Response in harbour due to incidence of second-order low-frequency waves

Using the method of multiple scales and matched asymptotic expansions, the response in a harbour due to the incidence of the set-down wave of a group wave train with narrow spectrum is investigated. The envelope of the incident wave train is assumed to modulate slowly in the direction of wave propagation. The relationship between the envelopes of incident waves and transmitted or scattered waves is derived. It is shown that the second-order scattered wave generated by the set-down wave of the incident wave train propagates only at the longwave celerity √gh. This scattered wave in the harbour is discussed along with the response due to the set-down wave. The resonant response has been calculated for two types of geometries.