On the development of the theory of the solitary wave. A historical essay

Abstract

The mathematical evolution of the theory of a solitary wave is followed from its inception when analytical mechanics was formed. Very early already, it was recognized that the motion of a wave differs from that of the water particles, but the first exact solution of a wave theory, published by Gerstner in 1802, assumes that both motions coincide. With Russell's experimental observations in 1838, and with his classification of waves into four orders, the motion of transmission, i.e. the motion of the wave, was solidly established as the transfer of momentum from one spatial point to its neighbour. His discovery of a solitary wave of permanent form, whose velocity depends on its height started a mathematical dispute. Several authors developed their own theory of waves. The most successful contributions were brought by Airy in 1845 with a nonlinear shallow water theory of waves of non-permanent form, by Stokes in 1849 with a theory of deep water waves. Stokes concluded that his theory holds whenever the amplitude of the surface elevation relative to the water depth is smaller than the ratio of the waterdepth to the wavelength:ηmax/h < (h/λ)2, whereas Airy's results implyηmax/h < (h/λ)2 thus providing different physical phenomena. The first mathematical description of a wave of permanent form was obtained by Boussinesq in 1871; it was rediscovered by an other method by Lord Rayleigh in 1876. The next steps were to improve the stationary behaviour of the solitary wave. Rayleigh had remarked that in his solution the pressure distribution at the free surface is not constant, thus invalidating the stationarity of the solution. McCowan, 1894, Kortewegde Vries, 1895 gave more accurate results for the wave form and the maximum possible waveheight. Later on, systematic expansions were introduced and allowed an entire hierarchy of waves to be developed, a process presently still being under way. In this context proofs of convergence and existence of the forementioned approximations to an exact solution of a wave in an irrotational perfect fluid are the focus of the 20th century research.