests is carried out to study the wave–OWC interactions for different wave periods, wave heights and pneumatic damping factors. It is found that the horizontal wave force is usually larger than the vertical one. Also, there a direct relationship between the pneumatic and hydrodynamic vertical forces with a maximum vertical force almost at the device natural frequency, whereas the pneumatic damping has a little effect on the horizontal force. Additionally, simulating the turbine damping with an orifice plate induces higher vertical loads than utilizing a slot opening. Furthermore, 3D modeling significantly escalates and declines the predicted hydrodynamic vertical and horizontal wave loads, respectively.
On the structure of artificially generated water wave trains
Papadimitrakis, Yiannis A.
1986-01-01
The structure of an artificially generated sinusoidal water wave train of fixed frequency under the influence of wind is analyzed. Artificially generated waves of 1 Hz are studied at seven wind speeds in the range of 140-400 cm/s. It is observed that the water wave train deformed by wind consists of two components at both the fundamental mode and the harmonics. The amplitude and phase of the wave components are derived, and the dispersion relation and component phase speeds are examined. The data reveal that the amplitude of the forced and free-traveling second harmonics correlate with previous theories, and that the deviation of the measured phase speed from the linear theory is caused by the nonlinearity of the primary wave, the interaction between short gravity waves and the primary wave, and the advection effects of wind drift.
Local energy decay for wave equation in the absence of resonance at zero energy in 3D
Georgiev, Vladimir; Tarulli, Mirko
2011-01-01
In this paper we study spectral properties associated to Schrodinger operator with potential that is an exponential decaying function. As applications we prove local energy decay for solutions to the perturbed wave equation and lack of resonances for the NLS.
Dispersion equations for field-aligned cyclotron waves in axisymmetric magnetospheric plasmas
Directory of Open Access Journals (Sweden)
N. I. Grishanov
2006-03-01
Full Text Available In this paper, we derive the dispersion equations for field-aligned cyclotron waves in two-dimensional (2-D magnetospheric plasmas with anisotropic temperature. Two magnetic field configurations are considered with dipole and circular magnetic field lines. The main contribution of the trapped particles to the transverse dielectric permittivity is estimated by solving the linearized Vlasov equation for their perturbed distribution functions, accounting for the cyclotron and bounce resonances, neglecting the drift effects, and assuming the weak connection of the left-hand and right-hand polarized waves. Both the bi-Maxwellian and bi-Lorentzian distribution functions are considered to model the ring current ions and electrons in the dipole magnetosphere. A numerical code has been developed to analyze the dispersion characteristics of electromagnetic ion-cyclotron waves in an electron-proton magnetospheric plasma with circular magnetic field lines, assuming that the steady-state distribution function of the energetic protons is bi-Maxwellian. As in the uniform magnetic field case, the growth rate of the proton-cyclotron instability (PCI in the 2-D magnetospheric plasmas is defined by the contribution of the energetic ions/protons to the imaginary part of the transverse permittivity elements. We demonstrate that the PCI growth rate in the 2-D axisymmetric plasmasphere can be significantly smaller than that for the straight magnetic field case with the same macroscopic bulk parameters.
Wave front-ray synthesis for solving the multidimensional quantum Hamilton-Jacobi equation
Energy Technology Data Exchange (ETDEWEB)
Wyatt, Robert E.; Chou, Chia-Chun [Institute for Theoretical Chemistry and Department of Chemistry and Biochemistry, University of Texas at Austin, Austin, Texas 78712 (United States)
2011-08-21
A Cauchy initial-value approach to the complex-valued quantum Hamilton-Jacobi equation (QHJE) is investigated for multidimensional systems. In this approach, ray segments foliate configuration space which is laminated by surfaces of constant action. The QHJE incorporates all quantum effects through a term involving the divergence of the quantum momentum function (QMF). The divergence term may be expressed as a sum of two terms, one involving displacement along the ray and the other incorporating the local curvature of the action surface. It is shown that curvature of the wave front may be computed from coefficients of the first and second fundamental forms from differential geometry that are associated with the surface. Using the expression for the divergence, the QHJE becomes a Riccati-type ordinary differential equation (ODE) for the complex-valued QMF, which is parametrized by the arc length along the ray. In order to integrate over possible singularities in the QMF, a stable and accurate Moebius propagator is introduced. This method is then used to evolve rays and wave fronts for four systems in two and three dimensions. From the QMF along each ray, the wave function can be easily computed. Computational difficulties that may arise are described and some ways to circumvent them are presented.
Fast solution of elliptic partial differential equations using linear combinations of plane waves.
Pérez-Jordá, José M
2016-02-01
Given an arbitrary elliptic partial differential equation (PDE), a procedure for obtaining its solution is proposed based on the method of Ritz: the solution is written as a linear combination of plane waves and the coefficients are obtained by variational minimization. The PDE to be solved is cast as a system of linear equations Ax=b, where the matrix A is not sparse, which prevents the straightforward application of standard iterative methods in order to solve it. This sparseness problem can be circumvented by means of a recursive bisection approach based on the fast Fourier transform, which makes it possible to implement fast versions of some stationary iterative methods (such as Gauss-Seidel) consuming O(NlogN) memory and executing an iteration in O(Nlog(2)N) time, N being the number of plane waves used. In a similar way, fast versions of Krylov subspace methods and multigrid methods can also be implemented. These procedures are tested on Poisson's equation expressed in adaptive coordinates. It is found that the best results are obtained with the GMRES method using a multigrid preconditioner with Gauss-Seidel relaxation steps.
A correction function method for the wave equation with interface jump conditions
Abraham, David S.; Marques, Alexandre Noll; Nave, Jean-Christophe
2018-01-01
In this paper a novel method to solve the constant coefficient wave equation, subject to interface jump conditions, is presented. In general, such problems pose issues for standard finite difference solvers, as the inherent discontinuity in the solution results in erroneous derivative information wherever the stencils straddle the given interface. Here, however, the recently proposed Correction Function Method (CFM) is used, in which correction terms are computed from the interface conditions, and added to affected nodes to compensate for the discontinuity. In contrast to existing methods, these corrections are not simply defined at affected nodes, but rather generalized to a continuous function within a small region surrounding the interface. As a result, the correction function may be defined in terms of its own governing partial differential equation (PDE) which may be solved, in principle, to arbitrary order of accuracy. The resulting scheme is not only arbitrarily high order, but also robust, having already seen application to Poisson problems and the heat equation. By extending the CFM to this new class of PDEs, the treatment of wave interface discontinuities in homogeneous media becomes possible. This allows, for example, for the straightforward treatment of infinitesimal source terms and sharp boundaries, free of staircasing errors. Additionally, new modifications to the CFM are derived, allowing compatibility with explicit multi-step methods, such as Runge-Kutta (RK4), without a reduction in accuracy. These results are then verified through numerous numerical experiments in one and two spatial dimensions.
A metasurface carpet cloak for electromagnetic, acoustic and water waves.
Yang, Yihao; Wang, Huaping; Yu, Faxin; Xu, Zhiwei; Chen, Hongsheng
2016-01-29
We propose a single low-profile skin metasurface carpet cloak to hide objects with arbitrary shape and size under three different waves, i.e., electromagnetic (EM) waves, acoustic waves and water waves. We first present a metasurface which can control the local reflection phase of these three waves. By taking advantage of this metasurface, we then design a metasurface carpet cloak which provides an additional phase to compensate the phase distortion introduced by a bump, thus restoring the reflection waves as if the incident waves impinge onto a flat mirror. The finite element simulation results demonstrate that an object can be hidden under these three kinds of waves with a single metasurface cloak.
An extended K-dV equation for nonlinear magnetosonic wave in a multi-ion plasma
Energy Technology Data Exchange (ETDEWEB)
Ida, A. [Nagoya Univ. (Japan). School of Engineering; Sanuki, H.; Todoroki, J.
1995-06-01
Nonlinear magnetosonic waves propagating perpendicularly to a magnetic field are studied in two-ion plasma. It is shown that high frequency magnetosonic wave under the influence of finite cut-off frequency is described by an extended K-dV equation, rather than conventional K-dV equation. Modulational stability of this mode is strongly affected by the finite cut-off frequency in two-ion plasma. (author).
Modeling rapid mass movements using the shallow water equations
Hergarten, S.; Robl, J.
2014-11-01
We propose a new method to model rapid mass movements on complex topography using the shallow water equations in Cartesian coordinates. These equations are the widely used standard approximation for the flow of water in rivers and shallow lakes, but the main prerequisite for their application - an almost horizontal fluid table - is in general not satisfied for avalanches and debris flows in steep terrain. Therefore, we have developed appropriate correction terms for large topographic gradients. In this study we present the mathematical formulation of these correction terms and their implementation in the open source flow solver GERRIS. This novel approach is evaluated by simulating avalanches on synthetic and finally natural topographies and the widely used Voellmy flow resistance law. The results are tested against analytical solutions and the commercial avalanche model RAMMS. The overall results are in excellent agreement with the reference system RAMMS, and the deviations between the different models are far below the uncertainties in the determination of the relevant fluid parameters and involved avalanche volumes in reality. As this code is freely available and open source, it can be easily extended by additional fluid models or source areas, making this model suitable for simulating several types of rapid mass movements. It therefore provides a valuable tool assisting regional scale natural hazard studies.
Manning, Robert M.
2004-01-01
The extended wide-angle parabolic wave equation applied to electromagnetic wave propagation in random media is considered. A general operator equation is derived which gives the statistical moments of an electric field of a propagating wave. This expression is used to obtain the first and second order moments of the wave field and solutions are found that transcend those which incorporate the full paraxial approximation at the outset. Although these equations can be applied to any propagation scenario that satisfies the conditions of application of the extended parabolic wave equation, the example of propagation through atmospheric turbulence is used. It is shown that in the case of atmospheric wave propagation and under the Markov approximation (i.e., the delta-correlation of the fluctuations in the direction of propagation), the usual parabolic equation in the paraxial approximation is accurate even at millimeter wavelengths. The comprehensive operator solution also allows one to obtain expressions for the longitudinal (generalized) second order moment. This is also considered and the solution for the atmospheric case is obtained and discussed. The methodology developed here can be applied to any qualifying situation involving random propagation through turbid or plasma environments that can be represented by a spectral density of permittivity fluctuations.
Velocity flow field and water level measurements in shoaling and breaking water waves
CSIR Research Space (South Africa)
Mukaro, R
2010-01-01
Full Text Available In this paper we report on the laboratory investigations of breaking water waves. Measurements of the water levels and instantaneous fluid velocities were conducted in water waves breaking on a sloping beach within a glass flume. Instantaneous water...
Singular solutions to Protter's problem for the 3-D wave equation involving lower order terms
Directory of Open Access Journals (Sweden)
Myron K. Grammatikopoulos
2003-01-01
Full Text Available In 1952, at a conference in New York, Protter formulated some boundary value problems for the wave equation, which are three-dimensional analogues of the Darboux problems (or Cauchy-Goursat problems on the plane. Protter studied these problems in a 3-D domain $Omega_0$, bounded by two characteristic cones $Sigma_1$ and $Sigma_{2,0}$, and by a plane region $Sigma_0$. It is well known that, for an infinite number of smooth functions in the right-hand side, these problems do not have classical solutions. Popivanov and Schneider (1995 discovered the reason of this fact for the case of Dirichlet's and Neumann's conditions on $Sigma_0$: the strong power-type singularity appears in the generalized solution on the characteristic cone $Sigma_{2,0}$. In the present paper we consider the case of third boundary-value problem on $Sigma_0$ and obtain the existence of many singular solutions for the wave equation involving lower order terms. Especifica ally, for Protter's problems in $mathbb{R}^{3}$ it is shown here that for any $nin N$ there exists a $C^{n}({Omega}_0$-function, for which the corresponding unique generalized solution belongs to $C^{n}({Omega}_0slash O$ and has a strong power type singularity at the point $O$. This singularity is isolated at the vertex $O$ of the characteristic cone $Sigma_{2,0}$ and does not propagate along the cone. For the wave equation without lower order terms, we presented the exact behavior of the singular solutions at the point $O$.
A scattering theory for the wave equation on Kerr black hole exteriors
Dafermos, Mihalis; Shlapentokh-Rothman, Yakov
2014-01-01
We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|
An arbitrary-order staggered time integrator for the linear acoustic wave equation
Lee, Jaejoon; Park, Hyunseo; Park, Yoonseo; Shin, Changsoo
2018-02-01
We suggest a staggered time integrator whose order of accuracy can arbitrarily be extended to solve the linear acoustic wave equation. A strategy to select the appropriate order of accuracy is also proposed based on the error analysis that quantitatively predicts the truncation error of the numerical solution. This strategy not only reduces the computational cost several times, but also allows us to flexibly set the modelling parameters such as the time step length, grid interval and P-wave speed. It is demonstrated that the proposed method can almost eliminate temporal dispersive errors during long term simulations regardless of the heterogeneity of the media and time step lengths. The method can also be successfully applied to the source problem with an absorbing boundary condition, which is frequently encountered in the practical usage for the imaging algorithms or the inverse problems.
