Mathematical Modelling Of Ocean Waves

Abstract

This study is concerned about finding simple solution techniques to determine the wave-wave interactions of four progressive waves travelling with four different wavenumbers and frequencies in a deep ocean. It has been demonstrated by many previous researchers including Phillips [6] and Hasselmann et.al [I] that this set of four waves, called a quadruplet, could exchange energy if they interact nonlinearly such that the resonance conditions are satisfied. In this paper, however, we shall be concerned with the approximate solutions of the nonlinear transfer action functions which satisfy a set of first order ordinary differential equations using the JONS WAP spectrum as the initial input condition. We will demonstrate three simple methods to compute the nonlinear transfer action functions within an infinitesimal interaction phase-space element Ak. The solutions by two analytical methods namely, Picard's iteration and Bernoulli's integrating factor methods are compared with those obtained by the fourth order Runge-Kutta numerical scheme. The comparison shows favourable agreement. In this paper, the complete solution of the nonlinear source function Sni has been avoided.