Fairly-long water waves

Abstract

Water waves have fascinated artists and poets, fishermen and surfers, as well as mathematicians, scientists, and engineers. Recent developments in the understanding of some types of water waves are cast in terms sufficiently general that they can be applied to waves in media other than water, for example, in high temperature plasmas, in gases, and in solids, as well. Historically, many important advances in wave theory have been made with water waves motivating the research. For example, the work of Airy (1845) on long nonlinear water waves preceded the partially analogous work on nonlinear sound waves of Riemann (1858–9) and Earnshaw (1858). The research of Kelvin (Thomson, 1887) leading to the enunciation of the method of stationary phase, which has found wide application to problems of wave propagation in any dispersive media, is entitled, On the Waves Produced by a Single Impulse in Water of any Depth, or in a Dispersive Medium. More recently, the properties of resonant interactions among waves in dispersive systems are known best for water waves (Phillips, 1960; Longuet-Higgins, 1962; Longuet-Higgins and Phillips, 1962; Benney, 1962; Hasselman, 1962; 1963a, b; McGoldrick, 1965), where experiments can be performed relatively easily, (McGoldrick et al., 1966). Some contemporary works in resonant interactions among other types of waves include waves in plasmas (Fishman et al., 1960; Litvak, 1960), electromagnetic waves in solids (Armstrong et al., 1962), and many others. Solid state scientists should note that the fundamental paper on heat conduction in solids by Peierls (1929) was the direct ancestor to the work over the last decade on resonant wave-wave interactions. Finally, remarkable properties of the Korteweg deVries Equation are only now being brought to light, the equation having originally been derived for water waves (Korteweg and deVries, 1895), but applicable to a wide variety of physical situations. The hope of the author is that the methods and results of modern research on water waves can find application in other fields of science. Broadly speaking, if a problem in waves involves dispersion and/or nonlinearity, chances are better that a solution exists, or has been attempted, for water waves than for any other type.