Mixed Initial-Boundary Value Problem for the Capillary Wave Equation
Directory of Open Access Journals (Sweden)
B. Juarez Campos
2016-01-01
Full Text Available We study the mixed initial-boundary value problem for the capillary wave equation: iut+u2u=∂x3/2u, t>0, x>0; u(x,0=u0(x, x>0; u(0,t+βux(0,t=h(t, t>0, where ∂x3/2u=(1/2π∫0∞signx-y/x-yuyy(y dy. We prove the global in-time existence of solutions of IBV problem for nonlinear capillary equation with inhomogeneous Robin boundary conditions. Also we are interested in the study of the asymptotic behavior of solutions.
General decay of solutions of a nonlinear system of viscoelastic wave equations
Said-Houari, Belkacem
2011-04-16
This work is concerned with a system of two viscoelastic wave equations with nonlinear damping and source terms acting in both equations. Under some restrictions on the nonlinearity of the damping and the source terms, we prove that, for certain class of relaxation functions and for some restrictions on the initial data, the rate of decay of the total energy depends on those of the relaxation functions. This result improves many results in the literature, such as the ones in Messaoudi and Tatar (Appl. Anal. 87(3):247-263, 2008) and Liu (Nonlinear Anal. 71:2257-2267, 2009) in which only the exponential and polynomial decay rates are considered. © 2011 Springer Basel AG.
On global attraction to solitary waves for the Klein–Gordon equation with concentrated nonlinearity
Kopylova, Elena
2017-11-01
The global attraction is proved for the nonlinear three-dimensional Klein–Gordon equation with a nonlinearity concentrated at one point. Our main result is the convergence of each ‘finite energy solution’ to the manifold of all solitary waves as t\\to+/-∞ . This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation. We justify this mechanism by the following strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time-spectrum in the spectral gap [-m, m] and satisfies the original equation. Then the application of the Titchmarsh convolution theorem reduces the spectrum of each omega-limit trajectory to a single frequency \
Equation of state with scale-invariant hidden local symmetry and gravitational waves
Directory of Open Access Journals (Sweden)
Lee Hyun Kyu
2018-01-01
Full Text Available The equation of state (EoS for the effective theory proposed recently in the frame work of the scale-invariant hidden local symmetry is discussed briefly. The EoS is found to be relatively stiffer at lower density and but relatively softer at higher density. The particular features of EoS on the gravitational waves are discussed. A relatively stiffer EoS for the neutron stars with the lower density induces a larger deviation of the gravitational wave form from the point-particle-approximation. On the other hand, a relatively softer EoS for the merger remnant of the higher density inside might invoke a possibility of the immediate formation of a black hole for short gamma ray bursts or the appearance of the higher peak frequency for gravitational waves from remnant oscillations. It is anticipated that this particular features could be probed in detail by the detections of gravitational waves from the binary neutron star mergers.
Global Nonexistence of Solutions for Viscoelastic Wave Equations of Kirchhoff Type with High Energy
Directory of Open Access Journals (Sweden)
Gang Li
2012-01-01
Full Text Available We consider viscoelastic wave equations of the Kirchhoff type utt-M(∥∇u∥22Δu+∫0tg(t-sΔu(sds+ut=|u|p-1u with Dirichlet boundary conditions, where ∥⋅∥p denotes the norm in the Lebesgue space Lp. Under some suitable assumptions on g and the initial data, we establish a global nonexistence result for certain solutions with arbitrarily high energy, in the sense that limt→T*-(∥u(t∥22+∫0t∥u(s∥22ds=∞ for some 0
Skeletonized Wave Equation Inversion in VTI Media without too much Math
Feng, Shihang
2017-05-17
We present a tutorial for skeletonized inversion of pseudo-acoustic anisotropic VTI data. We first invert for the anisotropic models using wave equation traveltime inversion. Here, the skeletonized data are the traveltimes of transmitted and/or reflected arrivals that lead to simpler misfit functions and more robust convergence compared to full waveform inversion. This provides a good starting model for waveform inversion. The effectiveness of this procedure is illustrated with synthetic data examples and a marine data set recorded in the Gulf of Mexico.
Gerbi, Stéphane
2013-01-15
The goal of this work is to study a model of the wave equation with dynamic boundary conditions and a viscoelastic term. First, applying the Faedo-Galerkin method combined with the fixed point theorem, we show the existence and uniqueness of a local in time solution. Second, we show that under some restrictions on the initial data, the solution continues to exist globally in time. On the other hand, if the interior source dominates the boundary damping, then the solution is unbounded and grows as an exponential function. In addition, in the absence of the strong damping, then the solution ceases to exist and blows up in finite time.
Directory of Open Access Journals (Sweden)
Hassan Kamil Jassim
2016-01-01
Full Text Available We used the local fractional variational iteration transform method (LFVITM coupled by the local fractional Laplace transform and variational iteration method to solve three-dimensional diffusion and wave equations with local fractional derivative operator. This method has Lagrange multiplier equal to minus one, which makes the calculations more easily. The obtained results show that the presented method is efficient and yields a solution in a closed form. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new method.
Solitary and Jacobi elliptic wave solutions of the generalized Benjamin-Bona-Mahony equation
Belobo, Didier Belobo; Das, Tapas
2017-07-01
Exact bright, dark, antikink solitary waves and Jacobi elliptic function solutions of the generalized Benjamin-Bona-Mahony equation with arbitrary power-law nonlinearity will be constructed in this work. The method used to carry out the integration is the F-expansion method. Solutions obtained have fractional and integer negative or positive power-law nonlinearities. These solutions have many free parameters such that they may be used to simulate many experimental situations, and to precisely control the dynamics of the system.
On asymptotic stability of standing waves of discrete Schr\\"odinger equation in $\\Bbb Z$
Cuccagna, Scipio; Tarulli, Mirko
2008-01-01
We prove an analogue of a classical asymptotic stability result of standing waves of the Schr\\"odinger equation originating in work by Soffer and Weinstein. Specifically, our result is a transposition on the lattice Z of a result by Mizumachi and it involves a discrete Schr\\"odinger operator H. The decay rates on the potential are less stringent than in Mizumachi, since we require for the potential $q\\in \\ell ^{1,1}$. We also prove $|e^{itH}(n,m)|\\le C ^{-1/3}$ for a fixed $C$ requiring...
High frequency computation in wave equations and optimal design for a cavity
Lai, Jun
Two types of problems are studied in this thesis. One part of the thesis is devoted to high frequency computation. Motivated by fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beam methods which were originally designed for initial value problems of wave equations in the high frequency regime, we develop fast multiscale Gaussian beam methods for wave equations in bounded convex domains in the high frequency regime. To compute the wave propagation in bounded convex domains, we have to take into account reflecting multiscale Gaussian beams, which are accomplished by enforcing reflecting boundary conditions during beam propagation and carrying out suitable reflecting beam summation. To propagate multiscale beams efficiently, we prove that the ratio of the squared magnitude of beam amplitude and the beam width is roughly conserved, and accordingly we propose an effective indicator to identify significant beams. We also prove that the resulting multiscale Gaussian beam methods converge asymptotically. Numerical examples demonstrate the accuracy and efficiency of the method. The second part of the thesis studies the reduction of backscatter radar cross section (RCS) for a cavity embedded in the ground plane. One approach for RCS reduction is through the coating material. Assume the bottom of the cavity is coated by a thin, multilayered radar absorbing material (RAM) with possibly different permittivities. The objective is to minimize the backscatter RCS by the incidence of a plane wave over a single or a set of incident angles and frequencies. By formulating the scattering problem as a Helmholtz equation with artificial boundary condition, the gradient with respect to the material permittivities is determined efficiently by the adjoint state method, which is integrated into a nonlinear optimization scheme. Numerical example shows the RCS may be significantly reduced. Another approach is through shape optimization. By introducing a transparent
Super-Grid Modeling of the Elastic Wave Equation in Semi-Bounded Domains
Energy Technology Data Exchange (ETDEWEB)
Petersson, N. Anders; Sjögreen, Björn
2014-10-01
We develop a super-grid modeling technique for solving the elastic wave equation in semi-bounded two- and three-dimensional spatial domains. In this method, waves are slowed down and dissipated in sponge layers near the far-field boundaries. Mathematically, this is equivalent to a coordinate mapping that transforms a very large physical domain to a significantly smaller computational domain, where the elastic wave equation is solved numerically on a regular grid. To damp out waves that become poorly resolved because of the coordinate mapping, a high order artificial dissipation operator is added in layers near the boundaries of the computational domain. We prove by energy estimates that the super-grid modeling leads to a stable numerical method with decreasing energy, which is valid for heterogeneous material properties and a free surface boundary condition on one side of the domain. Our spatial discretization is based on a fourth order accurate finite difference method, which satisfies the principle of summation by parts. We show that the discrete energy estimate holds also when a centered finite difference stencil is combined with homogeneous Dirichlet conditions at several ghost points outside of the far-field boundaries. Therefore, the coefficients in the finite difference stencils need only be boundary modified near the free surface. This allows for improved computational efficiency and significant simplifications of the implementation of the proposed method in multi-dimensional domains. Numerical experiments in three space dimensions show that the modeling error from truncating the domain can be made very small by choosing a sufficiently wide super-grid damping layer. The numerical accuracy is first evaluated against analytical solutions of Lamb’s problem, where fourth order accuracy is observed with a sixth order artificial dissipation. We then use successive grid refinements to study the numerical accuracy in the more
Lubin, Pierre; Vincent, Stéphane; Caltagirone, Jean-Paul
2005-04-01
The scope of this Note is to show the results obtained for simulating the two-dimensional head-on collision of two solitary waves by solving the Navier-Stokes equations in air and water. The work is dedicated to the numerical investigation of the hydrodynamics associated to this highly nonlinear flow configuration, the first numerical results being analyzed. The original numerical model is proved to be efficient and accurate in predicting the main features described in experiments found in the literature. This Note also outlines the interest of this configuration to be considered as a test-case for numerical models dedicated to computational fluid mechanics. To cite this article: P. Lubin et al., C. R. Mecanique 333 (2005).
Shock wave focusing in water inside convergent structures
Directory of Open Access Journals (Sweden)
C Wang
2016-09-01
Full Text Available Experiments on shock focusing in water-filled convergent structures have been performed. A shock wave in water is generated by means of a projectile, launched from a gas gun, which impacts a water-filled convergent structure. Two types of structures have been tested; a bulk material and a thin shell structure. The geometric shape of the convergent structures is given by a logarithmic spiral, and this particular shape is chosen because it maximizes the amount of energy reaching the focal region. High-speed schlieren photography is used to visualize the shock dynamics during the focusing event. Results show that the fluid-structure interaction between the thin shell structure and the shock wave in the water is different from that of a bulk structure; multiple reflections of the shock wave inside the thin shell are reflected back into the water, thus creating a wave train, which is not observed for shock focusing in a bulk material.
Buffoni, Boris; Groves, Mark D.; Wahlén, Erik
2017-12-01
Fully localised solitary waves are travelling-wave solutions of the three- dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as `lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number {β} greater than {1/3} ) has recently been given. In this article we present an existence theory for the physically more realistic case {0 < β < 1/3} . A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey-Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.
An efficient flexible-order model for coastal and ocean water waves
DEFF Research Database (Denmark)
Engsig-Karup, Allan Peter; Bingham, Harry B.; Lindberg, Ole
Current work are directed toward the development of an improved numerical 3D model for fully nonlinear potential water waves over arbitrary depths. The model is high-order accurate, robust and efficient for large-scale problems, and support will be included for flexibility in the description...... of structures. The mathemathical equations for potential waves in the physical domain is transformed through $\\sigma$-mapping(s) to a time-invariant boundary-fitted domain which then becomes a basis for an efficient solution strategy. The improved 3D numerical model is based on a finite difference method...
Hydrodynamic analysis of oscillating water column wave energy devices
DEFF Research Database (Denmark)
Bingham, Harry B.; Ducasse, Damien; Nielsen, Kim
2015-01-01
for wave-body interactions, 2014, http://www.wamit.com) is used for the basic wave-structure interaction analysis. The damping applied to each chamber by the power take off is modeled in the experiment by forcing the air through a hole with an area of about 1 % of the chamber water surface area...
Poznanski, R R
2010-09-01
A reaction-diffusion model is presented to encapsulate calcium-induced calcium release (CICR) as a potential mechanism for somatofugal bias of dendritic calcium movement in starburst amacrine cells. Calcium dynamics involves a simple calcium extrusion (pump) and a buffering mechanism of calcium binding proteins homogeneously distributed over the plasma membrane of the endoplasmic reticulum within starburst amacrine cells. The system of reaction-diffusion equations in the excess buffer (or low calcium concentration) approximation are reformulated as a nonlinear Volterra integral equation which is solved analytically via a regular perturbation series expansion in response to calcium feedback from a continuously and uniformly distributed calcium sources. Calculation of luminal calcium diffusion in the absence of buffering enables a wave to travel at distances of 120 μm from the soma to distal tips of a starburst amacrine cell dendrite in 100 msec, yet in the presence of discretely distributed calcium-binding proteins it is unknown whether the propagating calcium wave-front in the somatofugal direction is further impeded by endogenous buffers. If so, this would indicate CICR to be an unlikely mechanism of retinal direction selectivity in starburst amacrine cells.
