Abstract
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 C_{g}, 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
^{*} Current affiliation: Civil and Environmental Engineering, University of Maine, Orono, Maine.