Homogenization Method in the Problem of Long Wave Propagation from a Localized Source in a Basin over an Uneven Bottom

In the framework of the linearized shallow water equations, the homogenization method for wave type equations with rapidly oscillating coefficients that generally cannot be represented as periodic functions of the fast variables is applied to the Cauchy problem for the wave equation describing the evolution of the free surface elevation for long waves propagating in a basin over an uneven bottom. Under certain conditions on the function describing the basin depth, we prove that the solution of the homogenized equation asymptotically approximates the solution of the original equation. Model homogenized wave equations are constructed for several examples of one-dimensional sections of the real ocean bottom profile, and their numerical and asymptotic solutions are compared with numerical solutions of the original equations.     

A Higher-Order Internal Wave Model Accounting for Large Bathymetric Variations

A higher-order strongly nonlinear model is derived to describe the evolution of large amplitude internal waves over arbitrary bathymetric variations in a two-layer system where the upper layer is shallow while the lower layer is comparable to the characteristic wavelength. The new system of nonlinear evolution equations with variable coefficients is a generalization of the deep configuration model proposed by Choi and Camassa [ 1 ] and accounts for both a higher-order approximation to pressure coupling between the two layers and the effects of rapidly varying bottom variation. Motivated by the work of Rosales and Papanicolaou [ 2 ], an averaging technique is applied to the system for weakly nonlinear long internal waves propagating over periodic bottom topography. It is shown that the system reduces to an effective Intermediate Long Wave (ILW) equation, in contrast to the Korteweg-de Vries (KdV) equation derived for the surface wave case.     

广义缓坡方程

运用表面波Hamilton方法和缓坡逼近假定,分析缓变三维流场和非平整海底对波浪传播的影响,推导出广义缓坡方程。海底地形由两个分量组成:慢变分量,其水平长度尺度大于表面波的波长;快变分量,其振幅与表面波的波长相比为一小量,但是其频率与表面波频率相当。该广义缓坡方程具有广泛的适用范围,以下著名的缓坡方程成为它的特例:经典的Berkhoff缓坡方程;包含环境流效应的Kirby缓坡方程;描述波状海底特征的Dingemans缓坡方程。另外,同时也得到了描述环境流场和快变海底效应的广义浅水方程。     

A Long Wave Approximation for Capillary-Gravity Waves and an Effect of the Bottom

The Korteweg–de Vries (KdV) equation is known as a model of long waves in an infinitely long canal over a flat bottom and approximates the 2-dimensional water wave problem, which is a free boundary problem for the incompressible Euler equation with the irrotational condition. In this article, we consider the validity of this approximation in the case of the presence of the surface tension. Moreover, we consider the case where the bottom is not flat and study an effect of the bottom to the long wave approximation. We derive a system of coupled KdV like equations and prove that the dynamics of the full problem can be described approximately by the solution of the coupled equations for a long time interval. We also prove that if the initial data and the bottom decay at infinity in a suitable sense, then the KdV equation takes the place of the coupled equations.     

Do peaked solitary water waves indeed exist?

Many models of shallow water waves, such as the famous Camassa–Holm equation, admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa–Holm equation. Besides, it is proved that Kelvin’s theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, unlike the traditional smooth waves that are dispersive with wave height, the phase speed of the peaked solitary waves has nothing to do with wave height, but depends (for a fixed wave height) on its decay length, i.e., the actual wavelength: in fact, the peaked solitary waves are dispersive with the actual wavelength when wave height is fixed. In addition, unlike traditional smooth waves whose kinetic energy decays exponentially from free surface to bottom, the kinetic energy of the peaked solitary waves either increases or almost keeps the same. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.     

Water waves over a rough bottom in the shallow water regime

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.     

Exact Free Surface Flows for Shallow Water Equations I: The Incompressible Case

A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well-known self-similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.     

Rotational waves generated by current‐topography interaction

We study nonlinear free‐surface rotational waves generated through the interaction of a vertically sheared current with a topography. Equivalently, the waves may be generated by a pressure distribution along the free surface. A forced Korteweg–de Vries equation (fKdV) is deduced incorporating these features. The weakly nonlinear, weakly dispersive reduced model is valid for small amplitude topographies. To study the effect of gradually increasing the topography amplitude, the free surface Euler equations are formulated in the presence of a variable depth and a sheared current of constant vorticity. Under constant vorticity, the harmonic velocity component is formulated in a simplified canonical domain, through the use of a conformal mapping which flattens both the free surface as well as the bottom topography. Critical, supercritical, and subcritical Froude number regimes are considered, while the bottom amplitude is gradually increased in both the irrotational and rotational wave regimes. Solutions to the fKdV model are compared to those from the Euler equations. We show that for rotational waves the critical Froude number is shifted away from 1. New stationary solutions are found and their stability tested numerically.     