Chen, Wen; Fang, Jun; Pang, Guofei; Holm, Sverre
2017-01-01
This paper proposes a fractional biharmonic operator equation model in the time-space domain to describe scattering attenuation of acoustic waves in heterogeneous media. Compared with the existing models, the proposed fractional model is able to describe arbitrary frequency-dependent scattering attenuation, which typically obeys an empirical power law with an exponent ranging from 0 to 4. In stark contrast to an extensive and rapidly increasing application of the fractional derivative models for wave absorption attenuation in the literature, little has been reported on frequency-dependent scattering attenuation. This is largely because the order of the fractional Laplacian is from 0 to 2 and is infeasible for scattering attenuation. In this study, the definition of the fractional biharmonic operator in space with an order varying from 0 to 4 is proposed, as well as a fractional biharmonic operator equation model of scattering attenuation which is consistent with arbitrary frequency power-law dependency and obeys the causal relation under the smallness approximation. Finally, the correlation between the fractional order and the ratio of wavelength to the diameter of the scattering heterogeneity is investigated and an expression on exponential form is also provided.
Extended common-image-point gathers for anisotropic wave-equation migration
Sava, Paul C.
2010-01-01
In regions characterized by complex subsurface structure, wave-equation depth migration is a powerful tool for accurately imaging the earth’s interior. The quality of the final image greatly depends on the quality of the model which includes anisotropy parameters (Gray et al., 2001). In particular, it is important to construct subsurface velocity models using techniques that are consistent with the methods used for imaging. Generally speaking, there are two possible strategies for velocity estimation from surface seismic data in the context of wavefield-based imaging (Sava et al., 2010). One possibility is to formulate an objective function in the data space, prior to migration, by matching the recorded data with simulated data. Techniques in this category are known by the name of waveform inversion. Another possibility is to formulate an objective function in the image space, after migration, by measuring and correcting image features that indicate model inaccuracies. Techniques in this category are known as wave-equation migration velocity analysis (MVA).
Eigenvalue solution for the convected wave equation in a circular soft wall duct
Alonso, Jose S.; Burdisso, Ricardo A.
2008-09-01
A numerical approach to find the eigenvalues of the wave equation applied in a circular duct with convective flow and soft wall boundary conditions is proposed. The characteristic equation is solved in the frequency domain as a function of a locally reacting acoustic impedance and the flow Mach number. In addition, the presence of the convective flow couples the solution of the eigenvalues with the axial propagation constants. The unknown eigenvalues are also related to these propagation constants by a quadratic expression that leads to two solutions. These two solutions replaced into the characteristic equation generate two separate eigenvalue problems depending on the direction of propagation. Given that the resulting nonlinear complex-valued equations do not provide the solution explicitly, a numerical technique must be used. The proposed approach is based on the minimization of the absolute value of the characteristic equation by the Nelder-Mead simplex method. The main advantage of this method is that it only uses function evaluations, rather than derivatives, and geometric reasoning. The minimization is performed starting from very low frequencies and increasing by small steps to the particular frequency of interest. The initial guess for the first frequency of calculation is provided as the hard wall eigenvalue solution. Then, the solution from the previous step is used as the initial value for the next calculation. This approach was specifically developed for applications with resonator-type liners commonly used in the commercial aviation industry, where the low-frequency behavior resembles that of a hard wall and agrees with the first initial guess for the first frequency of calculation. The numerical technique was found to be very robust in terms of convergence and stability. Also, the method provides a physical meaning for each eigenvalue since the variation as a function of frequency can be clearly followed with respect to the values that are originally
On exact solutions of a heat-wave type with logarithmic front for the porous medium equation
Kazakov, A. L.; Lempert, A. A.; Orlov, S. S.; Orlov, S. S.
2017-10-01
The paper deals with a nonlinear second-order parabolic equation with partial derivatives, which is usually called “the porous medium equation”. It describes the processes of heat and mass transfer as well as filtration of liquids and gases in porous media. In addition, it is used for mathematical modeling of growth and migration of population. Usually this equation is studied numerically like most other nonlinear equations of mathematical physics. So, the construction of exact solution in an explicit form is important to verify the numerical algorithms. The authors deal with a special solutions which are usually called “heat waves”. A new class of heat-wave type solutions of one-dimensional (plane-symmetric) porous medium equation is proposed and analyzed. A logarithmic heat wave front is studied in details. Considered equation has a singularity at the heat wave front, because the factor of the highest (second) derivative vanishes. The construction of these exact solutions reduces to the integration of a nonlinear second-order ordinary differential equation (ODE). Moreover, the Cauchy conditions lead us to the fact that this equation has a singularity at the initial point. In other words, the ODE inherits the singularity of the original problem. The qualitative analysis of the solutions of the ODE is carried out. The obtained results are interpreted from the point of view of the corresponding heat waves’ behavior. The most interesting is a damped solitary wave, the length of which is constant, and the amplitude decreases.
Derakhti, Morteza; Kirby, James T.; Shi, Fengyan; Ma, Gangfeng
2016-11-01
We examine wave-breaking predictions ranging from shallow- to deep-water conditions using a non-hydrostatic σ-coordinate RANS model NHWAVE as described in Derakhti et al. (2016a), comparing results both with corresponding experiments and with the results of a volume-of-fluid (VOF)/Navier-Stokes solver (Ma et al., 2011; Derakhti and Kirby, 2014a,b). Our study includes regular and irregular depth-limited breaking waves on planar and barred beaches as well as steepness-limited unsteady breaking focused wave packets in intermediate and deep water. In Part 1 of this paper, it is shown that the model resolves organized wave motions in terms of free-surface evolution, spectral evolution, organized wave velocity evolution and wave statistics, using a few vertical σ-levels. In addition, the relative contribution of modeled physical dissipation and numerical dissipation to the integral breaking-induced wave energy loss is discussed. In steepness-limited unsteady breaking focused wave packets, the turbulence model has not been triggered, and all the dissipation is imposed indirectly by the numerical scheme. Although the total wave-breaking-induced energy dissipation is underestimated in the unsteady wave packets, the model is capable of predicting the dispersive and nonlinear properties of different wave packet components before and after the break point, as well as the overall wave height decay and the evolution of organized wave velocity field and power spectrum density over the breaking region. In Part 2 (Derakhti et al., 2016b), model reproduction of wave-breaking-induced turbulence and mean circulation is examined in detail. The same equations and numerical methods are used for the various depth regimes, and no ad-hoc treatment, such as imposing hydrostatic conditions, is involved in triggering breaking. Vertical grid resolution in all simulated cases is at least an order of magnitude coarser than that of typical VOF-based simulations.
Third order wave equation in Duffin-Kemmer-Petiau theory: Massive case
Markov, Yu. A.; Markova, M. A.; Bondarenko, A. I.
2015-11-01
Within the framework of the Duffin-Kemmer-Petiau (DKP) formalism a more consistent approach to the derivation of the third order wave equation obtained earlier by M. Nowakowski [1] on the basis of heuristic considerations is suggested. For this purpose an additional algebraic object, the so-called q -commutator (q is a primitive cubic root of unity) and a new set of matrices ημ instead of the original matrices βμ of the DKP algebra are introduced. It is shown that in terms of these ημ matrices we have succeeded in reducing a procedure of the construction of cubic root of the third order wave operator to a few simple algebraic transformations and to a certain operation of the passage to the limit z →q , where z is some complex deformation parameter entering into the definition of the η -matrices. A corresponding generalization of the result obtained to the case of the interaction with an external electromagnetic field introduced through the minimal coupling scheme is carried out and a comparison with M. Nowakowski's result is performed. A detailed analysis of the general structure for a solution of the first order differential equation for the wave function ψ (x ;z ) is performed and it is shown that the solution is singular in the z →q limit. The application to the problem of construction within the DKP approach of the path integral representation in parasuperspace for the propagator of a massive vector particle in a background gauge field is discussed.
Analytical approximation and numerical simulations for periodic travelling water waves
Kalimeris, Konstantinos
2017-12-01
We present recent analytical and numerical results for two-dimensional periodic travelling water waves with constant vorticity. The analytical approach is based on novel asymptotic expansions. We obtain numerical results in two different ways: the first is based on the solution of a constrained optimization problem, and the second is realized as a numerical continuation algorithm. Both methods are applied on some examples of non-constant vorticity. This article is part of the theme issue 'Nonlinear water waves'.
Sun, Wen-Rong; Wang, Lei
2018-01-01
To show the existence and properties of matter rogue waves in an F =1 spinor Bose-Einstein condensate (BEC), we work on the three-component Gross-Pitaevskii (GP) equations. Via the Darboux-dressing transformation, we obtain a family of rational solutions describing the extreme events, i.e. rogue waves. This family of solutions includes bright-dark-bright and bright-bright-bright rogue waves. The algebraic construction depends on Lax matrices and their Jordan form. The conditions for the existence of rogue wave solutions in an F =1 spinor BEC are discussed. For the three-component GP equations, if there is modulation instability, it is of baseband type only, confirming our analytic conditions. The energy transfers between the waves are discussed.
Sun, Wen-Rong; Wang, Lei
2018-01-01
To show the existence and properties of matter rogue waves in an F=1 spinor Bose-Einstein condensate (BEC), we work on the three-component Gross-Pitaevskii (GP) equations. Via the Darboux-dressing transformation, we obtain a family of rational solutions describing the extreme events, i.e. rogue waves. This family of solutions includes bright-dark-bright and bright-bright-bright rogue waves. The algebraic construction depends on Lax matrices and their Jordan form. The conditions for the existence of rogue wave solutions in an F=1 spinor BEC are discussed. For the three-component GP equations, if there is modulation instability, it is of baseband type only, confirming our analytic conditions. The energy transfers between the waves are discussed.
WAVE EQUATION DATUMING TO CORRECT TOPOGRAPHY EFFECT ON FOOTHILL SEISMIC DATA
Directory of Open Access Journals (Sweden)
Montes Vides Luis Alfredo
2005-08-01
Full Text Available The current seismic processing applies Static Corrections to overcome the effects associated to rough topography, based in the assumption that velocity in near surface is lower than in the substratum, which force going up rays travel near to vertical. However, when the velocity contrast between these layers is not large enough, the trajectory of the up going rays deviate from vertical raveling the reflectors erroneously. A better alternative to correct this is to continue the wave field to a datum, because it does not assume vertical ray trajectory and solves the acoustic wave equation to extrapolate sources and receivers. The Kirchhoff approach was tested in synthetic shots continuing their wave field to a datum and finally it was applied instead of Static Corrections in real data acquired in foothill zones. First shot and receiver gathers were downward continued to the base of weathering layer and later upward continued to a final flat datum. Comparing the obtained results we observed that continuation approach provides a noticeable enhancement of reflectors in seismic records, displaying a better continuity of the reflectors and an increment in frequency content.
A stochastic collocation method for the second order wave equation with a discontinuous random speed
Motamed, Mohammad
2012-08-31
In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems. © 2012 Springer-Verlag.
Resonant Interactions of Capillary-Gravity Water Waves
Martin, Calin Iulian
2017-12-01
We show here that capillary-gravity wave trains can propagate at the free surface of a rotational water flow of constant non-zero vorticity over a flat bed only if the flow is two-dimensional. Moreover, we also show that the vorticity must have only one non zero component which points in the horizontal direction orthogonal to the direction of wave propagation. This result is of relevance in the study of nonlinear resonances of wave trains. We perform such a study for three- and four wave interactions.
Mechanical balance laws for fully nonlinear and weakly dispersive water waves
Kalisch, Henrik; Mitsotakis, Dimitrios
2015-01-01
The Serre-Green-Naghdi system is a coupled, fully nonlinear system of dispersive evolution equations which approximates the full water wave problem. The system is an extension of the well known shallow-water system to the situation where the waves are long, but not so long that dispersive effects can be neglected. In the current work, the focus is on deriving mass, momentum and energy densities and fluxes associated with the Serre-Green-Naghdi system. These quantities arise from imposing balance equations of the same asymptotic order as the evolution equations. In the case of an even bed, the conservation equations are satisfied exactly by the solutions of the Serre-Green-Naghdi system. The case of variable bathymetry is more complicated, with mass and momentum conservation satisfied exactly, and energy conservation satisfied only in a global sense. In all cases, the quantities found here reduce correctly to the corresponding counterparts in both the Boussinesq and the shallow-water scaling. One consequence o...
Deng, Gao-Fu; Gao, Yi-Tian
2017-06-01
Under investigation in this paper is a generalized (3+1)-dimensional varible-coefficient nonlinear-wave equation, which has been presented for nonlinear waves in liquid with gas bubbles. The bilinear form, Bäcklund transformation, Lax pair and infinitely-many conservation laws are obtained via the binary Bell polynomials. One-, two- and three-soliton solutions are generated by virtue of the Hirota method. Travelling-wave solutions are derived with the aid of the polynomial expansion method. The one-periodic wave solutions are constructed by the Hirota-Riemann method. Discussions among the soliton, periodic- and travelling-wave solutions are presented: I) the soliton velocities are related to the variable coefficients, while the soliton amplitudes are unaffected; II) the interaction between the solitons is elastic; III) there are three cases of the travelling-wave solutions, i.e., the triangle-type periodical, bell-type and soliton-type travelling-wave solutions, while we notice that bell-type travelling-wave solutions can be converted into one-soliton solutions via taking suitable parameters; IV) the one-periodic waves approach to the solitary waves under some conditions and can be viewed as a superposition of overlapping solitary waves, placed one period apart.
