Surface and Interfacial Gravity Waves Induced by an Impulsive Disturbance in a Two-Layer Inviscid Fluid

The transient gravity waves due to an impulsive source in a two-layer fluid system are investigated analytically. The fluid is assumed to be inviscid and incompressible. The density of each of the two layers is constant. Five different boundary conditions are considered. The depth of each of the two layers is infinite or finite. The upper fluid of finite depth is covered by a rigid lid or a free surface. Based on the assumption of small-amplitude waves, a linear system is established. The integral solutions for the free-surface and interfacial waves are obtained by means of the Fourier-Laplace transform. The corresponding asymptotic representations are derived for large time with a fixed distance-to-time ratio by the Stokes and Scorer methods of stationary phase. The analytical solutions show that there are two different modes, namely the free-surface and interfacial wave modes. The wave profiles observed depend on the relation between the distance-to-time ratio and the maximal group velocities and on the limiting values of the second derivatives of the frequencies as well.