Well-Posedness of a Problem of Propagation of Nonlinear Acoustic-Gravity Waves in the Atmosphere Generated by Surface Pressure Variations

Currently there are many international microbarograph networks for high-resolution recording of wave pressure variations on the Earth’s surface. This arouses interest in wave propagation in the atmosphere generated by atmospheric pressure variations. A full system of nonlinear hydrodynamic equations for atmospheric gases with lower boundary conditions in the form of wavelike pressure variations on the Earth’s surface is considered. Since the wave amplitudes near the Earth’s surface are small, linearized equations are used in the analysis of well-posedness of the problem. With the help of a wave energy functional method, it is shown that in the non-dissipative case the solution to the boundary value problem is uniquely determined by the variable pressure field on the Earth’s surface. The corresponding dissipative problem is well-posed if, in addition to the pressure field, appropriate conditions on the velocity and temperature on the Earth’s surface are given. In the case of an isothermal atmosphere, the problem admits analytical solutions that are harmonic in the variables x and t. A good agreement between the numerical and analytical solutions is obtained. The study shows that the temperature and density can rapidly vary at the lower boundary of the boundary value problem. An example of solving the three-dimensional problem with variable pressure on the Earth’s surface taken from experimental observations is given. The developed algorithms and computer programs can be used to simulate atmospheric waves generated by pressure variations on the Earth’s surface.     

Breakdown of the Whitham Modulation Theory and the Emergence of Dispersion

The Whitham modulation theory for periodic traveling waves of PDEs generated by a Lagrangian produces first‐order dispersionless PDEs that are, generically, either hyperbolic or elliptic. In this paper, degeneracy of the Whitham equations is considered where one of the characteristic speeds is zero. In this case, the Whitham equations are no longer valid. Reformulation and rescaling show that conservation of wave action morphs into the Korteweg–de Vries (KdV) equation on a longer time scale thereby generating dispersion in the Whitham modulation equations even for finite amplitude waves.     

Approximation of solutions of the two-dimensional wave equation with variable velocity and localized right-hand side using some “simple” solutions

Asymptotic solutions based on the characteristics and the modified Maslov canonical operator of the two-dimensional wave equation with variable coefficients and right-hand side corresponding to: (a) an instantaneous source; (b) a rapidly acting, but “time spread,” source, are compared. An algorithm for approximating a (more complicated) solution of problem (b) by linear combinations of the derivatives of the (simpler) solution of problem (a) is proposed. Numerical calculations showing the accuracy of this approximation are presented. The replacement of the solutions of problem (b) by those of problem (a) becomes especially important in the case where the wave equation is considered in the domain with boundary on which the velocity of the wave equation vanishes. Then the characteristics of the problem become singular (nonstandard) and solutions of type (a) generalize to the case referred to above in a much simpler and effective way than solutions of type (b). Such a situation arises in problems where long waves (for example, tsunami waves) are incident on a sloping seashore.     

Light scattering by a homogeneous sphere near a plane boundary

The scattering of electromagnetic waves by a homogeneous sphere near a plane boundary is presented in this paper. The vector wave equations derived from Maxwell’s equations are solved by means of the two orthogonal solutions to the scalar wave equation. Hankel transformation and Erdélyi’s formula are used to satisfy the planar boundary conditions and the determination of the unknown coefficients in the scattered field and internal fields is achieved by matching the electromagnetic boundary conditions on the surface of the sphere. Existence and uniqueness of the solution of the series involving these unknown coefficients are shown.     

Weak Formulation of Free-Surface Wave Equations

An alternative method for deriving water wave dispersion relations and evolution equations is to use a weak formulation. The free-surface displacement η is written as an eigenfunction expansion, [equation] where the an(t) are time-dependent coefficients. For a tank with vertical sides the E_n are eigenfunctions of the eigenvalue problem, [equation] on the tank side walls. Evolution equations for the a_n(t) can be obtained by taking inner products of the lin-earised equation of motion, [equation] with the normal irrotational wave modes. For unforced waves each evolution equation is a simple harmonic oscillator, but the method is most useful when the body force F is something more exotic than gravity. It can always be represented by a forcing term in the SHM evolution equation, and it is not necessary to assume F irrotational. Several applications are considered: the Faraday experiment, generation of surface waves by an unsteady magnetic field, and the metal-pad instability in aluminium reduction cells.     

The long‐wave limit for the water wave problem I. The case of zero surface tension

The Korteweg‐de Vries equation, Boussinesq equation, and many other equations can be formally derived as approximate equations for the two‐dimensional water wave problem in the limit of long waves. Here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension. We prove that the solutions of the water wave problem in the long‐wave limit split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg‐de Vries equation. Our result allows us to describe the nonlinear interaction of solitary waves. © 2000 John Wiley & Sons, Inc.     