An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation
Zhan, Ge
2013-02-19
The pure P-wave equation for modelling and migration in tilted transversely isotropic (TTI) media has attracted more and more attention in imaging seismic data with anisotropy. The desirable feature is that it is absolutely free of shear-wave artefacts and the consequent alleviation of numerical instabilities generally suffered by some systems of coupled equations. However, due to several forward-backward Fourier transforms in wavefield updating at each time step, the computational cost is significant, and thereby hampers its prevalence. We propose to use a hybrid pseudospectral (PS) and finite-difference (FD) scheme to solve the pure P-wave equation. In the hybrid solution, most of the cost-consuming wavenumber terms in the equation are replaced by inexpensive FD operators, which in turn accelerates the computation and reduces the computational cost. To demonstrate the benefit in cost saving of the new scheme, 2D and 3D reverse-time migration (RTM) examples using the hybrid solution to the pure P-wave equation are carried out, and respective runtimes are listed and compared. Numerical results show that the hybrid strategy demands less computation time and is faster than using the PS method alone. Furthermore, this new TTI RTM algorithm with the hybrid method is computationally less expensive than that with the FD solution to conventional TTI coupled equations. © 2013 Sinopec Geophysical Research Institute.
Energy Technology Data Exchange (ETDEWEB)
Williamson, D.L.; Hack, J.J.; Jakob, R.; Swarztrauber, P.N. (National Center for Atmospheric Research, Boulder, CO (United States)); Drake, J.B. (Oak Ridge National Lab., TN (United States))
1991-08-01
A suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry. The shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth. These cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade-offs which must always be made in numerical modeling. Before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods. The cases are presented in order of complexity. They consist of advection across the poles, steady state geostrophically balanced flow of both global and local scales, forced nonlinear advection of an isolated low, zonal flow impinging on an isolated mountain, Rossby-Haurwitz waves and observed atmospheric states. One of the cases is also identified as a computer performance/algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers. 31 refs.
Numerical analysis of regular waves over an onshore oscillating water column
Energy Technology Data Exchange (ETDEWEB)
Davyt, D.P.; Teixeira, P.R.F. [Universidade Federal do Rio Grande (FURG), RS (Brazil)], E-mail: pauloteixeira@furg.br; Ramalhais, R. [Universidade Nova de Lisboa, Caparica (Portugal). Fac. de Ciencias e Tecnologia; Didier, E. [Laboratorio Nacional de Engenharia Civil, Lisboa (Portugal)], E-mail: edidier@lnec.pt
2010-07-01
The potential of wave energy along coastal areas is a particularly attractive option in regions of high latitude, such as the coasts of northern Europe, North America, New Zealand, Chile and Argentina where high densities of annual average wave energy are found (typically between 40 and 100 kW/m of wave front). Power estimated in the south of Brazil is 30kW/m, creating a possible alternative of source energy in the region. There are many types and designs of equipment to capture energy from waves under analysis, such as the oscillating water column type (OWC) which has been one of the first to be developed and installed at sea. Despite being one of the most analyzed wave energy converter devices, there are few case studies using numerical simulation. In this context, the numerical analysis of regular waves over an onshore OWC is the main objective of this paper. The numerical models FLUINCO and FLUENT are used for achieving this goal. The FLUINCO model is based on RANS equations which are discretized using the two-step semi-implicit Taylor-Galerkin method. An arbitrary Lagrangian Eulerian formulation is used to enable the solution of problems involving free surface movements. The FLUENT code (version 6.3.26) is based on the finite volume method to solve RANS equations. Volume of Fluid method (VOF) is used for modeling free surface flows. Time integration is achieved by a second order implicit scheme, momentum equations are discretized using MUSCL scheme and HRIC (High Resolution Interface Capturing) scheme is used for convective term of VOF transport equation. The case study consists of a 10.m deep channel with a 10 m wide chamber at its end. One meter high waves with different periods are simulated. Comparisons between FLUINCO and FLUENT results are presented. Free surface elevation inside the chamber; velocity distribution and streamlines; amplification factor (relation between wave height inside the chamber and incident wave height); phase angle (angular
Standing Waves and Inquiry Using Water Droplets
Sinclair, Dina; Vondracek, Mark
2015-01-01
Most high school and introductory college physics classes study simple harmonic motion and various wave phenomena. With the majority of states adopting the Next Generation Science Standards and pushing students to explore the scientific process for themselves, there is a growing demand for hands-on inquiry activities that involve and develop more…
Revil, A.
2017-05-01
I developed a model of cross-coupled flow in partially saturated porous media based on electrokinetic coupling including the effect of ion filtration (normal and reverse osmosis) and the multi-component nature of the pore water (wetting) phase. The model also handles diffusion and membrane polarization but is valid only for saturations above the irreducible water saturation. I start with the local Nernst-Planck and Stokes equations and I use a volume-averaging procedure to obtain the generalized Ohm, Fick, and Darcy equations with cross-coupling terms at the scale of a representative elementary volume of the porous rock. These coupling terms obey Onsager's reciprocity, which is a required condition, at the macroscale, to keep the total dissipation function of the system positive. Rather than writing the electrokinetic terms in terms of zeta potential (the double layer electrical potential on the slipping plane located in the pore water), I developed the model in terms of an effective charge density dragged by the flow of the pore water. This effective charge density is found to be strongly controlled by the permeability and the water saturation. I also developed an electrical conductivity equation including the effect of saturation on both bulk and surface conductivities, the surface conductivity being associated with electromigration in the electrical diffuse layer coating the grains. This surface conductivity depends on the CEC of the porous material.
The Mode Solution of the Wave Equation in Kasner Spacetimes and Redshift
Energy Technology Data Exchange (ETDEWEB)
Petersen, Oliver Lindblad, E-mail: lindblad@uni-potsdam.de [Universität Potsdam, Institut für Mathematik (Germany)
2016-12-15
We study the mode solution to the Cauchy problem of the scalar wave equation □φ = 0 in Kasner spacetimes. As a first result, we give the explicit mode solution in axisymmetric Kasner spacetimes, of which flat Kasner spacetimes are special cases. Furthermore, we give the small and large time asymptotics of the modes in general Kasner spacetimes. Generically, the modes in non-flat Kasner spacetimes grow logarithmically for small times, while the modes in flat Kasner spacetimes stay bounded for small times. For large times, however, the modes in general Kasner spacetimes oscillate with a polynomially decreasing amplitude. This gives a notion of large time frequency of the modes, which we use to model the wavelength of light rays in Kasner spacetimes. We show that the redshift one obtains in this way actually coincides with the usual cosmological redshift.
AN ANALYTICAL SOLUTION OF KINEMATIC WAVE EQUATIONS FOR OVERLAND FLOW UNDER GREEN-AMPT INFILTRATION
Directory of Open Access Journals (Sweden)
Giorgio Baiamonte
2010-03-01
Full Text Available This paper deals with the analytical solution of kinematic wave equations for overland flow occurring in an infiltrating hillslope. The infiltration process is described by the Green-Ampt model. The solution is derived only for the case of an intermediate flow regime between laminar and turbulent ones. A transitional regime can be considered a reliable flow condition when, to the laminar overland flow, is also associated the effect of the additional resistance due to raindrop impact. With reference to the simple case of an impervious hillslope, a comparison was carried out between the present solution and the non-linear storage model. Some applications of the present solution were performed to investigate the effect of main parameter variability on the hillslope response. Particularly, the effect of hillslope geometry and rainfall intensity on the time to equilibrium is shown.
The new potential for the Weyl tensor in N dimensions: gauge and wave equation
Edgar, S. Brian; Senovilla, J. M. M.
Although the Lanczos potential (a (2,1) form, Lab{}^c) for the Weyl tensor does not exist in dimensions greater than four, a new potential (a (2,3) form, Pab{}cde, which coincides with the double dual of Lab{}c in four dimensions) has recently been shown to exist in all dimensions n≥ 4. In this talk we investigate the question of gauge and discuss the structure of the new potential's wave equation which is obtained from the Bianchi identities; identifying the gauge supplies us with a new direct proof of the existence of Pab{}cde via the Cauchy-Kowaleski theorem, as well as the foundation for more general investigations of the first order symmetric hyperbolic structure.
Existence and asymptotic stability of a viscoelastic wave equation with a delay
Kirane, Mokhtar
2011-07-07
In this paper, we consider the viscoelastic wave equation with a delay term in internal feedbacks; namely, we investigate the following problem, together with initial conditions and boundary conditions of Dirichlet type. Here (x, t) ∈ Ω × (0, ∞), g is a positive real valued decreasing function and μ1, μ2 are positive constants. Under an hypothesis between the weight of the delay term in the feedback and the weight of the term without delay, using the Faedo-Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Under the same assumptions, general decay results of the energy are established via suitable Lyapunov functionals. © 2011 Springer Basel AG.
Existence and asymptotic behavior of the wave equation with dynamic boundary conditions
Graber, Philip Jameson
2012-03-07
The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time. © 2012 Springer Science+Business Media, LLC.
Hilditch, David; Bugner, Marcus; Rueter, Hannes; Bruegmann, Bernd
2016-01-01
A long-standing problem in numerical relativity is the satisfactory treatment of future null-infinity. We propose an approach for the evolution of hyperboloidal initial data in which the outer boundary of the computational domain is placed at infinity. The main idea is to apply the `dual foliation' formalism in combination with hyperboloidal coordinates and the generalized harmonic gauge formulation. The strength of the present approach is that, following the ideas of Zenginoglu, a hyperboloidal layer can be naturally attached to a central region using standard coordinates of numerical relativity applications. Employing a generalization of the standard hyperboloidal slices, developed by Calabrese et. al., we find that all formally singular terms take a trivial limit as we head to null-infinity. A byproduct is a numerical approach for hyperboloidal evolution of nonlinear wave equations violating the null-condition. The height-function method, used often for fixed background spacetimes, is generalized in such a...
Wave equation tomography using the unwrapped phase - Analysis of the traveltime sensitivity kernels
Djebbi, Ramzi
2013-01-01
Full waveform inversion suffers from the high non-linearity in the misfit function, which causes the convergence to a local minimum. In the other hand, traveltime tomography has a quasi-linear misfit function but yields low- resolution models. Wave equation tomography (WET) tries to improve on traveltime tomography, by better adhering to the requirements of our finite-frequency data. However, conventional (WET), based on the crosscorelaion lag, yields the popular hallow banana sensitivity kernel indicating that the measured wavefield at a point is insensitive to perturbations along the ray theoretical path at certain finite frequencies. Using the instantaneous traveltime, the sensitivity kernel reflects more the model-data dependency we grown accustom to in seismic inversion (even phase inversion). Demonstrations on synthetic and the Mamousi model support such assertions.
Konno, Hidetoshi
2017-06-01
The paper presents the birth-death stochastic process of an optical rogue wave with a long memory described by a fractional master equation. An exact analytic expression for the probability generating function is obtained with an integral representation of the confluent Heun function. This enables a full statistical analysis under any initial condition. It is demonstrated that the present mathematical approach can be utilized for the analysis of birth-death stochastic processes when the generating function can be described by a class of Heun differential equations.
Directory of Open Access Journals (Sweden)
H. O. Bakodah
2013-01-01
Full Text Available A method of lines approach to the numerical solution of nonlinear wave equations typified by the regularized long wave (RLW is presented. The method developed uses a finite differences discretization to the space. Solution of the resulting system was obtained by applying fourth Runge-Kutta time discretization method. Using Von Neumann stability analysis, it is shown that the proposed method is marginally stable. To test the accuracy of the method some numerical experiments on test problems are presented. Test problems including solitary wave motion, two-solitary wave interaction, and the temporal evaluation of a Maxwellian initial pulse are studied. The accuracy of the present method is tested with and error norms and the conservation properties of mass, energy, and momentum under the RLW equation.
Zhang, Sanzong
2015-05-26
Full-waveform inversion requires the accurate simulation of the dynamics and kinematics of wave propagation. This is difficult in practice because the amplitudes cannot be precisely reproduced for seismic waves in the earth. Wave-equation reflection traveltime tomography (WT) is proposed to avoid this problem by directly inverting the reflection-traveltime residuals without the use of the high-frequency approximation. We inverted synthetic traces and recorded seismic data for the velocity model by WT. Our results demonstrated that the wave-equation solution overcame the high-frequency approximation of ray-based tomography, was largely insensitive to the accurate modeling of amplitudes, and mitigated problems with ambiguous event identification. The synthetic examples illustrated the effectiveness of the WT method in providing a highly resolved estimate of the velocity model. A real data example from the Gulf of Mexico demonstrated these benefits of WT, but also found the limitations in traveltime residual estimation for complex models.
Analysis and computation of the elastic wave equation with random coefficients
Motamed, Mohammad
2015-10-21
We consider the stochastic initial-boundary value problem for the elastic wave equation with random coefficients and deterministic data. We propose a stochastic collocation method for computing statistical moments of the solution or statistics of some given quantities of interest. We study the convergence rate of the error in the stochastic collocation method. In particular, we show that, the rate of convergence depends on the regularity of the solution or the quantity of interest in the stochastic space, which is in turn related to the regularity of the deterministic data in the physical space and the type of the quantity of interest. We demonstrate that a fast rate of convergence is possible in two cases: for the elastic wave solutions with high regular data; and for some high regular quantities of interest even in the presence of low regular data. We perform numerical examples, including a simplified earthquake, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo sampling method for approximating quantities with high stochastic regularity.
Higher-Order Wave Equation Within the Duffin-Kemmer-Petiau Formalism
Markov, Yu. A.; Markova, M. A.; Bondarenko, A. I.
