Scattering of long waves at the mouth of estuaries bordering on a continental shelf. I.

Abstract

Long waves at an estuary which is bordered by a continental shelf are considered with emphasis on the behaviors of waves in an endless channel and a harbour when plane periodic waves from the open sea and edge waves along the shelf are incident upon them. The most interesting feature of our work is a suppressing mechanism of the shelf upon the waves in the channel or harbour during a resonance period of the plane leaky mode on the shelf (in contradiction to our expectation). The energy of the waves scattered around the estuary is primarily trapped, during the above resonance period, on the shelf as the plane leaky mode and is radiated into the deep water. Partitioning of generated energies to each domain (channel, shelf or deep water) is nearly independent of the shelf width (which is comparatively large). In the model of an endless channel, the plane mode of the transmission coefficient in the channel is considerably affected by the presence of a shelf, while the higher modes are not influenced. Upon incidence by edge waves, the amplitude of the plane (or higher) mode in the endless channel is larger for a higher (or lower) mode incidence than that for a lower (or higher) mode incidence. Each mode of the edge waves generated from the estuary becomes maximum at a wave number slightly larger than the resonance wave number of the plane leaky mode on the shelf. In a model of a rectangular harbour, the amplitude of the longitudinal resonance oscillation in the presence of a shelf, if normalized by the amplitude of the incident waves along a straight coast facing the open sea, undulates along a straight line which denotes the variation of the resonance amplitude for a case without a shelf. The above resonance amplitude in the harbour in the presence of a shelf, if normalized by the amplitude of the incident waves in the deep water, becomes larger than that in the model without a shelf. The location of the node of the longitudinal resonance oscillation at the harbour mouth is very unstable during the resonance period of the plane leaky mode on the shelf owing to the presence of the overwhelming influence of the shelf resonance.