Introduction

In this chapter we consider waves at the interface between two fluids of different density and waves in multi-layer fluids. We derive the fully nonlinear equations for waves of arbitrary amplitude and then determine approximate formulae appropriate for small-amplitude waves. These are expressed through coupled linear partial differential equations for which analytic solutions may be found through standard methods. Approximate solutions for nonlinear waves are considered in Chapter 4.

A ‘one-layer fluid’ has uniform density and is bounded above by a free surface that may oscillate up and down, for example, due to gravitational forces. Waves in a one-layer fluid are specifically referred to here as ‘surface waves’. Waves on the ocean surface can be treated as existing in a one-layer fluid if the density variation with depth in the ocean negligibly affects their dynamics.

A ‘two-layer fluid’ describes a fluid of one density underlying a second fluid of smaller density. ‘Interfacial waves’ propagate at the interface between the two fluids. In one sense, surface waves on the ocean are interfacial waves in that they propagate at the interface between water and air. However, in this book we distinguish surface waves from interfacial waves by requiring for the latter that the density difference between the upper and lower layer fluids is a small fraction of the density of either layer. Thus undular displacements of an interface between fresh and salt water are considered for waves in a two-layer fluid.