Effect of the shape of initial perturbation on a reflected long wave from a beach

Abstract

Изучаются асимптотические решения двумерного волнового уравнения с переменной скоростью и вырождением на границе области (берега). Рассматривается задача Коши с локализованными начальными данными, отвечающая поршневой модели цунами. Приведена асимптотическая формула для решения, работающая в малой окрестности берега. Исследуется вопрос о симметрии набегающей и отраженной волн. Этот же вопрос изучается для волнового уравнения с правой частью, отвечающей распределенному по времени источнику. The paper addresses the two-dimensional wave equation with variable velocity in a bounded domain. The velocity is assumed to degenerate on the boundary of the domain (the shore) as a square root of the distance to the boundary. We consider the Cauchy problem with localized initial data corresponding to the piston tsunami waves model. This problem is studied from the viewpoint of the asymptotic theory, where the small parameter µ is set by the ratio of the characteristic wave length to the characteristic size of the domain (the ocean). We propose an asymptotic formula for the solution working in a neighborhood of the shore of order µ . We study the symmetry between an incoming and reflected wave profiles. It turns out that profile shape does not change if the Fourier transform of the initial source function is real. This happens because the wave profile is close to an eigenfunction of the Hilbert transform. We also study the symmetry of profiles for the inhomogeneous wave equation. The right-hand side of this equation corresponds to a time spread source as opposed to instantaneous one in the piston model. This linear problem is a first step in studying more complicated system of the shallow water equations. The latter system is nonlinear, however in view of the results due to Carrier and Greenspan, its solution can be found if the solution of the linearized problem is known.