16 - Coupled wave equations

Summary

Introduction

For wave propagation in a stratified medium, the idea of writing the governing differential equations as a set of coupled equations has already been used in §§ 7.8, 7.15, for the study of W.K.B. solutions. The term ‘coupled equations’ is usually given to a set of simultaneous ordinary linear differential equations with the following properties:

1. (1) There is one independent variable which in this book is the height z or a linear function of it.

2. (2) The number of equations is the same as the number of dependent variables.

3. (3) In each equation one dependent variable appears in derivatives up to a higher order than any other. The terms in this variable are called ‘principal’ terms and the remaining terms are called ‘coupling’ terms.

4. (4) The principal terms contain a different dependent variable in each equation, so that each dependent variable appears in the principal terms of one and only one equation.

It is often possible to choose the dependent variables so that the coupling terms are small over some range of z. Then the equations may be solved by successive approximations. As a first approximation the coupling terms are neglected, and the resulting equations can then be solved. The values thus obtained for the dependent variables are substituted in the coupling terms and the resulting equations are solved to give a better approximation. Some examples of this process are given in § 16.13.