Deep Water Limit Research Articles

The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves

We analyze the self-focusing of gravity-capillary surface waves modeled by the Davey-Stewartson equations. This system governs the coupling of the wave amplitude to the induced mean flow and is an anisotropic two-dimensional Schrödinger equation with a non local cubic nonlinearity. With accurate numerical simulations based on dynamic rescaling and asymptotic analysis we show that the dynamics of singularity formation is critical, as in the usual two-dimensional cubic Schrödinger equation, which is the deep water limit of the Davey-Stewartson equations. We also derive a sharp upper bound for the initial amplitude of the wave that prevents singularity formation.

Two-dimensional evolution of surface gravity waves on a fluid of arbitrary depth.

Nonlinear evolution equations (NEE's) are derived that describe weakly-two-dimensional motion of surface-gravity waves on a three-dimensional incompressible and inviscid fluid. When compared with the Kadomtsev-Petviashvili (KP) equation, the NEE's presented here have the following two advantages: (a) they are applicable to wave phenomena on a fluid of arbitrary depth; and (b) they can deal with head-on collisions of various wave structures. A discussion is then made of NEE's arising from the shallow- and deep-water limits. It is shown that the KP equation is a special case of our equations.

Nonlinear evolution of surface gravity waves over an uneven bottom

A unified theory is developed which describes nonlinear evolution of surface gravity waves propagating over an uneven bottom in the case of two-dimensional incompressible and inviscid fluid of arbitrary depth. Under the assumptions that the bottom of the fluid has a slowly varying profile and the wave steepness is small, a system of approximate nonlinear evolution equations (NEEs) for the surface elevation and the horizontal component of surface velocity is derived on the basis of a systematic perturbation method with respect to the steepness parameter. A single NEE for the surface elevation is also presented. These equations are expressed in terms of original coordinate variables and therefore they have a direct relevance to physical systems. Since the formalism does not rely on the often used assumptions of shallow water and long waves, the NEEs obtained are uniformly valid from shallow water to deep water and have wide applications in various wave phenomena of physical and engineering importance. The shallow- and deep-water limits of the equations are discussed and the results are compared with existing theories. It is found that our theory includes as specific cases almost all approximate theories known at present.

Nonlinear evolutions of surface gravity waves on fluid of finite depth.

New types of nonlinear evolution equations are derived that describe finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of finite depth. The novelty is as follows: (a) The equation can be expressed as a single equation with respect to the surface elevation, and (b) the expansion is in a steepness parameter and does not involve other approximations such as long waves or shallow water. Both the shallow- and deep-water limits of the equations are discussed.

Wave Numbers of Linear Progressive Waves

An explicit solution for wave number and wave speed is derived. The method of solution is to expand the linear wave dispersion relation in Taylor series in the neighborhoods of deep and shallow water limits. The two series are matched at the inflection point of the dispersion function, where the shoaling wave height is a minimum value. Numerical comparisons indicate that the present solution is computationally efficient and has an error of less than 0.05%. Two applications are made to solve problems, involving nearshore spectral analysis and refraction of ocean waves.

Forces and moment on a horizontal plate due to wave scattering

Wave reflection and transmission from a fixed horizontal plate have been widely studied but theoretical solutions are only available for certain limiting conditions. A general solution for this wave scattering problem is presented using the finite-element method, covering the whole range of relative depth ratio from shallow to deep water limits and submergence depth ratio from the water surface to the bed. Existing long-wave solutions for the surface plate and the submerged plate have been extended to obtain the hydrodynamic forces and overturning moment exerted on the plate. Results from the finite-element program compare well with these solutions. Variations of the reflection coefficient, wave forces and moment, with the plate width to wave length ratio, relative depth ratio and submergence depth ratio are discussed.

SHALLOW WATER WAVES: A SPECTRAL APPROACH

Bouws et al. (1983, 1984) have shown that wind sea spectra in finite depth water can be described by a self-similar spectral equation that in the deep water limit is the JONSWAP spectrum (Hasselmann et al. 1973). This paper shows that the spectral parameter a is linked to wave steepness, for wind sea and swell; presents a simple model for wave transformation across the surf zone; and shows that the spectral theory provides data similar to the results of Bretschneider (1958) for shallow water wave growth.

