Introduction

Steady free-streamline flows of water when gravitational forces can be neglected have been discussed in §3.7. Most unsteady free-streamline problems are intractable except by numerical means and generally become more so when gravitational forces are important. However, flows involving gravity where the unsteady motion is a ‘small’ perturbation of a relatively simple mean state occur frequently in the form of surface waves. In the absence of motion the free surface of a liquid in equilibrium under gravity is often ‘horizontal’. A disturbance applied locally that distorts the surface brings into play gravitational restoring forces that cause the disturbance to spread out over the surface in the form of ‘waves’. The waves carry energy away from the source region, propagating parallel to the mean free surface. The agitation produced by a passing wave and the energy flux is generally in the form of a transient disturbance of the fluid particles (around approximately closed particle paths), which are not in themselves transported to any great extent by the wave, and the influence of the wave on fluid at depths exceeding a characteristic wavelength tends to be negligible. In this section these general properties of surface gravity waves are discussed and illustrated by simple examples.

Conditions at the free surface

Consider the simplest case of water whose free surface in equilibrium can be regarded as horizontal and in the plane z = 0 of the coordinate axes (x, y, z), where z increases vertically upwards (Figure 5.1.1).