Three-Dimensional Water Waves

We study the evolution of small-amplitude water waves when the fluid motion is three dimensional. An isotropic pseudodifferential equation that governs the evolution of the free surface of a fluid with arbitrary, uniform depth is derived. It is shown to reduce to the Benney-Luke equation, the Korteweg-de Vries (KdV) equation, the Kadomtsev-Petviashvili (KP) equation, and to the nonlinear shallow water theory in the appropriate limits. We compute, numerically, doubly periodic solutions to this equation. In the weakly two-dimensional long wave limit, the computed patterns and nonlinear dispersion relations agree well with those of the doubly periodic theta function solutions to the KP equation. These solutions correspond to traveling hexagonal wave patterns, and they have been compared with experimental measurements by Hammack, Scheffner, and Segur. In the fully two-dimensional long wave case, the solutions deviate considerably from those of KP, indicating the limitation of that equation. In the finite depth case, both resonant and nonresonant traveling wave patterns are obtained.     

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 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.     

The Propagation of Finite-Amplitude Waves in a Model Boundary Layer

Finite-amplitude wave propagation is considered in flows of boundary-layer type when the wavelength is long compared to the boundary layer thickness. In this limit, the evolution of the amplitude is governed by the Benjamin-Ono equation and we have computed the coefficients of its nonlinear and dispersive terms for the specific case of Tietjens's model. The propagation of wave packets is also considered, and it is found that for packets centered about an O(1) wavenumber questions again arise relative to long waves, except that now the packet-induced mean flow is the “long wave.” By introducing an appropriate scaling for the far field and employing multiple scales in the direction transverse to the flow, it is shown how the mean-flow distortion can be made to vanish at infinity.     

Well-posedness of the Ostrovsky–Hunter Equation under the combined effects of dissipation and short-wave dispersion

The Ostrovsky–Hunter equation provides a model for small-amplitude long waves in a rotating fluid of finite depth. It is a nonlinear evolution equation. In this paper we study the well-posedness for the Cauchy problem associated with this equation in presence of some weak dissipation effects.     

Long Internal Waves in Fluids of Great Depth

An equation is derived that governs the evolution in two spatial dimensions of long internal waves in fluids of great depth. The equation is a natural generalization of Benjamin's (1967) one-dimensional equation, and relates to it in the same way that the equation of Kadomtsev and Petviashvili relates to the Kortewegde-Vries equation. The stability of one-dimensional solitons with respect to long transverse disturbances is studied in the context of this equation. Solitons are found to be unstable with respect to such perturbations in any system in which the phase speed is a minimum (rather than a maximum) for the longest waves. Internal waves do not have this property, and are not unstable with respect to such perturbations.     

A model equation for nonlinear wavelength selection and amplitude evolution of frontal waves

Summary We study a model equation describing the temporal evolution of nonlinear finite-amplitude waves on a density front in a rotating fluid. The linear spectrum includes an unstable interval where exponential growth of the amplitude is expected. It is shown that the length scale of the waves in the nonlinear situation is determined by the linear instabilities; the effect of the nonlinearities is to limit the amplitude's growth, leaving the wavelength unchanged. When linearly stable waves are prescribed as initial data, a short interval of rapid decrease in amplitude is encountered first, followed by a transfer of energy to the unstable part of the spectrum, where the fastest growing mode starts to dominate. A localized disturbance is broken up into its Fourier components, the linearly unstable modes grow at the expense of all other modes, and final amplitudes are determined by the nonlinear term. Periodic evolution of linearly unstable waves in the nonlinear situation is also observed. Based on the numerical results, the existence of low-order chaos in the partial differential equation governing weakly nonlinear wave evolution is conjectured.     

Evolution Equations for Long,Nonlinear Internal Waves in Stratified Shear Flows

Evolution equations for long, nonlinear internal waves are derived when the basic stratified shear flow has a slow temporal and spatial variation as well as the usual dependence on the vertical coordinate. When the horizontal waveguide has a limited vertical extent the evolution equation is a variable coefficient Korteweg-deVries equation, while in the deep fluid case the evolution equation is a variable coefficient Benjamin-Davis-Ono equation. Explicit expressions are obtained for the coefficients of these equations.     

Stability on Time-Dependent Domains

We explore the key differences in the stability picture between extended systems on time-fixed and time-dependent spatial domains. As a paradigm, we take the complex Swift–Hohenberg equation, which is the simplest nonlinear model with a finite critical wavenumber, and use it to study dynamic pattern formation and evolution on time-dependent spatial domains in translationally invariant systems, i.e., when dilution effects are absent. In particular, we discuss the effects of a time-dependent domain on the stability of spatially homogeneous and spatially periodic base states, and explore its effects on the Eckhaus instability of periodic states. New equations describing the nonlinear evolution of the pattern wavenumber on time-dependent domains are derived, and the results compared with those on fixed domains. Pattern coarsening on time-dependent domains is contrasted with that on fixed domains with the help of the Cahn–Hilliard equation extended here to time-dependent domains. Parallel results for the evolution of the Benjamin–Feir instability on time-dependent domains are also given.     