On Lie symmetry analysis,conservation laws and solitary waves to a longitudinal wave motion equation

Under investigation in this work is a longitudinal wave motion equation, which describes the solitary waves propagation with dispersion caused by transverse Poisson’s effect in a magneto-electro-elastic circular rod. The Lie symmetry method is employed to study its vector fields and optimal systems, respectively. Furthermore, the symmetry reductions and eight families of soliton wave solutions of the equation are obtained on the basis of the optimal systems, including hyperbolic-type and trigonometric-type solutions. Two of reduced equations are Painlevé-like equations. Finally, by virtue of conservation law multiplier, the complete set of local conservation laws of the equation for the arbitrary constant coefficients is well constructed with a detailed derivation.     

A note on the wave propagation in water of variable depth

In the present work, utilizing the two dimensional equations of an incompressible inviscid fluid and the reductive perturbation method we studied the propagation of weakly nonlinear waves in water of variable depth. For the case of slowly varying depth, the evolution equation is obtained as the variable coefficient Korteweg-de Vries (KdV) equation. Due to the difficulties for the analytical solutions, a numerical technics so called “the method of integrating factor” is used and the evolution equation is solved under a given initial condition and the bottom topography. It is observed the parameters of bottom topography causes to the changes in wave amplitude, wave profile and the wave speed.     

Navier Stokes model of solitary wave collision

Wave collision and its interaction characteristics is one of the important challenges in coastal engineering. This article concerns the collision of solitary waves over a horizontal bottom considering unsteady, incompressible viscous flow with free surface. The method solves the two dimensional Naiver–Stokes equations for conservation of momentum, continuity equation, and full nonlinear kinematic free-surface equation for Newtonian fluids, as the governing equations in a vertical plan. A mapping was developed to trace the deformed free surface encountered during wave propagation, transforms and interaction by transferring the governing equations from the physical domain to a computational domain. Also a numerical scheme is developed using finite element modeling technique in order to predict the solitary wave collision. Consequently results compared with other researches and show the inelastic behavior of solitary wave collision.     

Model Equations and Traveling Wave Solutions for Shallow-Water Waves with the Coriolis Effect

In the present study, we start by formally deriving the simplified phenomenological models of long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the Earth’s rotation. These new model equations are analogous to the Green–Naghdi equations, the first-order approximations of the KdV-, or BBM type, respectively. We then justify rigorously that in the long-wave limit, unidirectional solutions of a class of KdV- or BBM type are well approximated by the solutions of the Camassa–Holm equation in a rotating setting. The modeling and analysis of those mathematical models then illustrate that the Coriolis forcing in the propagation of shallow-water waves can not be neglected. Indeed, the CH-approximation with the Coriolis effect captures stronger nonlinear effects than the nonlinear dispersive rotational KdV type. Furthermore, we demonstrate nonexistence of the Camassa–Holm-type peaked solution and classify various localized traveling wave solutions to the Camassa–Holm equation with the Coriolis effect depending on the range of the rotation parameter.

     

Interfacial long traveling waves in a two‐layer fluid with variable depth

Long wave propagation in a two‐layer fluid with variable depth is studied for specific bottom configurations, which allow waves to propagate over large distances. Such configurations are found within the linear shallow‐water theory and determined by a family of solutions of the second‐order ordinary differential equation (ODE) with three arbitrary constants. These solutions can be used to approximate the true bottom bathymetry. All such solutions represent smooth bottom profiles between two different singular points. The first singular point corresponds to the point where the two‐layer flow transforms into a uniform one. In the vicinity of this point nonlinear shallow‐water theory is used and the wave breaking criterion, which corresponds to the gradient catastrophe is found. The second bifurcation point corresponds to an infinite increase in water depth, which contradicts the shallow‐water assumption. This point is eliminated by matching the “nonreflecting” bottom profile with a flat bottom. The wave transformation at the matching point is described by the second‐order Fredholm equation and its approximated solution is then obtained. The results extend the theory of internal waves in inhomogeneous stratified fluids actively developed by Prof. Roger Grimshaw, to the new solutions types.     

Internal Kelvin waves in a two-layer liquid model

The subinertial internal Kelvin wave solutions of a linearized system of the ocean dynamics equations for a semi-infinite two-layer f-plane model basin of constant depth bordering a straight, vertical coast are imposed. A rigid lid surface condition and no-slip wall boundary condition are imposed. Some trapped wave equations are presented and approximate solutions using an asymptotic method are constructed. In the absence of bottom friction, the solution consists of a frictionally modified Kelvin wave and a vertical viscous boundary layer. With a no-slip bottom boundary condition, the solution consists of a modified Kelvin wave, two vertical viscous boundary layers, and a large cross-section scale component. The numerical solutions for Kelvin waves are obtained for model parameters that take account of a joint effect of lateral viscosity, bottom friction, and friction between the layers.