2017-03-01
Within the framework of the Duffin-Kemmer-Petiau (DKP) formalism a consistent approach to derivation of the third-order wave equation is suggested. For this purpose, an additional algebraic object, the so-called q-commutator ( q is a primitive cubic root of unity) and a new set of matrices ημ instead of the original matrices βμ of the DKP algebra are introduced. It is shown that in terms of these η-matrices, we have succeeded to reduce the procedure of the construction of cubic root of the third-order wave operator to a few simple algebraic transformations and to a certain operation of passage to the limit z → q, where z is some complex deformation parameter entering into the definition of the ημ-matrices. A corresponding generalization of the result obtained to the case of interaction with an external electromagnetic field introduced through the minimal coupling scheme is performed. The application to the problem of construction within the DKP approach of the path integral representation in parasuperspace for the propagator of a massive vector particle in a background gauge field is discussed.
Ulku, Huseyin Arda
2014-07-06
Effects of material nonlinearities on electromagnetic field interactions become dominant as field amplitudes increase. A typical example is observed in plasmonics, where highly localized fields “activate” Kerr nonlinearities. Naturally, time domain solvers are the method of choice when it comes simulating these nonlinear effects. Oftentimes, finite difference time domain (FDTD) method is used for this purpose. This is simply due to the fact that explicitness of the FDTD renders the implementation easier and the material nonlinearity can be easily accounted for using an auxiliary differential equation (J.H. Green and A. Taflove, Opt. Express, 14(18), 8305-8310, 2006). On the other hand, explicit marching on-in-time (MOT)-based time domain integral equation (TDIE) solvers have never been used for the same purpose even though they offer several advantages over FDTD (E. Michielssen, et al., ECCOMAS CFD, The Netherlands, Sep. 5-8, 2006). This is because explicit MOT solvers have never been stabilized until not so long ago. Recently an explicit but stable MOT scheme has been proposed for solving the time domain surface magnetic field integral equation (H.A. Ulku, et al., IEEE Trans. Antennas Propag., 61(8), 4120-4131, 2013) and later it has been extended for the time domain volume electric field integral equation (TDVEFIE) (S. B. Sayed, et al., Pr. Electromagn. Res. S., 378, Stockholm, 2013). This explicit MOT scheme uses predictor-corrector updates together with successive over relaxation during time marching to stabilize the solution even when time step is as large as in the implicit counterpart. In this work, an explicit MOT-TDVEFIE solver is proposed for analyzing electromagnetic wave interactions on scatterers exhibiting Kerr nonlinearity. Nonlinearity is accounted for using the constitutive relation between the electric field intensity and flux density. Then, this relation and the TDVEFIE are discretized together by expanding the intensity and flux - sing half
Directory of Open Access Journals (Sweden)
Huanhe Dong
2014-01-01
Full Text Available We introduce how to obtain the bilinear form and the exact periodic wave solutions of a class of (2+1-dimensional nonlinear integrable differential equations directly and quickly with the help of the generalized Dp-operators, binary Bell polynomials, and a general Riemann theta function in terms of the Hirota method. As applications, we solve the periodic wave solution of BLMP equation and it can be reduced to soliton solution via asymptotic analysis when the value of p is 5.
Ding, Min; Li, Yachun
2017-08-01
We consider the 1-D piston problem for the isentropic relativistic Euler equations when the total variations of the initial data and the speed of the piston are both sufficiently small. By a modified wave front tracking method, we establish the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength. Meanwhile, we study the convergence of the entropy solutions to the corresponding entropy solutions of the classical non-relativistic isentropic Euler equations as the light speed c →+∞ .
Fast volumetric integral-equation solver for acoustic wave propagation through inhomogeneous media.
Bleszynski, E; Bleszynski, M; Jaroszewicz, T
2008-07-01
Elements are described of a volumetric integral-equation-based algorithm applicable to accurate large-scale simulations of scattering and propagation of sound waves through inhomogeneous media. The considered algorithm makes possible simulations involving realistic geometries characterized by highly subwavelength details, large density contrasts, and described in terms of several million unknowns. The algorithm achieves its competitive performance, characterized by O(N log N) solution complexity and O(N) memory requirements, where N is the number of unknowns, through a fast and nonlossy fast Fourier transform based matrix compression technique, the adaptive integral method, previously developed for solving large-scale electromagnetic problems. Because of its ability of handling large problems with complex geometries, the developed solver may constitute an efficient and high fidelity numerical simulation tool for calculating acoustic field distributions in anatomically realistic models, e.g., in investigating acoustic energy transfer to the inner ear via nonairborne pathways in the human head. Examples of calculations of acoustic field distribution in a human head, which require solving linear systems of equations involving several million unknowns, are presented.
A time-domain Kirchhoff formula for the convective acoustic wave equation
Ghorbaniasl, Ghader; Siozos-Rousoulis, Leonidas; Lacor, Chris
2016-03-01
Kirchhoff's integral method allows propagated sound to be predicted, based on the pressure and its derivatives in time and space obtained on a data surface located in the linear flow region. Kirchhoff's formula for noise prediction from high-speed rotors and propellers suffers from the limitation of the observer located in uniform flow, thus requiring an extension to arbitrarily moving media. This paper presents a Kirchhoff formulation for moving surfaces in a uniform moving medium of arbitrary configuration. First, the convective wave equation is derived in a moving frame, based on the generalized functions theory. The Kirchhoff formula is then obtained for moving surfaces in the time domain. The formula has a similar form to the Kirchhoff formulation for moving surfaces of Farassat and Myers, with the presence of additional terms owing to the moving medium effect. The equation explicitly accounts for the influence of mean flow and angle of attack on the radiated noise. The formula is verified by analytical cases of a monopole source located in a moving medium.
Time-fractional wave-diffusion equation in an inhomogeneous half-space
Liemert, André; Kienle, Alwin
2015-06-01
We consider the fundamental solution of the time-fractional wave-diffusion equation in a three-dimensional half-space medium which contains an inhomogeneity in form of a plane parallel layer. The corresponding Green’s function which is derived by means of the Fourier and Laplace transforms can be accurately and efficiently evaluated without recourse to the Mittag-Leffler or the Fox H-function. Moreover, it is shown that in the one-dimensional case the fundamental solution in an inhomogeneous half-space is no longer a probability density function. In addition, we consider the advection equation for the fractional Laplacian {{(-Δ )}\\frac{1{2}}} and the Caputo time-fractional derivative of orders 0\\lt β ≤slant 1 on a bounded domain. Simple algorithms for accurate evaluation of the M-Wright function {{M}β }(x) and the Mittag-Leffler function {{E}β }(-x) are enclosed at the end of this article.
Angle gathers in wave-equation imaging for transversely isotropic media
Alkhalifah, Tariq Ali
2010-11-12
In recent years, wave-equation imaged data are often presented in common-image angle-domain gathers as a decomposition in the scattering angle at the reflector, which provide a natural access to analysing migration velocities and amplitudes. In the case of anisotropic media, the importance of angle gathers is enhanced by the need to properly estimate multiple anisotropic parameters for a proper representation of the medium. We extract angle gathers for each downward-continuation step from converting offset-frequency planes into angle-frequency planes simultaneously with applying the imaging condition in a transversely isotropic with a vertical symmetry axis (VTI) medium. The analytic equations, though cumbersome, are exact within the framework of the acoustic approximation. They are also easily programmable and show that angle gather mapping in the case of anisotropic media differs from its isotropic counterpart, with the difference depending mainly on the strength of anisotropy. Synthetic examples demonstrate the importance of including anisotropy in the angle gather generation as mapping of the energy is negatively altered otherwise. In the case of a titled axis of symmetry (TTI), the same VTI formulation is applicable but requires a rotation of the wavenumbers. © 2010 European Association of Geoscientists & Engineers.
Charland, Jenna; Touboul, Julien; Rey, Vincent
2013-04-01
Wave propagation against current : a study of the effects of vertical shears of the mean current on the geometrical focusing of water waves J. Charland * **, J. Touboul **, V. Rey ** jenna.charland@univ-tln.fr * Direction Générale de l'Armement, CNRS Délégation Normandie ** Université de Toulon, 83957 La Garde, France Mediterranean Institute of Oceanography (MIO) Aix Marseille Université, 13288 Marseille, France CNRS/INSU, IRD, MIO, UM 110 In the nearshore area, both wave propagation and currents are influenced by the bathymetry. For a better understanding of wave - current interactions in the presence of a 3D bathymetry, a large scale experiment was carried out in the Ocean Basin FIRST, Toulon, France. The 3D bathymetry consisted of two symmetric underwater mounds on both sides in the mean wave direction. The water depth at the top the mounds was hm=1,5m, the slopes of the mounds were of about 1:3, the water depth was h=3 m elsewhere. For opposite current conditions (U of order 0.30m/s), a huge focusing of the wave up to twice its incident amplitude was observed in the central part of the basin for T=1.4s. Since deep water conditions are verified, the wave amplification is ascribed to the current field. The mean velocity fields at a water depth hC=0.25m was measured by the use of an electromagnetic current meter. The results have been published in Rey et al [4]. The elliptic form of the "mild slope" equation including a uniform current on the water column (Chen et al [1]) was then used for the calculations. The calculated wave amplification of factor 1.2 is significantly smaller than observed experimentally (factor 2). So, the purpose of this study is to understand the physical processes which explain this gap. As demonstrated by Kharif & Pelinovsky [2], geometrical focusing of waves is able to modify significantly the local wave amplitude. We consider this process here. Since vertical velocity profiles measured at some locations have shown significant
Ma, Xiao; Yang, Dinghui
2017-06-01
The finite-difference method, which is an important numerical tool for solving seismic wave equations, is widely applied in simulation, wave-equation-based migration and inversion. As the seismic wave phase plays a critical role in forward simulation and inversion, it should be preserved during wavefield simulation. In this paper, we propose a type of phase-preserving stereomodelling method, which is simultaneously symplectic and low numerical dispersive. First, we propose three new time-marching schemes for solving wave equations that are optimal symplectic partitioned Runge-Kutta schemes with minimized phase errors. Relevant simulations on a harmonic oscillator show that even after 200 000 temporal iterations, our schemes can still avoid the phase drifting issue that appears in other symplectic schemes. We use these symplectic schemes as time integrators, and a numerically low dispersive operator called the stereomodelling discrete operator as a spatial discretization approach to solve seismic wave equations. Theoretical analysis on the stability conditions shows that the new methods are more stable than previous methods. We also investigate the numerical dispersion relations of the methods proposed in this study. To further investigate phase accuracy, we compare the numerical solutions generated by the proposed methods with analytic solutions. Several numerical experiments indicate that our proposed methods are efficient for various models and perform well with perfectly matched layer boundary conditions.
Shebalin, John V.
1988-01-01
An exact analytic solution is found for a basic electromagnetic wave-charged particle interaction by solving the nonlinear equations of motion. The particle position, velocity, and corresponding time are found to be explicit functions of the total phase of the wave. Particle position and velocity are thus implicit functions of time. Applications include describing the motion of a free electron driven by an intense laser beam..
A Novel 3D Viscoelastic Acoustic Wave Equation Based Update Method for Reservoir History Matching
Katterbauer, Klemens
2014-12-10
The oil and gas industry has been revolutionized within the last decade, with horizontal drilling and hydraulic fracturing enabling the extraction of huge amounts of shale gas in areas previously considered impossible and the recovering of hydrocarbons in harsh environments like the arctic or in previously unimaginable depths like the off-shore exploration in the South China sea and Gulf of Mexico. With the development of 4D seismic, engineers and scientists have been enabled to map the evolution of fluid fronts within the reservoir and determine the displacement caused by the injected fluids. This in turn has led to enhanced production strategies, cost reduction and increased profits. Conventional approaches to incorporate seismic data into the history matching process have been to invert these data for constraints that are subsequently employed in the history matching process. This approach makes the incorporation computationally expensive and requires a lot of manual processing for obtaining the correct interpretation due to the potential artifacts that are generated by the generally ill-conditioned inversion problems. I have presented here a novel approach via including the time-lapse cross-well seismic survey data directly into the history matching process. The generated time-lapse seismic data are obtained from the full wave 3D viscoelastic acoustic wave equation. Furthermore an extensive analysis has been performed showing the robustness of the method and enhanced forecastability of the critical reservoir parameters, reducing uncertainties and exhibiting the benefits of a full wave 3D seismic approach. Finally, the improved performance has been statistically confirmed. The improvements illustrate the significant improvements in forecasting that are obtained via readily available seismic data without the need for inversion. This further optimizes oil production in addition to increasing return-on-investment on oil & gas field development projects, especially
New function of Mittag-Leffler type and its application in the fractional diffusion-wave equation
Energy Technology Data Exchange (ETDEWEB)
Yu Rui [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)]. E-mail: joyfm810909@yahoo.com.cn; Zhang Hongqing [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)
2006-11-15
The classical Mittag-Leffler (M-L) functions have already proved their efficiency as solutions of fractional-order differential and integral equations. In this paper we introduce a modified M-L type function and deduce its important integral transforms. Then the solution of the initial-boundary value problem for the so-called fractional diffusion-wave equation with real-order time and space derivatives is given by using the inverse Fourier transform of the new function.
