Generation, Transformation, and Scattering of Long Waves Induced by a Short-Wave Group over Finite Topography

摘要

Second-order analytical solutions are constructed for various long waves generated by a gravity wave train propagating over finite variable depth h(x) using a multiphase Wentzel-Kramers-Brillouin (WKB) method. It is found that, along with the well-known long wave, locked to the envelope of the wave train and traveling at the group velocity Q, a forced long wave and free long waves are induced by the depth variation in this region. The forced long wave depends on the depth derivatives h_x and h_xx and travels at C_g, whereas the free long waves depend on h, h_x, and h_xx and travel in the opposite directions at √gh. They interfere with each other and generate free long waves radiating away from this region. The author found that this topography-induced forced long wave is in quadrature with the short-wave group and that a secondary long-wave orbital velocity is generated by variable water depth, which is in quadrature with its horizontal bottom counterpart. Both these processes play an important role in the energy transfer between the short-wave groups and long waves. These behaviors were not revealed by previous studies on long waves induced by a wave group over finite topography, which calculated the total amplitude of long-wave components numerically without consideration of the phase of the long waves. The analytical solutions here also indicate that the discontinuity of h_x and h_xx at the topography junctions has a significant effect on the scattered long waves. The controlling factors for the amplitudes of these long waves are identified and the underlying physical processes systematically investigated in this presentation.