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.     

Method of Propagating Waves for a One-Dimensional Inhomogeneous Medium

The aim of this work is to develop a method of propagating waves based on the idea of a wave as a changing state of a medium. This method allows us to represent a solution of the one-dimensional wave equation in an inhomogeneous medium as the sum of two constantly deformed waves, the right wave and the left wave, transported from point to point with coefficients depending on the points and the transport time. By the propagating-wave method we obtain explicit (as far as possible) formulas for solutions of the mixed problem with homogeneous and inhomogeneous boundary conditions and solutions of the Goursat problem. The derivation of these formulas is based on special convolution formulas for the transport coefficients that are similar to the addition identities for trigonometric functions.__________Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 24, pp. 3–43, 2004.     

周期底地形上内波的Hamilton长波展开

给出了周期底部边界条件下两层密度成层流体中2-维非线性长波问题的Hamilton公式．从该公式出发,应用Hamilton摄动理论,导出了底地形短尺度变化下描述双向长波运动的有效Boussinesq方程和描述单向长波运动的近似KdV方程．结果的推导都是在多重尺度算子渐近分析理论框架下完成的．     

有限深两层流中内孤立波的高阶解*

本文研究有限水深两层流中孤立波的三阶近似理论,并考虑了自由表面对孤立波的影响,运用坐标变形方法得到了三阶内孤立波的发展方程,求得波速的解析表达式。对方程进行了数值计算,得到了几种参数下三阶解曲线,指出自由表面对波型和波速的影响是二阶的。计算表明三阶解对一阶、二阶解有明显的改进,使其更加接近试验结果。     

Generalized Eigenfunctions for Waves in Inhomogeneous Media

Many wave propagation phenomena in classical physics are governed by equations that can be recast in Schrödinger form. In this approach the classical wave equation (e.g., Maxwell's equations, acoustic equation, elastic equation) is rewritten in Schrödinger form, leading to the study of the spectral theory of its classical wave operator, a self-adjoint, partial differential operator on a Hilbert space of vector-valued, square integrable functions. Physically interesting inhomogeneous media give rise to nonsmooth coefficients. We construct a generalized eigenfunction expansion for classical wave operators with nonsmooth coefficients. Our construction yields polynomially bounded generalized eigenfunctions, the set of generalized eigenvalues forming a subset of the operator's spectrum with full spectral measure.     

Operator Function Method in the Problem of Normal Waves in an Inhomogeneous Waveguide

The problem of normal waves in a closed (screened) regular waveguiding structure of arbitrary cross-section is considered by reducing it to a boundary value problem for the longitudinal electromagnetic field components in Sobolev spaces. The variational statements of the problem is used to determine the solution. The problem is reduced to studying an operator function. The properties of the operators contained in the operator function necessary to analyze its spectral properties are studied. Theorems on the spectrum discreteness and the distribution of characteristic numbers of the operator function on the complex plane are proved. The problem of completeness of the system of root vectors of the operator function is considered. The theorem on the double completeness of the system of root vectors of the operator function with finite deficiency is proved.     

Long Nonlinear Surface Waves in the Presence of a Variable Current

We investigate the eigenvalue problem governing the propagation of long nonlinear surface waves when there is a current beneath the surface, y being the vertical coordinate. The amplitude of such waves evolves according to the KdV equation and it was proved by Burns [ 1 ] that their speed of propagation c is such that there is no critical layer (i.e., c lies outside the range of ). If, however, the critical layer is nonlinear, the result of Burns does not necessarily apply because the phase change of linear theory then vanishes. In this paper, we consider specific velocity profiles and determine c as a function of Froude number for modes with nonlinear critical layers. Such modes do not always exist, the case of the asymptotic suction profile being a notable example. We find, however, that singular modes can be obtained for boundary layer profiles of the Falkner–Skan similarity type, including the Blasius case. These and other examples are treated and we examine singular solutions of the Rayleigh equation to gain insight about the long wave limit of such solutions.     

密度和刚度线性变化对非均匀地壳层中扭转表面波传播的影响

研究了覆盖在非均匀半无限空间上的非均匀地壳层中,扭转表面波传播的可能性.地壳层的非均匀性随着厚度线性变化,非均匀半无限空间的非均匀性表现为3种类型,即指数型、二次型和双曲型.采用封闭形式,可以分别推导出上述3种类型非均匀性的色散方程.对于覆盖在半空间上的同一地壳层,色散方程与经典案例的方程一致.研究发现,随着非均匀地壳层中密度线性变化的非均匀参数的增大,相速度减小,而由刚度引起的非均匀因素对相速度的影响相反.     

A Numerical Study of the Exact Evolution Equations for Surface Waves in Water of Finite Depth

We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate the exact evolution of nonlinear free surface waves governed by the two-dimensional Euler equations. These equations are written in the transformed plane where the free surface is mapped onto a flat surface and do not require the common assumption that the waves have small amplitude used in deriving the weakly nonlinear Korteweg–de Vries and Boussinesq long-wave equations. We compare the solution of the exact reduced equations with these weakly nonlinear long-wave models and with the nonlinear long-wave equations of Su and Gardner that do not assume the waves have small amplitude. The Su and Gardner solutions are in remarkably close agreement with the exact Euler solutions for large amplitude solitary wave interactions while the interactions of low-amplitude solitary waves of all four models agree. The simulations demonstrate that our method is an efficient and accurate approach to integrate all of these equations and conserves the mass, momentum, and energy of the Euler equations over very long simulations.     

