Water Waves

The phenomena of water waves historically provided a great deal of impetus and framework for the development of the theory of dispersive waves. The purpose of this chapter is to give a brief account of the mathematical theory of wavemotion in fluids with a free surface subjected to gravitational and other forces. There are two types of surface‐wave motions. Shallow‐water waves arise when the wavelength of the oscillations is much greater than the depth of the fluid. Here, the vertical acceleration of the fluid is small in comparison to the horizontal acceleration. Surface waves correspond to disturbances that do not extend far below the surface. The wavelength is much less than the depth of the fluid, and the vertical acceleration is, then, no longer negligible. The features that make an analysis of water waves difficult are the presence of nonlinearities; the free surface being unknown a priori , besides being variant with time. In order to make progress with the theory of water waves, it is, in general, necessary to simplify the model by making special hypotheses of one kind or another. These suggest themselves on the basis of general physical circumstances contemplated in a given class of problems. Thus, two approximate theories result when the amplitude of the wave is considered to be small (surface waves); the depth of the water is considered to be small with respect to the wavelength (shallow‐water waves). The first hypothesis leads to a linear theory for boundary‐value problems of nearly classical type. The second leads to a nonlinear theory for initial‐value problems, which, in the lowest order is of the type corresponding to the wave propagation in compressible fluids (see Chapter 15).