Resonant reflection of water waves in a long channel with corrugated boundaries

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Abstract
A one-dimensional wave equation is derived for water-wave propagations in a long channel with corrugated boundaries. The amplitude and the wavelength of boundary undulations are assumed to be smaller than and in the same order of magnitude as the incident wavelength, respectively. When the Bragg reflection condition (i.e. the wavenumber of the boundary undulations is twice that of the incident wavenumber) is nearly satisfied, significant wave reflection could occur. Coupled equations for transmitted and reflected wave fields are derived for the near resonant coupling. The detuning mechanism is attributed to the slight deviation in the wavenumber of the corrugated boundaries from the Bragg wavenumber. Analytical solutions are obtained for the cases where the boundary undulations are within a finite region. The application of the present theory to the design of a harbour resonator is discussed.

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