Direct and inverse scattering problems of the nonlinear intermediate long wave equation

The inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed. The equation describes long internal gravity waves in a stratified fluid of finite depth, and reduces to the Korteweg–de Vries equation as shallow water limit and the Benjamin–Ono equation as deep water limit. Both limits of the method and novel aspects of the theory are also discussed.

N-Soliton Solution of the Higher Order Wave Equation for a Fluid of Finite Depth

Exact N -soliton solution of the higher order wave equation for a stratified fluid layer of finite depth is obtained by employing Hirota's bilinear transformation method. The solution reduces to the N -soliton solution of the higher order Korteweg-de Vries equation in the shallow water limit and to that of the higher order Benjamin-One equation in the deep water limit. The asymptotic form of the solution for large time is derived.

Exact One-and Two-Periodic Wave Solutions of Fluids of Finite Depth

Exact one- and two-periodic wave solutions of a nonlinear integro-differential equation describing the propagation of waves in a stratified fluid of finite depth is presented. The solution reduces to the Korteweg-de Vries (KdV) periodic wave solution in the shallow water limit and to the Benjamin-Ono (B-O) periodic wave solution in the deep water limit.

Comment on "Internal-wave solitons of fluids with finite depth"

The assertion by Chen and Lee that known finite-depth-soliton solutions do not smoothly tend to Benjamin-Ono solitons in the deep-water limit is incorrect.

Exact multi-soliton solution for nonlinear waves in a stratified fluid of finite depth

The exact multi-soliton solution of a nonlinear integro-differential equation describing the propagation of waves in a stratified fluid of finite depth is presented. The solution reduces to the Korteweg-de Vries (KdV) soliton solution is the shallow water limit and to the Benjamin-Ono (BO) soliton solution in the deep water limit. The asymptotic form of the solution for large times is derived.

On an internal wave equation describing a stratified fluid with finite depth

In this note we give a Bäcklund transformation, a recursion scheme for the infinite number of conservation laws and an inverse scattering problem for an integro-differential equation governing long internal gravity waves in a stratified fluid of finite total depth. They are shown to reduce to those of the Korteweg—de Vries equation in the shallow water limit and those of the Benjamin—Ono equation in the deep water limit.

Weakly-Nonlinear, Long Internal Gravity Waves in Stratified Fluids of Finite Depth

This paper presents an analytical investigation of the propagation of a weakly-nonlinear, long internal gravity wave in a stratified medium of finite total depth. The governing equation is derived and shown to reduce to the KdV equation in the shallow-water limit and to the Benjamin/Ono equation in the deep-water limit. The equation is also shown to possess four conserved quantities, just as was the case for the deep-water waves considered by Ono. The equation suggests the existence of a steady-state waveform described by two parameters which degenerates into the one-parameter, steady-state waveforms discussed by Benjamin. A numerical approach using Fornberg's pseudospectral method is used to examine the solution. The results demonstrate the existence of a solitary wavelike steady-state solution with a solitonlike behavior. The effect of finite water depth on the waveform and the wave speed of the steady-state solitary waves is discussed.

The wave height variation for regular waves in shoaling water

The theory of cnoidal wave shoaling has previously been connected to data in deeper water (described by sinusoidal wave theory) by assuming continuity in energy flux E f at the matching point between the two theories (Svendsen and Brink-Kjær, 1972) resulting in a small shift in wave height H at the matching point. In this paper we assume directly continuity in wave height and tables are given from which the wave height variation can be determined when the wave is specified at any water depth (including deep water). It is also shown that a deep water limit for cnoidal waves exists. The nature of this limit and the behaviour of the waves close to it are analysed. Experimental data for waves with no free second harmonic components are compared with a shoaling model based on linear theory for h/L 0 > 0.10 and cnoidal theory for h/L 0 < 0.10 . The agreement is very good for deep water wave steepnesses H 0 /L 0 up to 3–4%. For larger steepnesses the linear theory fails to predict the variation for h/L 0 > 0.10 .

Mass transport in water waves

The theory of mass transport in water waves is re-examined. Boundary-layer arguments are incorporated into the Lagrangian equations of motion, and the mass transport velocity is conveniently derived for both monochromatic and random waves. For finite depth, Longuet-Higgins' result is confirmed. Conditions at the far end of a wave tank are found to be related to a mean pressure gradient, and results are also obtained for a truly infinite channel. The nonuniformity of the solution for the deep-water limit is discussed.