The Cauchy-Poisson problem in an inviscid stratified liquid

Based upon the Boussinesq approximation, an initial value investigation is made of the axisymmetric free surface response of a nonrotating inviscid stratified liquid of finite or infinite depth to the initial displacement of the free surface. The asymptotic analysis of the integral solution is carried out by the stationary phase method to describe the solution for large time and distance from the origin of disturbance. It is shown that the asymptotic solution consists of the classical free surface gravity waves and the internal waves.