Hydrodynamic analysis of oscillating water column wave energy devices
DEFF Research Database (Denmark)
Bingham, Harry B.; Ducasse, Damien; Nielsen, Kim
2015-01-01
A 40-chamber I-Beam attenuator-type, oscillating water column, wave energy converter is analyzed numerically based on linearized potential flow theory, and experimentally via model test experiments. The high-order panel method WAMIT by Newman and Lee (WAMIT; a radiation–diffraction panel program...... for wave-body interactions, 2014, http://www.wamit.com) is used for the basic wave-structure interaction analysis. The damping applied to each chamber by the power take off is modeled in the experiment by forcing the air through a hole with an area of about 1 % of the chamber water surface area....... In the numerical model, this damping is modeled by an equivalent linearized damping coefficient which extracts the same amount of energy over one cycle as the experimentally measured quadratic damping coefficient. The pressure in each chamber in regular waves of three different height-to-length ratios is measured...
Metamaterial Absorber for Electromagnetic Waves in Periodic Water Droplets
Yoo, Young Joon; Ju, Sanghyun; Park, Sang Yoon; Ju Kim, Young; Bong, Jihye; Lim, Taekyung; Kim, Ki Won; Rhee, Joo Yull; Lee, YoungPak
2015-01-01
Perfect metamaterial absorber (PMA) can intercept electromagnetic wave harmful for body in Wi-Fi, cell phones and home appliances that we are daily using and provide stealth function that military fighter, tank and warship can avoid radar detection. We reported new concept of water droplet-based PMA absorbing perfectly electromagnetic wave with water, an eco-friendly material which is very plentiful on the earth. If arranging water droplets with particular height and diameter on material surface through the wettability of material surface, meta-properties absorbing electromagnetic wave perfectly in GHz wide-band were shown. It was possible to control absorption ratio and absorption wavelength band of electromagnetic wave according to the shape of water droplet–height and diameter– and apply to various flexible and/or transparent substrates such as plastic, glass and paper. In addition, this research examined how electromagnetic wave can be well absorbed in water droplets with low electrical conductivity unlike metal-based metamaterials inquiring highly electrical conductivity. Those results are judged to lead broad applications to variously civilian and military products in the future by providing perfect absorber of broadband in all products including transparent and bendable materials. PMID:26354891
Metamaterial Absorber for Electromagnetic Waves in Periodic Water Droplets.
Yoo, Young Joon; Ju, Sanghyun; Park, Sang Yoon; Ju Kim, Young; Bong, Jihye; Lim, Taekyung; Kim, Ki Won; Rhee, Joo Yull; Lee, YoungPak
2015-09-10
Perfect metamaterial absorber (PMA) can intercept electromagnetic wave harmful for body in Wi-Fi, cell phones and home appliances that we are daily using and provide stealth function that military fighter, tank and warship can avoid radar detection. We reported new concept of water droplet-based PMA absorbing perfectly electromagnetic wave with water, an eco-friendly material which is very plentiful on the earth. If arranging water droplets with particular height and diameter on material surface through the wettability of material surface, meta-properties absorbing electromagnetic wave perfectly in GHz wide-band were shown. It was possible to control absorption ratio and absorption wavelength band of electromagnetic wave according to the shape of water droplet-height and diameter- and apply to various flexible and/or transparent substrates such as plastic, glass and paper. In addition, this research examined how electromagnetic wave can be well absorbed in water droplets with low electrical conductivity unlike metal-based metamaterials inquiring highly electrical conductivity. Those results are judged to lead broad applications to variously civilian and military products in the future by providing perfect absorber of broadband in all products including transparent and bendable materials.
Metamaterial Absorber for Electromagnetic Waves in Periodic Water Droplets
Yoo, Young Joon; Ju, Sanghyun; Park, Sang Yoon; Ju Kim, Young; Bong, Jihye; Lim, Taekyung; Kim, Ki Won; Rhee, Joo Yull; Lee, Youngpak
2015-09-01
Perfect metamaterial absorber (PMA) can intercept electromagnetic wave harmful for body in Wi-Fi, cell phones and home appliances that we are daily using and provide stealth function that military fighter, tank and warship can avoid radar detection. We reported new concept of water droplet-based PMA absorbing perfectly electromagnetic wave with water, an eco-friendly material which is very plentiful on the earth. If arranging water droplets with particular height and diameter on material surface through the wettability of material surface, meta-properties absorbing electromagnetic wave perfectly in GHz wide-band were shown. It was possible to control absorption ratio and absorption wavelength band of electromagnetic wave according to the shape of water droplet-height and diameter- and apply to various flexible and/or transparent substrates such as plastic, glass and paper. In addition, this research examined how electromagnetic wave can be well absorbed in water droplets with low electrical conductivity unlike metal-based metamaterials inquiring highly electrical conductivity. Those results are judged to lead broad applications to variously civilian and military products in the future by providing perfect absorber of broadband in all products including transparent and bendable materials.
Fu, Lei
2017-05-11
Full-waveform inversion of land seismic data tends to get stuck in a local minimum associated with the waveform misfit function. This problem can be partly mitigated by using an initial velocity model that is close to the true velocity model. This initial starting model can be obtained by inverting traveltimes with ray-tracing traveltime tomography (RT) or wave-equation traveltime (WT) inversion. We have found that WT can provide a more accurate tomogram than RT by inverting the first-arrival traveltimes, and empirical tests suggest that RT is more sensitive to the additive noise in the input data than WT. We present two examples of applying WT and RT to land seismic data acquired in western Saudi Arabia. One of the seismic experiments investigated the water-table depth, and the other one attempted to detect the location of a buried fault. The seismic land data were inverted by WT and RT to generate the P-velocity tomograms, from which we can clearly identify the water table depth along the seismic survey line in the first example and the fault location in the second example.
SHALLOW WATER EQUATION SOLUTION IN 2D USING FINITE DIFFERENCE METHOD WITH EXPLICIT SCHEME
Directory of Open Access Journals (Sweden)
Nuraini Nuraini
2017-09-01
Full Text Available Abstract. Modeling the dynamics of seawater typically uses a shallow water model. The shallow water model is derived from the mass conservation equation and the momentum set into shallow water equations. A two-dimensional shallow water equation alongside the model that is integrated with depth is described in numerical form. This equation can be solved by finite different methods either explicitly or implicitly. In this modeling, the two dimensional shallow water equations are described in discrete form using explicit schemes. Keyword: shallow water equation, finite difference and schema explisit. REFERENSI 1. Bunya, S., Westerink, J. J. dan Yoshimura. 2005. Discontinuous Boundary Implementation for the Shallow Water Equations. Int. J. Numer. Meth. Fluids. 47: 1451-1468. 2. Kampf Jochen. 2009. Ocean Modelling For Beginners. Springer Heidelberg Dordrecht. London New York. 3. Rezolla, L 2011. Numerical Methods for the Solution of Partial Diferential Equations. Trieste. International Schoolfor Advanced Studies. 4. Natakussumah, K. D., Kusuma, S. B. M., Darmawan, H., Adityawan, B. M. Dan Farid, M. 2007. Pemodelan Hubungan Hujan dan Aliran Permukaan pada Suatu DAS dengan Metode Beda Hingga. ITB Sain dan Tek. 39: 97-123. 5. Casulli, V. dan Walters, A. R. 2000. An unstructured grid, three-dimensional model based on the shallow water equations. Int. J. Numer. Meth. Fluids. 32: 331-348. 6. Triatmodjo, B. 2002. Metode Numerik Beta Offset. Yogyakarta.
Effects of water saturation on P-wave propagation in fractured coals: An experimental perspective
Liu, Jie; Liu, Dameng; Cai, Yidong; Gan, Quan; Yao, Yanbin
2017-09-01
Internal structure of coalbed methane (CBM) reservoirs can be evaluated through ultrasonic measurements. The compressional wave that propagates in a fractured coal reservoir may indicate the internal coal structure and fluid characteristics. The P-wave propagation was proposed to study the relations between petrophysical parameters (including water saturation, fractures, porosity and permeability) of coals and the P-wave velocity (Vp), using a KON-NM-4A ultrasonic velocity meter. In this study, the relations between Vps and water saturations were established: Type I is mainly controlled by capillary of developed seepage pores. The controlling factors on Type II and Type III are internal homogeneity of pores/fractures and developed micro-fractures, respectively. Micro-fractures density linearly correlates with the Vp due to the fracture volume and dispersion of P-wave; and micro-fractures of types C and D have a priority in Vp. For dry coals, no clear relation exists between porosity, permeability and the Vp. However, as for water-saturated coals, the correlation coefficients of porosity, permeability and Vp are slightly improved. The Vp of saturated coals could be predicted with the equation of Vp-saturated = 1.4952Vp-dry-26.742 m/s. The relation between petrophysical parameters of coals and Vp under various water saturations can be used to evaluate the internal structure in fractured coals. Therefore, these relations have significant implications for coalbed methane (CBM) exploration.
Shallow Water Wave Models with and without Singular Kernel: Existence, Uniqueness, and Similarities
Directory of Open Access Journals (Sweden)
Emile Franc Doungmo Goufo
2017-01-01
Full Text Available After the recent introduction of the Caputo-Fabrizio derivative by authors of the same names, the question was raised about an eventual comparison with the old version, namely, the Caputo derivative. Unlike Caputo derivative, the newly introduced Caputo-Fabrizio derivative has no singular kernel and the concern was about the real impact of this nonsingularity on real life nonlinear phenomena like those found in shallow water waves. In this paper, a nonlinear Sawada-Kotera equation, suitable in describing the behavior of shallow water waves, is comprehensively analyzed with both types of derivative. In the investigations, various fixed-point theories are exploited together with the concept of Piccard K-stability. We are then able to obtain the existence and uniqueness results for the models with both versions of derivatives. We conclude the analysis by performing some numerical approximations with both derivatives and graphical simulations being presented for some values of the derivative order γ. Similar behaviors are pointed out and they concur with the expected multisoliton solutions well known for the Sawada-Kotera equation. This great observation means either of both derivatives is suitable to describe the motion of shallow water waves.
Impulse waves generated by snow avalanches: Momentum and energy transfer to a water body
Zitti, Gianluca; Ancey, Christophe; Postacchini, Matteo; Brocchini, Maurizio
2016-12-01
When a snow avalanche enters a body of water, it creates an impulse wave whose effects may be catastrophic. Assessing the risk posed by such events requires estimates of the wave's features. Empirical equations have been developed for this purpose in the context of landslides and rock avalanches. Despite the density difference between snow and rock, these equations are also used in avalanche protection engineering. We developed a theoretical model which describes the momentum transfers between the particle and water phases of such events. Scaling analysis showed that these momentum transfers were controlled by a number of dimensionless parameters. Approximate solutions could be worked out by aggregating the dimensionless numbers into a single dimensionless group, which then made it possible to reduce the system's degree of freedom. We carried out experiments that mimicked a snow avalanche striking a reservoir. A lightweight granular material was used as a substitute for snow. The setup was devised so as to satisfy the Froude similarity criterion between the real-world and laboratory scenarios. Our experiments in a water channel showed that the numerical solutions underestimated wave amplitude by a factor of 2 on average. We also compared our experimental data with those obtained by Heller and Hager (2010), who used the same relative particle density as in our runs, but at higher slide Froude numbers.
Multipeakedness and groupiness of shallow water waves along Indian coast
Digital Repository Service at National Institute of Oceanography (India)
SanilKumar, V.; Anand, N.M.; AshokKumar, K.; Mandal, S.
and amplitudes of sea waves. JDurnal of GeDphysical Re search, 80(18), 2688-2694. MASSON, D. and CHAro.lJLER, P., 1993. Wave groups, a closer look at spectral methods. CDastal Engineering, 20, 249-275. MATHEW, J.; BABA. M., and KURIAN, N.P., 1995. Mudbanks... Research 1052-1065 West Palm Beach, Florida Fall 2003 Multipeakedness a,nd Groupiness of Shallow Water Waves Along Indian Coast v. Sanil Kumar, N.M. Anand, K. Ashok Kumar, and S. MandaI Ocean Engineering Division National Institute of Oceanography Goa-403...
Calming the Waters or Riding the Waves?
DEFF Research Database (Denmark)
Rydén, Pernille; Kottika, Efthymia; Hossain, Muhammad Ismail
that strengthens their brand. The consumers are empowered by ‘letting this anger out’, from which firms can gain huge attention. Companies can utilize such situations to inform people on their brands’ core values, and initiate discussions of larger societal relevance, which improves the brand awareness and value......Traditional consumer anger management tends to be compromising rather than empowering the brand. This paper conceptualizes and provides a case example on how consumer empowerment and negative emotions can in fact create opportunities for companies to ride the waves of consumer anger in a way...
System for harvesting water wave energy
Wang, Zhong Lin; Su, Yanjie; Zhu, Guang; Chen, Jun
2016-07-19
A generator for harvesting energy from water in motion includes a sheet of a hydrophobic material, having a first side and an opposite second side, that is triboelectrically more negative than water. A first electrode sheet is disposed on the second side of the sheet of a hydrophobic material. A second electrode sheet is disposed on the second side of the sheet of a hydrophobic material and is spaced apart from the first electrode sheet. Movement of the water across the first side induces an electrical potential imbalance between the first electrode sheet and the second electrode sheet.
Mcaninch, G. L.; Myers, M. K.
