Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind. Most of the new material presented relates to the initial-value problem for the equation u t + u x + u u x − u x x t = 0 , ( a ) , whose solution u ( x,t ) is considered in a class of real nonperiodic functions defined for ࢤ∞ < x < ∞, t ≥0. As an approximation derived for moderately long waves of small but finite amplitude in particular physical systems, this equation has the same formal justification as the Korteweg-de Vries equation u t + u x + u u x − u x x x = 0 , ( b ) with which ( a ) is to be compared in various ways. It is contended that ( a ) is in important respects the preferable model, obviating certain problematical aspects of ( b ) and generally having more expedient mathematical properties. The paper divides into two parts where respectively the emphasis is on descriptive and on rigorous mathematics In §2 the origins and immediate properties of equations ( a ) and ( b ) are discussed in general terms, and the comparative shortcomings of ( b ) are reviewed. In the remainder of the paper (§§ 3,4) - which can be read independently Preceding discussion _ an exact theory of ( a ) is developed. In § 3 the existence of classical solutions is proved: and following our main result, theorem 1, several extensions and sidelights are presented. In § 4 solutions are shown to be unique, to depend continuously on their initial values, and also to depend continuously on forcing functions added to the right-hand side of ( a ). Thus the initial-value problem is confirmed to be classically well set in the Hadamard sense. In appendix 1 a generalization of ( a ) is considered, in which dispersive effects within a wide class are represented by an abstract pseudo-differential operator. The physical origins of such an equation are explained in the style of § 2, two examples are given deriving from definite physical problems, and an existence theory is outlined. In appendix 2 a technical fact used in § 3 is established.

An instability mechanism, leading to the generation of cross-waves in a closed channel, was examined recently by Garrett (1970). His theory is not applicable to long channels where the wavemaker produces a primary field which is a progressive wave train. In such cases, the heaving of the mean surface, of considerable significance in the instability mechanism, is confined to the nonpropagating field near the wavemaker. Here the theory of resonant interactions is extended to describe the energy transfer from this forced localized field to the cross-wave field. There are close analogies between the present results and Garrett's, although the resonant bandwidth estimated here is an order of magnitude smaller. The theory indicates that nonlinear effects may control the decay of cross-waves down the channel.

The usual way of posing the problem for the reflexion of wave trains from beaches seems inevitably to imply perfect reflexion. Energy considerations show that wave absorption must be associated with the degradation of mechanical energy either through wave breaking or viscous effects. Some experiments reported here showed substantial wave absorption in the absence of any breaking.We describe some theoretical and experimental work aimed at assessing the role played by friction at the bottom in determining the reflexion coefficient of a beach. The results suggest that, if the parameter (νω3)½½gα2 is not too small, bottom friction can be a significant factor in the absorption process for waves on beaches. Here ν represents the kinematic viscosity (or perhaps an ‘eddy’ viscosity) of the fluid, ω is the frequency of the motions, α is the slope of the beach and g is the acceleration due to gravity.

It has been observed that standing surface waves in water may be excited by acoustic fields of very much higher frequency. No special relationship between the two frequencies appears to be required, but there is such a relationship between the spatial variations of the acoustic and surface wave modes. Another requirement is that the lower frequency should be comparable with the resonant and width of the acoustic response of the system. An explanation of such phenomena is proposed and is tested on a somewhat idealized model by the use of techniques which could be extended to deal with more realistic situations. The model serves to explain qualitatively the available experimental observations. It is suggested that the phenomenon of spatial resonance is not confined to the interaction between water waves and acoustic fields, but may occur generally in systems having modes with related spatial patterns but greatly different frequencies.