The wave equation in a medium with a time-dependent boundary

Abstract

It is shown that the Helmholtz equation is not exactly correct for a medium with a time-dependent boundary. The equation can be used with very good approximation when the time-derivative of the surface elevation is much smaller than the speed of the waves through the medium. For underwater sound waves, reflected and scattered by an ocean surface that can be described by the Pierson-Moskowitz spectrum, this means that the wind speed has to be much less than the sound speed.