Water Waves: Relating Modern Theory to Advanced Engineering Applications

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.

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

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 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.

Long waves in shallow water propagating over a background shear flow towards a sloping beach are being investigated. The classical shallow-water equations are extended to incorporate both a background shear flow and a linear beach profile, resulting in a non-reducible hyperbolic system. Nevertheless, it is shown how several changes of variables based on the hodograph transform may be used to transform the system into a linear equation which may be solved exactly using the method of separation of variables. This method can be used to investigate the run-up of a long wave on a planar beach including the development of the waterline.

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.

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.

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.

: LONG-TERM GOAL. The first goal of this effort is to achieve a better understanding of the role of capillary waves in ocean surface phenomena. A second goal is to investigate the propagation of ocean waves into shallow water, as on beaches and reefs, and the generation of capillary waves from these. OBJECTIVES. Objectives of this research are to demonstrate a physical model and a practical computational model for investigating capillary wave interactions with longer waves and to apply this to applications, such as generation and spectra in both wavenumber and frequency. Further objectives are to model nonlinear waves in shallow water, including the generation of capillary waves by these.

Nonlinear water waves in shallow water in the presence of constant vorticity: A Whitham approach