Nonlinear wave propagation through a stratified atmosphere

We examine the propagation of sound waves through a stratified atmosphere. The method of multiple scales is employed to obtain an asymptotic equation which describes the evolution of sound waves in an atmosphere with spatially dependant density and entropy fields. The evolution equation is an inviscid Burger-like equation which contains quadratic and cubic nonlinearities, and a curvature term all of which are functions of the space variables. A model equation is derived when the modulations of the signal in a direction transverse to the direction of propagation become significant.     

A Note on the Transmission of Water Waves Through Small Apertures

An integral equation formulation is used to evaluate the transmissioncoefficient when small amplitude water waves impinge on an aperturein a vertical wall whose maximum dimension is assumed to besmall compared with both the depth of submersion of the apertureand the wave-length of the incident wave.     

Analytic study on the Sawada–Kotera equation with a nonvanishing boundary condition in fluids

Under investigation in this paper is the Sawada–Kotera equation with a nonvanishing boundary condition, which describes the evolution of steeper waves of shorter wavelength than those described by the Korteweg–de Vries equation does. With the binary-Bell-polynomial, Hirota method and symbolic computation, the bilinear form and N-soliton solutions for this model are derived. Meanwhile, propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis. Via Bell-polynomial approach, the Bäcklund transformation is constructed in both the binary-Bell-polynomial and bilinear forms. Based on the binary-Bell-polynomial-type Bäcklund transformation, we obtain the Lax pair and conservation laws associated.     

Wave-Mean Flow Interactions in Thermally Stratified Poiseuille Flow

We consider nonlinear wave motions in thermally stratified Poiseuille flow. Attention is focused on short wavelength wave modes for which the neutral Reynolds number scales as the square of the wave number. The nonlinear evolution of a single monochromatic wave is governed by a first harmonic/mean-flow interaction theory in which the wave-induced mean flow is comparable in size to the wave component of the flow. An integrodifferential equation is derived which governs the normal variation of the wave amplitude. This equation admits finite-amplitude solutions which bifurcate supercritically from the linear neutral point(s).     

Oleinik type estimates for the Ostrovsky–Hunter equation

The Ostrovsky–Hunter equation provides a model of small-amplitude long waves in a rotating fluid of finite depth. This is a nonlinear evolution equation. In this study, we consider the well-posedness of the Cauchy problem associated with this equation within a class of bounded discontinuous solutions. We show that we can replace the Kruzkov-type entropy inequalities with an Oleinik-type estimate and we prove the uniqueness via a nonlocal adjoint problem. This implies that a shock wave in an entropy weak solution to the Ostrovsky–Hunter equation is admissible only if it jumps down in value (similar to the inviscid Burgers' equation).     

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.     

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.     

High-frequency asymptotics for the numerical solution of the Helmholtz equation

It is often noted that the Helmholtz equation is extremely difficult to solve, in particular, for high-frequency solutions for heterogeneous media. Since stability for second-order discretization methods requires one to choose at least 10–12 grid points per wavelength, the discrete problem on the possible coarsest mesh is huge. In a realistic simulation, one is required to choose 20–30 points per wavelength to achieve a reasonable accuracy; this problem is hard to solve. This article is concerned with the high-frequency asymptotic decomposition of the wavefield for an efficient and accurate simulation for the high-frequency numerical solution of the Helmholtz equation. It has been numerically verified that the new method is accurate enough even when one chooses 4–5 grid points per wavelength.     

Evolution of Packets of Surface Gravity Waves over Strong Smooth Topography

Wave packets in a smoothly inhomogeneous medium are governed by a nonlinear Schrödinger (NLS) equation with variable coefficients. There are two spatial scales in the problem: the spatial scale of the inhomogeneities and the distance over which nonlinearity and dispersion affect the packet. Accordingly, there are two limits where the problem can be approached asymptotically: when the former scale is much larger than the latter, and vice versa. In this paper, we examine the limit where the spatial scale of (periodic or random) inhomogeneities is much smaller than that of nonlinearity/dispersion (i.e., the latter effects are much weaker than the former). In this case, the packet undergoes rapid oscillations of the geometric-optical type, and also evolves slowly due to nonlinearity and dispersion. We demonstrate that the latter evolution is governed by an NLS equation with constant (averaged) coefficients. The general theory is illustrated by the example of surface gravity waves in a channel of variable depth. In particular, it is shown that topography increases the critical frequency, for which the nonlinearity coefficient of the NLS equation changes sign (in such cases, no steady solutions exist, i.e., waves with frequencies lower than the critical one disperse and cannot form packets).