1980-01-01
The parabolic approximation for the acoustic equations of motion is applied to the study of the sound field generated by a plane wave at or near grazing incidence to a finite impedance boundary. It is shown how this approximation accounts for effects neglected in the usual plane wave reflection analysis which, at grazing incidence, erroneously predicts complete cancellation of the incident field by the reflected field. Examples are presented which illustrate that the solution obtained by the parabolic approximation contains several of the physical phenomena known to occur in wave propagation near an absorbing boundary.
Directory of Open Access Journals (Sweden)
2016-01-01
Full Text Available A description of the Galilean symmetry invariant solutions to the KdV-Burgers equation is reduced to studying of phase trajectories of the corresponding ODE depending on a parameter (the velocity of a shock wave propagation. Exact invariant solutions are simple shock waves that become separatrixes on the phase portrait which always has two singular points for a given value of the parameter. For nonlinear superposition of shock waves the phase portrait contains four singular points; its consequent bifurcations lead to oscillations.
Directory of Open Access Journals (Sweden)
Antonio Gledson Goulart
2013-12-01
Full Text Available In this paper, the equation for the gravity wave spectra in mean atmosphere is analytically solved without linearization by the Adomian decomposition method. As a consequence, the nonlinear nature of problem is preserved and the errors found in the results are only due to the parameterization. The results, with the parameterization applied in the simulations, indicate that the linear solution of the equation is a good approximation only for heights shorter than ten kilometers, because the linearization the equation leads to a solution that does not correctly describe the kinetic energy spectra.
On the Nonlinear Perturbation K(n,m Rosenau-Hyman Equation: A Model of Nonlinear Scattering Wave
Directory of Open Access Journals (Sweden)
Abdon Atangana
2015-01-01
Full Text Available We investigate a nonlinear wave phenomenon described by the perturbation K(n,m Rosenau-Hyman equation within the concept of derivative with fractional order. We used the Caputo fractional derivative and we proposed an iteration method in order to find a particular solution of the extended perturbation equation. We proved the stability and the convergence of the suggested method for solving the extended equation without any restriction on (m,n and also on the perturbations terms. Using the inner product we proved the uniqueness of the special solution. By choosing randomly the fractional orders and m, we presented the numerical solutions.
Zhang, Xiaoen; Chen, Yong
2017-11-01
In this paper, a combination of stripe soliton and lump soliton is discussed to a reduced (3+1)-dimensional Jimbo-Miwa equation, in which such solution gives rise to two different excitation phenomena: fusion and fission. Particularly, a new combination of positive quadratic functions and hyperbolic functions is considered, and then a novel nonlinear phenomenon is explored. Via this method, a pair of resonance kink stripe solitons and rogue wave is studied. Rogue wave is triggered by the interaction between lump soliton and a pair of resonance kink stripe solitons. It is exciting that rogue wave must be attached to the stripe solitons from its appearing to disappearing. The whole progress is completely symmetry, the rogue wave starts itself from one stripe soliton and lose itself in another stripe soliton. The dynamic properties of the interaction between one stripe soliton and lump soliton, rogue wave are discussed by choosing appropriate parameters.
Liu, Lei; Tian, Bo; Wu, Xiao-Yu; Sun, Yan
2018-02-01
Under investigation in this paper is the higher-order rogue wave-like solutions for a nonautonomous nonlinear Schrödinger equation with external potentials which can be applied in the nonlinear optics, hydrodynamics, plasma physics and Bose-Einstein condensation. Based on the Kadomtsev-Petviashvili hierarchy reduction, we construct the Nth order rogue wave-like solutions in terms of the Gramian under the integrable constraint. With the help of the analytic and graphic analysis, we exhibit the first-, second- and third-order rogue wave-like solutions through the different dispersion, nonlinearity and linear potential coefficients. We find that only if the dispersion and nonlinearity coefficients are proportional to each other, heights of the background of those rogue waves maintain unchanged with time increasing. Due to the existence of complex parameters, such nonautonomous rogue waves in the higher-order cases have more complex features than those in the lower.
Wave Loads on Ships Sailing in Restricted Water Depth
DEFF Research Database (Denmark)
Vidic-Perunovic, Jelena; Jensen, Jørgen Juncher
2003-01-01
moment a ship may be subjected to during its operational lifetime. Whereas the influence of forward speed and ship heading with respect to the waves usually is accounted for, the effect of water depth is seldom considered, except in non-linear time domain formulations where a confined water domain must...... be specified anyhow. Usually, two-dimensional strip theories, either linear or non-linear, are applied for actual design cases and these theories are normally based on incident deep-water waves and furthermore apply added mass and damping calculations based on infinite water depth. Only a few papers have...... in the past addressed the influence of water depth on the ship response. In an early work Kim (1968) presented results for the variation of the added mass and hydrodynamic damping and for the heave and pitch motion for a Series 60 model using a relative motion strip theory formulation. A significant reduction...
Analytical approximation and numerical simulations for periodic travelling water waves.
Kalimeris, Konstantinos
2018-01-28
We present recent analytical and numerical results for two-dimensional periodic travelling water waves with constant vorticity. The analytical approach is based on novel asymptotic expansions. We obtain numerical results in two different ways: the first is based on the solution of a constrained optimization problem, and the second is realized as a numerical continuation algorithm. Both methods are applied on some examples of non-constant vorticity.This article is part of the theme issue 'Nonlinear water waves'. © 2017 The Author(s).
Balance-characteristic scheme as applied to the shallow water equations over a rough bottom
Goloviznin, V. M.; Isakov, V. A.
2017-07-01
The CABARET scheme is used for the numerical solution of the one-dimensional shallow water equations over a rough bottom. The scheme involves conservative and flux variables, whose values at a new time level are calculated by applying the characteristic properties of the shallow water equations. The scheme is verified using a series of test and model problems.
Student Misconceptions in Writing Balanced Equations for Dissolving Ionic Compounds in Water
Naah, Basil M.; Sanger, Michael J.
2012-01-01
The goal of this study was to identify student misconceptions and difficulties in writing symbolic-level balanced equations for dissolving ionic compounds in water. A sample of 105 college students were asked to provide balanced equations for dissolving four ionic compounds in water. Another 37 college students participated in semi-structured…
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Shahnam Javadi
2013-07-01
Full Text Available In this paper, the $(G'/G$-expansion method is applied to solve a biological reaction-convection-diffusion model arising in mathematical biology. Exact traveling wave solutions are obtained by this method. This scheme can be applied to a wide class of nonlinear partial differential equations.
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Giai Giang Vo
2015-01-01
Full Text Available This paper is devoted to the study of a wave equation with a boundary condition of many-point type. The existence of weak solutions is proved by using the Galerkin method. Also, the uniqueness and the stability of solutions are established.
Guo, Changhong; Fang, Shaomei
2017-10-01
This paper studied the planar, solitary, and spiral waves of the coupled Burgers-complex Ginzburg-Landau (Burgers-CGL) equations, which were derived from the nonlinear evolution of the coupled long-scale oscillatory and monotonic instabilities of a uniformly propagating combustion wave governed by a sequential chemical reaction having two flame fronts corresponding to two reaction zones with a finite separation distance between them. First, some exact solutions including the planar and solitary waves for the one-dimensional Burgers-CGL equations that are obtained by subtle transforms and the hyperbolic tangent function expansion method. Second, some spiral waves for the two-dimensional Burgers-CGL equations are investigated. The existence of the spiral waves is proved rigorously by Schauder's fixed point theorem applied to a class of functions, and the approximate solutions are also obtained by the power series expansion method. Furthermore, some numerical simulations are carried out near 0 mathematics, and the results verify the theoretical analysis.
Water vapor estimation using digital terrestrial broadcasting waves
Kawamura, S.; Ohta, H.; Hanado, H.; Yamamoto, M. K.; Shiga, N.; Kido, K.; Yasuda, S.; Goto, T.; Ichikawa, R.; Amagai, J.; Imamura, K.; Fujieda, M.; Iwai, H.; Sugitani, S.; Iguchi, T.
2017-03-01
A method of estimating water vapor (propagation delay due to water vapor) using digital terrestrial broadcasting waves is proposed. Our target is to improve the accuracy of numerical weather forecast for severe weather phenomena such as localized heavy rainstorms in urban areas through data assimilation. In this method, we estimate water vapor near a ground surface from the propagation delay of digital terrestrial broadcasting waves. A real-time delay measurement system with a software-defined radio technique is developed and tested. The data obtained using digital terrestrial broadcasting waves show good agreement with those obtained by ground-based meteorological observation. The main features of this observation are, no need for transmitters (receiving only), applicable wherever digital terrestrial broadcasting is available and its high time resolution. This study shows a possibility to estimate water vapor using digital terrestrial broadcasting waves. In the future, we will investigate the impact of these data toward numerical weather forecast through data assimilation. Developing a system that monitors water vapor near the ground surface with time and space resolutions of 30 s and several kilometers would improve the accuracy of the numerical weather forecast of localized severe weather phenomena.
Fourth-order wave equation in Bhabha-Madhavarao spin-3 2 theory
Markov, Yu. A.; Markova, M. A.; Bondarenko, A. I.
2017-09-01
Within the framework of the Bhabha-Madhavarao formalism, a consistent approach to the derivation of a system of the fourth-order wave equations for the description of a spin-3 2 particle is suggested. For this purpose an additional algebraic object, the so-called q-commutator (q is a primitive fourth root of unity) and a new set of matrices ημ, instead of the original matrices βμ of the Bhabha-Madhavarao algebra, are introduced. It is shown that in terms of the ημ matrices we have succeeded in reducing a procedure of the construction of fourth root of the fourth-order wave operator to a few simple algebraic transformations and to some operation of the passage to the limit z → q, where z is some (complex) deformation parameter entering into the definition of the η-matrices. In addition, a set of the matrices 𝒫1/2 and 𝒫3/2(±)(q) possessing the properties of projectors is introduced. These operators project the matrices ημ onto the spins 1/2- and 3/2-sectors in the theory under consideration. A corresponding generalization of the obtained results to the case of the interaction with an external electromagnetic field introduced through the minimal coupling scheme is carried out. The application to the problem of construction of the path integral representation in para-superspace for the propagator of a massive spin-3 2 particle in a background gauge field within the Bhabha-Madhavarao approach is discussed.
Travelling-wave amplitudes as solutions of the phase-field crystal equation
Nizovtseva, I. G.; Galenko, P. K.
2018-01-01
The dynamics of the diffuse interface between liquid and solid states is analysed. The diffuse interface is considered as an envelope of atomic density amplitudes as predicted by the phase-field crystal model (Elder et al. 2004 Phys. Rev. E 70, 051605 (doi:10.1103/PhysRevE.70.051605); Elder et al. 2007 Phys. Rev. B 75, 064107 (doi:10.1103/PhysRevB.75.064107)). The propagation of crystalline amplitudes into metastable liquid is described by the hyperbolic equation of an extended Allen-Cahn type (Galenko & Jou 2005 Phys. Rev. E 71, 046125 (doi:10.1103/PhysRevE.71.046125)) for which the complete set of analytical travelling-wave solutions is obtained by the http://www.w3.org/1999/xlink" xlink:href="RSTA20170202IM1"/> method (Malfliet & Hereman 1996 Phys. Scr. 15, 563-568 (doi:10.1088/0031-8949/54/6/003); Wazwaz 2004 Appl. Math. Comput. 154, 713-723 (doi:10.1016/S0096-3003(03)00745-8)). The general http://www.w3.org/1999/xlink" xlink:href="RSTA20170202IM2"/> solution of travelling waves is based on the function of hyperbolic tangent. Together with its set of particular solutions, the general http://www.w3.org/1999/xlink" xlink:href="RSTA20170202IM3"/> solution is analysed within an example of specific task about the crystal front invading metastable liquid (Galenko et al. 2015 Phys. D 308, 1-10 (doi:10.1016/j.physd.2015.06.002)). The influence of the driving force on the phase-field profile, amplitude velocity and correlation length is investigated for various relaxation times of the gradient flow. This article is part of the theme issue `From atomistic interfaces to dendritic patterns'.
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Hafiz Abdul Wajid
2014-01-01
Full Text Available We construct modified forward, backward, and central finite difference schemes, specifically for the Helmholtz equation, by using the Bloch wave property. All of these modified finite difference approximations provide exact solutions at the nodes of the uniform grid for the second derivative present in the Helmholtz equation and the first derivative in the radiation boundary conditions for wave propagation. The most important feature of the modified schemes is that they work for large as well as low wave numbers, without the common requirement of a very fine mesh size. The superiority of the modified finite difference schemes is illustrated with the help of numerical examples by making a comparison with standard finite difference schemes.
Generation of Focused Shock Waves in Water for Biomedical Applications
Lukeš, Petr; Šunka, Pavel; Hoffer, Petr; Stelmashuk, Vitaliy; Beneš, Jiří; Poučková, Pavla; Zadinová, Marie; Zeman, Jan
The physical characteristics of focused two-successive (tandem) shock waves (FTSW) in water and their biological effects are presented. FTSW were generated by underwater multichannel electrical discharges in a highly conductive saline solution using two porous ceramic-coated cylindrical electrodes of different diameter and surface area. The primary cylindrical pressure wave generated at each composite electrode was focused by a metallic parabolic reflector to a common focal point to form two strong shock waves with a variable time delay between the waves. The pressure field and interaction between the first and the second shock waves at the focus were investigated using schlieren photography and polyvinylidene fluoride (PVDF) shock gauge sensors. The largest interaction was obtained for a time delay of 8-15 μs between the waves, producing an amplitude of the negative pressure phase of the second shock wave down to -80 MPa and a large number of cavitations at the focus. The biological effects of FTSW were demonstrated in vitro on damage to B16 melanoma cells, in vivo on targeted lesions in the thigh muscles of rabbits and on the growth delay of sarcoma tumors in Lewis rats treated in vivo by FTSW, compared to untreated controls.
