Mathematical Aspects of the Cauchy-Poisson Problem in Three Dimensions

Abstract

Recent work on the three-dimensional Cauchy-Poisson problem (Clarisse Newman and Ursell 1995) is concerned with the development in time of the region near the wave front. The solution obtained in that work involves uniform asymptotic expansions and our mathematical arguments are complicated. In the present note an attempt is made to explain these arguments. We shall be concerned with incompressible inviscid fluid of finite constant depth h under gravity. The fluid is initially at rest and is set in motion at timet = 0 by an axially symmetric impulse distributed over the free surface. It is assumed that the linearized equations of motion are applicable. This is the famous Cauchy-Poisson problem, first treated by these authors in 1815, and described in Lamb (1932), sections 238–241, 255. Clearly the resulting motion has axial symmetry about the centre of disturbance; ring waves travel away from the centre, with long waves travelling faster than short waves. This solution can be expressed as an integral and can be approximated by Kelvin’s method of stationary phase, which tells us that after a long time each wave component (i.e. each wave frequency) travels away from the centre with the group velocity appropriate to that frequency. The maximum group velocity corresponds to infinite wavelength and is equal to (gh)^1/2, where g is the gravitational acceleration; the cylindrical surface travelling outwards with this velocity will be described as the wave front..