7. Irregular and Non-linear Waves

The surface waves of the sea are almost always random in the sense that detailed configuration of the surface varies in an irregular manner in both space and time. Section 7.1 contains a brief description of the Wiener spectrum in connection with the generalised Fourier representations for the surface waves (S. Bochner, Vorlesungen über Fouriersche Integrale, Chelsea, 1948 and N. Wiener, The Fourier Integral and certain of Its Applications, Dover, 1933). In this way we see how one may represent the surface elevation by a superposition of harmonic waves with amplitudes being a stochastic process.

The remaining sections in the chapter are devoted to non-linear waves. In Sect. 7.2 we give a systematic derivation of the shallow water theory from the exact hydrodynamical equations as the approximation of lowest order in a perturbation procedure. Here the relevant small parameter is the ratio of the depth of water to some characteristic length associated with the horizontal direction such as the wave length; the water is considered shallow when this parameter is small. It is a different kind of approximation from the previous linear theory for waves of small amplitude. The resulting equations here are quasi-linear and are exactly analogous to the ones in gas dynamics. Second order approximations are included in the last Sect. 7.3. In particular, an asymptotic theory will be developed for slowly varying wave trains, which may be considered as nearly uniform in the regions of order of magnitude of a small number of wave lengths and periods. Some non-linear dispersive wave phenomena will be discussed and more details can be found in H.W. Hoogstraten (Thesis, Delft University of Technology, 1968).