The Superharmonic Instability of Finite-Amplitude Surface Waves on Water of Finite Depth

Saffman's (1985) theory of the superharmonic stability of two-dimensional irrotational waves on fluid of infinite depth has been generalized to solitary and periodic waves of permanent form on finite uniform depth. The frame of reference for the calculation of the Hamiltonian for periodic waves of finite depth is found to be the frame in which the mean horizontal velocity is zero.     

缓变深度分层流体中的准周期波和准孤立波

本文讨论具缓变深度二流体系统中的非线性波,该系统由一不规则底部与一水平固壁间的两层常密度无粘流体所组成.文中用约化摄动法导出了所考虑模型的变系数Korteweg-de Vries方程,并用多重尺度法求出了该方程的近似解,发现底部固壁的不规则变化将产生所谓准周期波和准孤立波.它们的周期、波速和波形将发生缓慢变化,文中给出了准周期波的周期随深度的变化关系式以及准孤立波波幅、波速随深度的变化关系式,底部水平情形和单层流体情形可看成是本文的特例.     

Integrable Nonlinear Equations for Water Waves in Straits of Varying Depth and Width

A considerable amount of information is currently available on the creation and propagation of large solitary waves in marine straits. In order to be able to analyze such data we develop a theoretical model, extending previous one-dimensional models to the case of straits with varying width and depth, and nonvanishing vorticity. Starting from the Euler equations for a three-dimensional homogeneous incompressible inviscid fluid, we derive, in the quasi-one-dimensional long-wave and shallow-water approximation, a generalized KadomtsevPetviashvili (GKP) equation, together with its appropriate boundary conditions. In general, the coefficients of this equation depend on the form of the bottom and on the vorticity; the sides of the straits figure only in the boundary conditions. Under certain restrictions on the vorticity and the geometry of the straits we reduce the GKP equation to one of several completely integrable partial differential equations, in order to study the evolution of solitons which originate in the straits.     

Stability of Standing Waves for Nonlinear Schrödinger Equations with Inhomogeneous Nonlinearities

The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves e^{i ω t}ϕ_ω(x) for a nonlinear Schr?dinger equation with an inhomogeneous nonlinearity V (x)|u|^{p − 1}u, where V (x) is proportional to the electron density. Here, ω > 0 and ϕ_ω(x) is a ground state of the stationary problem. When V (x) behaves like |x|^−b at infinity, where 0 < b < 2, we show that e^{i ω t}ϕ_ω(x) is stable for p < 1 + (4 − 2b)/n and sufficiently small ω > 0. The main point of this paper is to analyze the linearized operator at standing wave solution for the case of V (x) = |x|^−b. Then, this analysis yields a stability result for the case of more general, inhomogeneous V (x) by a certain perturbation method. Communicated by Bernard Helffer submitted 14/07/04, accepted 28/02/05     

The Inverse Scattering Problem for Time-Harmonic Acoustic Waves in an Inhomogeneous Medium: Numerical Experiments

In this paper, the authors describe a new algorithm for solvingthe inverse scattering problem of determining the speed of soundin an inhomogeneous medium from far-field data. Limited testinghas shown that the algorithm has some capacity for reconstructingsimple sound profiles using data over a limited range of frequencies.     

Asymptotic Expansions for One-Phase Soliton-Type Solutions of the Korteweg-De Vries Equation with Variable Coefficients

We construct asymptotic expansions for a one-phase soliton-type solution of the Korteweg-de Vries equation with coefficients depending on a small parameter.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 111–124, January, 2005.     

Solitons in Shallow Seas of Variable Depth and in Marine Straits

A previously derived equation and boundary condition [the generalized Kadomtsev-Petviashvili system] is used to describe the propagation of stable solitary waves in open seas and marine straits. The GKP system is transformed, under specific geophysical conditions, into a simpler system that allows exact soliton type solutions. The curved wave crests corresponding to these solutions are plotted for several choices of the depth function and side boundaries.     

Inhibiting Shear Instability Induced by Large Amplitude Internal Solitary Waves in Two-Layer Flows with a Free Surface

We consider a strongly nonlinear long wave model for large amplitude internal waves in two-layer flows with the top free surface. It is shown that the model suffers from the Kelvin–Helmholtz (KH) instability so that any given shear (even if arbitrarily small) between the layers makes short waves unstable. Because a jump in tangential velocity is induced when the interface is deformed, the applicability of the model to describe the dynamics of internal waves is expected to remain rather limited. To overcome this major difficulty, the model is written in terms of the horizontal velocities at the bottom and the interface, instead of the depth-averaged velocities, which makes the system linearly stable for perturbations of arbitrary wavelengths as long as the shear does not exceed a certain critical value.