4 - Linear Water Waves

Summary

Water waves manifest themselves as disturbances to the free surface of an incompressible fluid with mean depth h and constant, uniform density ρ. Whether the disturbance is generated by the wind, the passage of a ship or a sub-sea earthquake, gravity and/or surface tension will act as restoring forces that tend to drive the fluid towards its equilibrium state. It is the balance between fluid inertia and restoring forces that gives rise to free surface waves. In British coastal waters, where typical sea depths range from a few metres to a hundred metres, 40% of observed waves have amplitudes of 2 m or less, and much longer wavelengths, up to nearly a kilometre in some cases. It would therefore seem worthwhile to develop a linear theory for such waves, based on the assumption that their amplitude is much smaller than their wavelength. We will assume that the flow is laminar, so that there are no breaking waves, no white water and no turbulence. The system is illustrated in figure 4.1. After studying basic linear gravity waves, both progressive and standing, we will move on to consider the generation and propagation of water waves in a variety of situations. These include the harnessing of wave power using a simple mechanical device, the generation of waves by a moving ship and the refraction of waves by changes in bed topography. We will also briefly consider the effects of surface tension and viscosity.