Resonant generation of internal waves on two muddy sea beds by a progressive surface wave

The present work is motivated by recent studies on the interaction between a progressive surface wave and the nearly standing subharmonic internal waves in a two-layer system. It is well known that the loading of progressive surface waves, a silty sediment bed was repeatedly and extensively fluidized. The great interest in understanding this phenomenon was induced by the practical applications in sediment transport, wave attenuation, and the design of marine structures. The nonlinear response of an initially flat sea bed, with two muddy sections, to a monochromatic surface progressive wave was investigated in the present study. Based on an analysis similar to that of Hill & Foda's (1998), the multiple scale perturbation method was adopted and the boundary value problem was expanded in a power series of the surface-wave steepness. The linear harmonics and the conditions for resonance were obtained by the leading order. While, the temporal evolution equations for the internal-wave amplitudes were investigated by a second-order analysis. It was found that result for equal density of two muddy sections is similar to that of Hill & Foda's (1998). Two opposite-traveling internal "mud" waves are selectively excited and formed a resonant triad with the progressive surface wave. However for a surface water wave progressing over two different muddy sections, the surface wave will also excite only two opposite-traveling short interfacial waves, forming a nearly standing wave at the interface of the fresh water and the muddy layer. Meanwhile, two opposite-outgoing "mud" waves each with very long wavelength will be simultaneously induced at the interface of two muddy sections. As a result, the amplitudes of the two short internal waves are found to grow exponentially in time. Furthermore, it will be much difficult to excite the internal waves when surface water wave progressing over two muddy sections with the large density gap.