A time-dependent nonlinear mild slope equation for water waves (original) (raw)

Fully Nonlinear Model for Water Wave Propagation from Deep to Shallow Waters

2011

A set of fully nonlinear Boussinessq-type equations (BTEs) with improved linear and nonlinear dispersive performance is presented. The equations are so that the highest order of the derivatives is three and they use the minimum number of unknowns: the free surface elevation and the horizontal velocity at a certain depth. The equations allow to reduce the errors both in linear frequency dispersion and shoaling below 0.30% for kh 5, and below 2.2% for kh 10, being k the wave number and h the water depth. The weakly nonlinear performance is also improved for kh 2. A simple fourth order explicit numerical scheme is presented so as to test the linear and nonlinear behavior of the model equations against analytical and experimental results.

A new nonlinear equation in the shallow water wave problem

Physica Scripta, 2014

In the paper a new nonlinear equation describing shallow water waves with the topography of the bottom directly taken into account is derived. This equation is valid in the weakly nonlinear, dispersive and long wavelength limit. Some examples of soliton motion for various bottom shapes obtained in numerical simulations according to the derived equation are presented.

A Fully Dispersive Weakly Nonlinear Model for Water Waves

Proceedings of the Royal Society of London a Mathematical Physical and Engineering Sciences, 1997

A fully dispersive weakly nonlinear water wave model is developed via a new approach named the multiterm-coupling technique, in which the velocity field is represented by a few vertical-dependence functions having different wave-numbers. This expression of velocity, which is approximately irrotational for variable depth, is used to satisfy the continuity and momentum equations. The Galerkin method is invoked to obtain a solvable set of coupled equations for the horizontal velocity components and shown to provide an optimum combination of the prescribed depth-dependence functions to represent a random wave-field with diversely varying wave-numbers. The new wave equations are valid for arbitrary ratios of depth to wavelength and therefore it is possible to recover all the well-known linear and weakly nonlinear wave models as special cases. Numerical simulations are carried out to demonstrate that a wide spectrum of waves, such as random deep water waves and solitary waves over constant depth as well as nonlinear random waves over variable depth, is well reproduced at affordable computational cost.

The Linearized Dispersion Relation of Navier-Stokes Equation in Shallow Water

— A few number of variants of Navier-Stokes equations and their higher order generalizations are derived to describe the two way propagation of small amplitude, wavelength, gravity waves on the surface of water in a canal. These systems are also seen to model the propagation of long-crested waves on large lakes or the ocean and in other contexts. Depending on linearized terms and positing a solution of the form e i(kx-wt) , the wave frequency is formulated in terms of wave number. Finally, the phase speed has been established with viscous term for long wavelength where the first three terms correspond to the expansion of the full linearized dispersion relation.

Dispersive Nonlinear Shallow-Water Equations

Studies in Applied Mathematics, 2009

A set of dispersive and hyperbolic depth-averaged equations is obtained using a hyperbolic approximation of a chosen set of fully nonlinear and weakly dispersive Boussinesq-type equations. These equations provide, at a reasonably reduced cost, both a physically sound description of the nearshore dynamics and a complete representation of dispersive and nonlinear wave phenomena. A detailed description of the conditioning of the dispersive terms and a physical interpretation of the hyperbolic approximation is provided. The dispersive and hyperbolic structure of the new set of equations is analyzed in depth and an analytical solitary-wave solution is found.

Nonlinear progressive waves in water of finite depth — an analytic approximation

Coastal Engineering, 2007

An analytical solution using homotopy analysis method is developed to describe the non-8 linear progressive waves in water of finite depth. The velocity potential of the wave is ex-9 pressed by Fourier series and the nonlinear free surface boundary conditions are satisfied by 10 continuous mapping. Unlike the perturbation method, the present approach is not dependent 11 on small parameters. Thus solutions are possible for steep waves. Furthermore, a significant 12 improvement of the convergence rate and region is achieved by applying Homotopy-Padé 13 Approximants. The calculated wave characteristics of the present solution agree well with 14 previous numerical and experimental results.

A Fully-Dispersive Nonlinear Wave Model and its Numerical Solutions

Coastal Engineering 1994, 1995

A set of fully-dispersive nonlinear wave equations is derived by introducing a velocity expression with a few vertical-dependence functions and then applying the Galerkin method, which provides an optimum combination of the verticaldependence functions to express an arbitrary velocity field under wave motion. The obtained equations can describe nonlinear non-breaking waves under general conditions, such as nonlinear random waves with a wide-banded spectrum at an arbitrary depth including very shallow and far deep water depths. The single component forms of the new wave equations, one of which is referred to here as "time-dependent nonlinear mild-slope equation", are shown to produce various existing wave equations such as Boussinesq and mild-slope equations as their degenerate forms. Numerical examples with comparison to experimental data are given to demonstrate the validity of the present wave equations and their high performance in expressing not only wave profiles but also velocity fields.

A Coupled-Mode, Fully-dispersive, Weakly-nonlinear Model for Water Waves over a General Bathymetry

In this paper a coupled-mode system of horizontal equations is derived with the aid of Luke's (1967) variational principle, which models the evolution of nonlinear water waves in intermediate depth over a general bathymetry. The vertical structure of the wave field is exactly represented by means of a local-mode series expansion of the wave potential, . This series contains the usual propagating and evanescent modes, plus two additional modes, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The system fully accounts for the effects of non-linearity and dispersion. In the present work the fully nonlinear coupled-mode system is simplified keeping only up to second-order terms, and the derived weakly non-linear model is applied to water waves propagating over a flat bottom and over an arbitrary bathymetry, in the time and in the frequency domain.

New Analytical Solution for Nonlinear Shallow Water-Wave Equations

Pure and Applied Geophysics, 2017

Majority of the hodograph transform solutions of the one-dimensional nonlinear shallow-water wave equations are obtained through integral transform techniques. This approach, however, might involve evaluation of elliptic integrals, which are highly singular. Here, we couple the hodograph transform approach with the classical eigenfunction expansion method rather than integral transform techniques and present a new analytical model for nonlinear long wave propagation over a plane beach. In contrast to classical initial or boundary value problem solutions, an initial-boundary value problem solution is formulated. In general, initial wave profile with nonzero initial velocity distribution is assumed and the flow variables are given in the form of Fourier-Bessel series. The spatial and temporal variation of the flow quantities, i.e., free-surface height and depth-averaged velocity, are estimated accurately through the developed method with much less computational effort compared to the existing integral transform techniques.