Helmholtz resonance of harbours

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Abstract
The resonant response of a harbour H of depth scale d and area A to excitation of frequency ω through a mouth M of width a is calculated in the joint limit a2/Aω2A/gd↓0. The results are relevant to the tsunami response of narrow-mouthed harbours. 16 is assumed that an adequate approximation to the radiation impedance of the external domain is available (Miles 1972). The boundary-value problem for H is reduced to the solution of ∇. (h∇ϕ) = −1/A, where h is the relative depth, the normal derivative of ϕ is prescribed in M and vanishes elsewhere on the boundary of H, and the spatial mean of ϕ must vanish. The kinetic energy in H is proportional to an inertial parameter M that is a quadratic functional of ϕ. It is demonstrated that decreasing/increasing h increases/decreases M. Explicit lower bounds to M are developed for both uniform and variable depth. The results are extended to coupled basins (inner and outer harbours). Several examples are considered, including a model of Long Beach Harbor, for which the calculated resonant frequency of the dominant mode is within 1% of the measured value. The effects of entry-separation and bottom-friction losses are considered; the latter are typically negligible, whereas the former may be comparable with, or dominate, radiation losses.
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