Generation of waves at an inertial interface

This paper is concerned with the generation of three-dimensional unsteady motion due to initial disturbances at the interface between two superposed fluids wherein the upper fluid extends infinitely upwards and the lower fluid is of uniform finite depth. The interface is composed of a thin but uniform distribution of a non-interacting material termed as an inertial surface. Assuming linear theory, the problem is formulated as a coupled initial value problem in the velocity potentials describing the motion in the two fluids. The Laplace transform in time and double Fourier transform in space have been utilised in the mathematical analysis to obtain the depression of the inertial interface in terms of a double integral, whose asymptotic form is obtained for large values of time and distance by the method of stationary phase. This is then depicted graphically for two types of initial disturbances in a number of figures to visualise the effect of the upper fluid and also the presence of the inertial surface at the interface on the wave motion. The time evolution of the interface depression is also shown graphically and the decaying phenomena are demonstrated by drawing the phase diagram.