Evolution of sub-harmonic interfacial waves generated by a surface wave through a three-wave interaction

Higher-order effects in three-wave resonant interaction of a surface wave, with a pair of interfacial waves in a two-layer fluid, are studied theoretically. Following an initial rapid growth, the interfacial waves approach a steady state of constant amplitude. An explicit solution is presented for transition to the ultimate state of the interaction. It is shown that for interaction in a wave flume, it is necessary to include a 2nd pair of the interfacial waves, resulting from a reflection of the original pair in the analysis. The effects of different parameters on the dynamics of the interaction are investigated. The results indicate that a faster initial growth does not necessarily lead to larger ultimate amplitude. Also, there are two angles, at which the interfacial waves continue to grow at an initial growth rate, possibly leading to wave breaking. The results are in qualitative agreement with previous experimental observations.