Experimental evidence and a theoretical formulation describing the interaction between a progressive surface wave and a nearly standing subharmonic internal wave in a two-layer system are presented. Laboratory investigations into the dynamics of an interface between water and a fluidized sediment bed reveal that progressive surface waves can excite short standing waves at this interface. The corresponding theoretical analysis is second order and specifically considers the case where the internal wave, composed of two oppositely travelling harmonics, is much shorter than the surface wave. Furthermore, the analysis is limited to the case where the internal waves are small, so that only the initial growth is described. Approximate solution to the nonlinear boundary value problem is facilitated through a perturbation expansion in surface wave steepness. When certain resonance conditions are imposed, quadratic interactions between any two of the harmonics are in phase with the third, yielding a resonant triad. At the second order, evolution equations are derived for the internal wave amplitudes. Solution of these equations in the inviscid limit reveals that, at this order, the growth rates for the internal waves are purely imaginary. The introduction of viscosity into the analysis has the effect of modifying the evolution equations so that the growth rates are complex. As a result, the amplitudes of the internal waves are found to grow exponentially in time. Physically, the viscosity has the effect of adjusting the phase of the pressure so that there is net work done on the internal waves. The growth rates are, in addition, shown to be functions of the density ratio of the two fluids, the fluid layer depths, and the surface wave conditions.
Hypoelliptic degenerate evolution equations of the second order
