4 - A first look at surface waves and instabilities

Summary

Introduction

One of the ironies of wave phenomena is that those waves which are most easily observed, such as waves on the surface of water, are more difficult to analyse theoretically than waves which propagate through a medium, such that their presence has to be inferred indirectly. Surface waves are more easily observed than bulk waves but mathematically it is more difficult to deal with them. The main reason for this is that for surface waves a boundary condition, such as continuity, must be satisfied at the surface of the wave which of course is not generally a simple plane. This was indicated briefly in the Introduction. Fortunately for linearized wave theory, the boundary condition has to be satisfied on the unperturbed surface which is usually planar. If the medium on either side of the boundary is homogeneous, the problem reduces to matching algebraic expressions at the interface. In this manner one obtains an algebraic dispersion relation for surface waves analogous to a bulk dispersion relation. However, in most practical situations the boundary is diffuse and this necessitates solving differential equations, with non-constant coefficients, for the behaviour perpendicular to the boundary. An algebraic dispersion relation is then replaced by an eigenvalue equation with a consequent increase in the difficulty of the problem.

The basic mathematical techniques are illustrated in Section 4.2 by considering the propagation of waves along the surface of a liquid.