Waves due to initial disturbances at the inertial surface in a stratified fluid of finite depth

Prity Ghosh,B N Mandal,Uma Basu

doi:10.1155/s0161171200001605

Abstract

This paper is concerned with a Cauchy-Poisson problem in a weakly stratified ocean of uniform finite depth bounded above by an inertial surface (IS). The inertial surface is composed of a thin but uniform distribution of noninteracting materials. The techniques of Laplace transform in time and either Green's integral theorem or Fourier transform have been utilized in the mathematical analysis to obtain the form of the inertial surface in terms of an integral. The asymptotic behaviour of the inertial surface is obtained for large time and distance and displayed graphically. The effect of stratification is discussed.

Highlights

The classical two-dimensional problem of generation of unsteady motion in deep water due to initial surface disturbances in the form of initial elevation or impulse concentrated at a point on the free surface was studied in the treatise of Lamb [4] and Stoker [9] assuming linear theory
Fourier transform technique was used in the mathematical analysis and the free surface elevation was obtained in the form of an infinite integral which was evaluated asymptotically for large time and distance by the method of stationary phase
We study the problem of wave generation due to initial disturbances in a weakly stratified fluid of finite depth covered by an inertial surface

Summary

Introduction

The classical two-dimensional problem of generation of unsteady motion in deep water due to initial surface disturbances in the form of initial elevation or impulse concentrated at a point on the free surface was studied in the treatise of Lamb [4] and Stoker [9] assuming linear theory. For a weakly stratified fluid with constant Brunt-Vaisala parameter, Debnath and Guha [2] formulated the problem of wave generation due to prescribed initial disturbance of the free surface in terms of an acceleration potential and obtained. The free surface profile asymptotically for large time and distance far away from the region of disturbance Their modelling a deep ocean by a stratified fluid of infinite depth with constant Brunt-Vaisala parameter is questionable since the density at very large depth becomes very large even if the Brunt-Vaisala parameter is small. The form of the inertial surface is obtained in terms of an integral This integral is evaluated asymptotically for large times and distances by the method of stationary phase when the initial disturbances at the inertial surface is concentrated at a point taken as the origin. P(x, y; s) is obtained by employing two methods, one based on an appropriate use of Green’s integral theorem and the other on Fourier integral transform

Methods of solution

Discussion