Chugunov, A. I.
2017-10-01
I suggest a novel approach for deriving evolution equations for rapidly rotating relativistic stars affected by radiation-driven Chandrasekhar-Friedman-Schutz instability. This approach is based on the multipolar expansion of gravitational wave emission and appeals to the global physical properties of the star (energy, angular momentum, and thermal state), but not to canonical energy and angular momentum, which is traditional. It leads to simple derivation of the Chandrasekhar-Friedman-Schutz instability criterion for normal modes and the evolution equations for a star, affected by this instability. The approach also gives a precise form to simple explanation of the Chandrasekhar-Friedman-Schutz instability; it occurs when two conditions are met: (a) gravitational wave emission removes angular momentum from the rotating star (thus releasing the rotation energy) and (b) gravitational waves carry less energy, than the released amount of the rotation energy. To illustrate the results, I take the r-mode instability in slowly rotating Newtonian stellar models as an example. It leads to evolution equations, where the emission of gravitational waves directly affects the spin frequency, being in apparent contradiction with widely accepted equations. According to the latter, effective spin frequency decrease is coupled with dissipation of unstable mode, but not with the instability as it is. This problem is shown to be superficial, and arises as a result of specific definition of the effective spin frequency applied previously. Namely, it is shown, that if this definition is taken into account properly, the evolution equations coincide with obtained here in the leading order in mode amplitude. I also argue that the next-to-leading order terms in evolution equations were not yet derived accurately and thus it would be more self-consistent to omit them.
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Nguyen Thanh Long
2005-12-01
Full Text Available In this paper we consider the nonlinear wave equation problem $$displaylines{ u_{tt}-Big(|u|_0^2,|u_{r}|_0^2ig(u_{rr}+frac{1}{r}u_{r} =f(r,t,u,u_{r},quad 0less than r less than 1,; 0 less than t less than T, ig|lim_{ro 0^+}sqrt{r}u_{r}(r,tig| less than infty, u_{r}(1,t+hu(1,t=0, u(r,0=widetilde{u}_0(r, u_{t}(r,0=widetilde{u}_1(r. }$$ To this problem, we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved, in weighted Sobolev using standard compactness arguments. In the latter part, we give sufficient conditions for quadratic convergence to the solution of the original problem, for an autonomous right-hand side independent on $u_{r}$ and a coefficient function $B$ of the form $B=B(|u|_0^2=b_0+|u|_0^2$ with $b_0$ greater than 0.
Wide-azimuth angle gathers for anisotropic wave-equation migration
Sava, Paul C.
2012-10-15
Extended common-image-point gathers (CIP) constructed by wide-azimuth TI wave-equation migration contain all the necessary information for angle decomposition as a function of the reflection and azimuth angles at selected locations in the subsurface. The aperture and azimuth angles are derived from the extended images using analytic relations between the space- and time-lag extensions using information which is already available at the time of migration, i.e. the anisotropic model parameters. CIPs are cheap to compute because they can be distributed in the image at the most relevant positions, as indicated by the geologic structure. If the reflector dip is known at the CIP locations, then the computational cost can be reduced by evaluating only two components of the space-lag vector. The transformation from extended images to angle gathers is a planar Radon transform which depends on the local medium parameters. This transformation allows us to separate all illumination directions for a given experiment, or between different experiments. We do not need to decompose the reconstructed wavefields or to choose the most energetic directions for decomposition. Applications of the method include illumination studies in complex areas where ray-based methods fail, and assuming that the subsurface illumination is sufficiently dense, the study of amplitude variation with aperture and azimuth angles. © 2012 European Association of Geoscientists & Engineers.
Energy decay for wave equations of phi-Laplacian type with weakly nonlinear dissipation
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Aissa Guesmia
2008-08-01
Full Text Available In this paper, first we prove the existence of global solutions in Sobolev spaces for the initial boundary value problem of the wave equation of $phi$-Laplacian with a general dissipation of the form $$ (|u'|^{l-2}u''-Delta_{phi}u+sigma(t g(u'=0 quadext{in } Omegaimes mathbb{R}_+ , $$ where $Delta_{phi}=sum_{i=1}^n partial_{x_i}igl(phi (|partial_{x_i}|^2partial_{x_i}igr$. Then we prove general stability estimates using multiplier method and general weighted integral inequalities proved by the second author in [18]. Without imposing any growth condition at the origin on $g$ and $phi$, we show that the energy of the system is bounded above by a quantity, depending on $phi$, $sigma$ and $g$, which tends to zero (as time approaches infinity. These estimates allows us to consider large class of functions $g$ and $phi$ with general growth at the origin. We give some examples to illustrate how to derive from our general estimates the polynomial, exponential or logarithmic decay. The results of this paper improve and generalize many existing results in the literature, and generate some interesting open problems.
Predoi, Mihai Valentin
2014-09-01
The dispersion curves for hollow multilayered cylinders are prerequisites in any practical guided waves application on such structures. The equations for homogeneous isotropic materials have been established more than 120 years ago. The difficulties in finding numerical solutions to analytic expressions remain considerable, especially if the materials are orthotropic visco-elastic as in the composites used for pipes in the last decades. Among other numerical techniques, the semi-analytical finite elements method has proven its capability of solving this problem. Two possibilities exist to model a finite elements eigenvalue problem: a two-dimensional cross-section model of the pipe or a radial segment model, intersecting the layers between the inner and the outer radius of the pipe. The last possibility is here adopted and distinct differential problems are deduced for longitudinal L(0,n), torsional T(0,n) and flexural F(m,n) modes. Eigenvalue problems are deduced for the three modes classes, offering explicit forms of each coefficient for the matrices used in an available general purpose finite elements code. Comparisons with existing solutions for pipes filled with non-linear viscoelastic fluid or visco-elastic coatings as well as for a fully orthotropic hollow cylinder are all proving the reliability and ease of use of this method. Copyright © 2014 Elsevier B.V. All rights reserved.
Automatic Wave Equation Migration Velocity Analysis by Focusing Subsurface Virtual Sources
Sun, Bingbing
2017-11-03
Macro velocity model building is important for subsequent pre-stack depth migration and full waveform inversion. Wave equation migration velocity analysis (WEMVA) utilizes the band-limited waveform to invert for the velocity. Normally, inversion would be implemented by focusing the subsurface offset common image gathers (SOCIGs). We re-examine this concept with a different perspective: In subsurface offset domain, using extended Born modeling, the recorded data can be considered as invariant with respect to the perturbation of the position of the virtual sources and velocity at the same time. A linear system connecting the perturbation of the position of those virtual sources and velocity is derived and solved subsequently by Conjugate Gradient method. In theory, the perturbation of the position of the virtual sources is given by the Rytov approximation. Thus, compared to the Born approximation, it relaxes the dependency on amplitude and makes the proposed method more applicable for real data. We demonstrate the effectiveness of the approach by applying the proposed method on both isotropic and anisotropic VTI synthetic data. A real dataset example verifies the robustness of the proposed method.
Said-Houari, Belkacem
2012-09-01
The goal of this work is to study a model of the viscoelastic wave equation with nonlinear boundary/interior sources and a nonlinear interior damping. First, applying the Faedo-Galerkin approximations combined with the compactness method to obtain existence of regular global solutions to an auxiliary problem with globally Lipschitz source terms and with initial data in the potential well. It is important to emphasize that it is not possible to consider density arguments to pass from regular to weak solutions if one considers regular solutions of our problem where the source terms are locally Lipschitz functions. To overcome this difficulty, we use an approximation method involving truncated sources and adapting the ideas in [13] to show that the existence of weak solutions can still be obtained for our problem. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term, then the solution ceases to exist and blows up in finite time provided that the initial data are large enough.
Dutta, Gaurav
2014-10-01
Strong subsurface attenuation leads to distortion of amplitudes and phases of seismic waves propagating inside the earth. Conventional acoustic reverse time migration (RTM) and least-squares reverse time migration (LSRTM) do not account for this distortion, which can lead to defocusing of migration images in highly attenuative geologic environments. To correct for this distortion, we used a linearized inversion method, denoted as Qp-LSRTM. During the leastsquares iterations, we used a linearized viscoacoustic modeling operator for forward modeling. The adjoint equations were derived using the adjoint-state method for back propagating the residual wavefields. The merit of this approach compared with conventional RTM and LSRTM was that Qp-LSRTM compensated for the amplitude loss due to attenuation and could produce images with better balanced amplitudes and more resolution below highly attenuative layers. Numerical tests on synthetic and field data illustrated the advantages of Qp-LSRTM over RTM and LSRTM when the recorded data had strong attenuation effects. Similar to standard LSRTM, the sensitivity tests for background velocity and Qp errors revealed that the liability of this method is the requirement for smooth and accurate migration velocity and attenuation models.
A perfectly matched layer for the time-dependent wave equation in heterogeneous and layered media
Duru, Kenneth
2014-01-01
A mathematical analysis of the perfectly matched layer (PML) for the time-dependent wave equation in heterogeneous and layered media is presented. We prove the stability of the PML for discontinuous media with piecewise constant coefficients, and derive energy estimates for discontinuous media with piecewise smooth coefficients. We consider a computational setup consisting of smaller structured subdomains that are discretized using high order accurate finite difference operators for approximating spatial derivatives. The subdomains are then patched together into a global domain by a weak enforcement of interface conditions using penalties. In order to ensure the stability of the discrete PML, it is necessary to transform the interface conditions to include the auxiliary variables. In the discrete setting, the transformed interface conditions are crucial in deriving discrete energy estimates analogous to the continuous energy estimates, thus proving stability and convergence of the numerical method. Finally, we present numerical experiments demonstrating the stability of the PML in a layered medium and high order accuracy of the proposed interface conditions. © 2013 Elsevier Inc.
Energy Technology Data Exchange (ETDEWEB)
Múnera, Héctor A., E-mail: hmunera@hotmail.com [Centro Internacional de Física (CIF), Apartado Aéreo 4948, Bogotá, Colombia, South America (Colombia); Retired professor, Department of Physics, Universidad Nacional de Colombia, Bogotá, Colombia, South America (Colombia)
2016-07-07
It is postulated that there exists a fundamental energy-like fluid, which occupies the flat three-dimensional Euclidean space that contains our universe, and obeys the two basic laws of classical physics: conservation of linear momentum, and conservation of total energy; the fluid is described by the classical wave equation (CWE), which was Schrödinger’s first candidate to develop his quantum theory. Novel solutions for the CWE discovered twenty years ago are nonharmonic, inherently quantized, and universal in the sense of scale invariance, thus leading to quantization at all scales of the universe, from galactic clusters to the sub-quark world, and yielding a unified Lorentz-invariant quantum theory ab initio. Quingal solutions are isomorphic under both neo-Galilean and Lorentz transformations, and exhibit nother remarkable property: intrinsic unstability for large values of ℓ (a quantum number), thus limiting the size of each system at a given scale. Unstability and scale-invariance together lead to nested structures observed in our solar system; unstability may explain the small number of rows in the chemical periodic table, and nuclear unstability of nuclides beyond lead and bismuth. Quingal functions lend mathematical basis for Boscovich’s unified force (which is compatible with many pieces of evidence collected over the past century), and also yield a simple geometrical solution for the classical three-body problem, which is a useful model for electronic orbits in simple diatomic molecules. A testable prediction for the helicoidal-type force is suggested.
Solutions of selected pseudo loop equations in water distribution ...
African Journals Online (AJOL)
This paper demonstrated the use of Microsoft Excel Solver (a computer package) in solving selected pseudo loop equations in pipe network analysis problems. Two pipe networks with pumps and overhead tanks were used to demonstrate the use of Microsoft Excel Solver in solving pseudo loops (open loops; networks with ...
Molding acoustic, electromagnetic and water waves with a single cloak.
Xu, Jun; Jiang, Xu; Fang, Nicholas; Georget, Elodie; Abdeddaim, Redha; Geffrin, Jean-Michel; Farhat, Mohamed; Sabouroux, Pierre; Enoch, Stefan; Guenneau, Sébastien
2015-06-09
We describe two experiments demonstrating that a cylindrical cloak formerly introduced for linear surface liquid waves works equally well for sound and electromagnetic waves. This structured cloak behaves like an acoustic cloak with an effective anisotropic density and an electromagnetic cloak with an effective anisotropic permittivity, respectively. Measured forward scattering for pressure and magnetic fields are in good agreement and provide first evidence of broadband cloaking. Microwave experiments and 3D electromagnetic wave simulations further confirm reduced forward and backscattering when a rectangular metallic obstacle is surrounded by the structured cloak for cloaking frequencies between 2.6 and 7.0 GHz. This suggests, as supported by 2D finite element simulations, sound waves are cloaked between 3 and 8 KHz and linear surface liquid waves between 5 and 16 Hz. Moreover, microwave experiments show the field is reduced by 10 to 30 dB inside the invisibility region, which suggests the multi-wave cloak could be used as a protection against water, sonic or microwaves.