On the speed and profile of steep solitary waves

Abstract

Previous estimates of the speed of solitary waves in shallow water unexpectedly showed that the speed and energy were greatest for waves of less than the maximum possible height. These calculations were based on Pade approximants. In the present paper we present some quite independent calculations based on an integral equation for the wave profile (Byatt-Smith 1970), now modified so that the wave speed appears as a dependent variable. There is remarkably good agreement with the previous method. In particular the existence of a maximum speed and energy are verified. The method also yields a more accurate profile for the free surface of steep solitary waves. As the wave amplitude increases, it is found that the point of intersection of neighbouring profiles moves up towards the crest. Hence the highest wave lies mostly beneath its neighbours, which helps to explain why its speed is less. Tables are given not only of the wave speed but also of the maximum surface slope as a function of wave amplitude. In no case does the slope exceed 30°, but for still higher waves this possibility is not excluded.