Shallow water dynamics on linear shear flows and plane beaches

A variational formulation of water waves is developed, based on the Hamiltonian theory of surface waves. An exact and unified description of the two-dimensional problem in the vertical plane is obtained in the form of a Hamiltonian functional, expressed in terms of surface quantities as canonical variables. The stability of the corresponding canonical equations can be ensured by using positive definite approximate energy functionals. While preserving full linear dispersion, the method distinguishes between short-wave nonlinearity, allowing the description of Stokes waves in deep water, and long-wave nonlinearity, applying to long waves in shallow water. Both types of nonlinearity are found necessary to describe accurately large-amplitude solitary waves.

The canonical Hamiltonian structure of the equations of fluid dynamics obtained in the Boussinesq approximation are considered. New variational formulations of these equations are proposed and it is found that, as in the case of the KdV equation and the equations governing long waves in shallow water, they are degenerate Lagrangian systems. Therefore, in order to cast these equations into canonical form it is again necessary to use Dirac’s theory of constraints. It is found that there are primary and secondary constraints which are second class and it is possible to construct the Hamiltonian in terms of canonical variables. Among the examples of Boussinesq equations that are discussed are the equations of Whitham–Broer–Kaup which Kupershmidt has recently expressed in symmetric form and shown to admit tri-Hamiltonian structure.

The purpose of this book is to present a self contained introduction to the mathematical and physical aspects of the theory of water waves. The book is aimed at undergraduate and graduate levels for engineers, physical scientists and mathematicians. Each chapter is concluded with practical problems expressed as exercises and accompanied by ample references for further studies. The book consists of ten chapters arranged into three parts: Part I: Basic Fluid Mechanics and Solutions Techniques which cover chapters 1-3. Part II: Water Waves covering chapters 4-7. Part III Advanced Water Waves which covers chapters 8-10. Parts I and II are elementary in nature; whereas Part III is more advanced. The first three chapters give the derivations of the fundamental mathematical equations. Chapter 2 outlines appropriate differential equations to describe the physical phenomena, and Chapter 3 reviews solution techniques of some simplified partial differential equations. Chapter 4 gives the developmental of wave equations, including the essential boundary conditons and describes small amplitude wave theory. Chapter 5 deals with finite amplitude wave theory and Chapter 6 outlines the study of tidal dynamics in shallow water. For random wave case, the deterministic methods described in previous chapters do not hold good. Therefore, chapter 7 is clearly devoted to wave statistics and wave energy spectrum. The application of wave theory is demonstrated in Chapter 8. Chapter 9 examines the nonlinear long waves in shallow water from a mathematical view point. The book concludes with Chapter 10 which illustrates the inverse scattering technique to solve solitary wave problem.

The equation of Camassa and Holm [2]2 is an approximate description of long waves in shallow water.

Two Distinct Approaches for Analytical Solutions and Painlevé Integrability of the $$(3+1)$$-Dimensional Variable Coefficient KP Equation Describing Long Waves in Shallow Water

Under investigation in this paper is a Kadomtsev–Petviashvili–Schrodinger system, which describes the long waves in shallow water. Integrability study has been made through the Painleve test. Via the Hirota method, the bilinear form and N-soliton solutions are obtained with an auxiliary variable. Collision of two solitons is found to be elastic by means of the asymptotic analysis. From the graphical descriptions of the two- and three-soliton solutions, it is found that both the bright and dark solitons collide with one another without any change in the physical quantities except for some small phase shifts during the process of each collision.

This paper adresses the problem of modelling long waves in shallow water, and compares two ways of approach, in the time and in the frequency domains. Finite Element techniques are applied in both cases, and special attentional is paid to the analysis of suitable boundary conditions for the open boundaries. The first part of the paper presents the mathematical models which have been used, i.e., the non-linear shallow water wave equations and its simplification for harmonic waves. This is followed by a description of the boundary conditions corresponding to incident wave, absorbing and reflecting boundaries. The Taylor-Galerkin explicit algorithm developed by Peraire, Morgan and Zienkiewicz [1,2] is used to discretize the non-linear shallow equations, and a classical weighted residual technique for the harmonic wave simplification. The last is solved by Precondicioned Conjugate Gradient technique which reduces the storage requirements. Finally, both methods are applied to simulate the response of Ciudadela harbour, giving similar results.

The difference between wave propagation in dispersive and non-dispersive media can be effectively demonstrated by observing the wave patterns invoked by uniformly moving surface disturbances. Although the dispersion relation of surface waves on water is complicated, there are some frequency intervals where the phase velocity of the waves reduces to the simple power law behavior cp∝ωκ. Among these cases are gravity waves short compared to the depth of the water (κ=−1), short capillary waves (κ=1/3), and long waves in shallow water (κ≈0). Making use of this power-law behavior, we vary the exponent and visualize the smooth transition from a non-dispersive to a dispersive medium.

In this work we study a Kaup-Boussinesq system, which is used in the analysis of long waves in shallow water. Travelling wave solutions are obtained by using direct integration. Secondly, conservation laws are derived by using the multiplier method.

<p>The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology.</p>

The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude long waves in shallow water and other dispersive media. Interest will be turned to the two-point boundary value problem wherein the wave motion is specified at both ends of a finite stretch of the medium of propagation. The principal new result is an exact theory of convergence of the two-point boundary value problem to the quarter-plane boundary value problem in which a semi-infinite stretch of the medium is disturbed at its finite end. The latter problem has been featured in modeling waves generated by a wavemaker in a flume and in describing the evolution of long crested, deep water waves propagating into the near shore zone of large bodies of water. In addition to their intrinsic interest, our results provide justification for the use of the two-point boundary value problem in numerical studies of the quarter plane problem.

formula omitted]-soliton solutions and localized structures for the [formula omitted]-dimensional Broer–Kaup–Kupershmidt system

Under investigation in this paper is the Whitham–Broer–Kaup (WBK) system, which describes the dispersive long wave in shallow water. Through a variable transformation, the WBK system is casted into a general Broer–Kaup system whose Lax pair can be derived by the Ablowitz–Kaup–Newell–Segur technology. With symbolic computation, based on the aforementioned Lax pair, the N-fold Darboux transformation is constructed with a gauge transformation and the multi-soliton solutions are obtained. Finally, the elastic interactions of the two-soliton solutions (including the head-on and overtaking collisions) for the WBK system are graphically studied. Those multi-soliton collisions can be used to illustrate the bidirectional propagation of the waves in shallow water.

Taking the approximate equations for long waves in shallow water as example, the quasi-wavelet discrete scheme is proposed for obtaining numerical solution of the (1+1) dimension nonlinear partial differential equation. In the method, the quasi-wavelet discrete scheme is adopted to discretize the spatial derivative discrete and the ordinary differential equation about time is obtained. Then the fourth order Rung-Katta method is employed to discretize the temporal derivative. Finally the quasi-wavelet solution is compared with the analytical solution, and the computations are validated.

Under investigation in this work is a generalized (2+1)-dimensional Boussinesq equation, which can be used to describe the propagation of small-amplitude, long wave in shallow water. By virtue of Bell's polynomials, an effective way is presented to succinctly construct its bilinear form. Furthermore, based on the bilinear formalism and the extended homoclinic test method, the breather wave solution, rogue-wave solution and solitary-wave solution of the equation are well constructed. Our results can be used to enrich the dynamical behavior of the generalized (2+1)-dimensional nonlinear wave fields.