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Abstract
A wave equation correct to first order in wave amplitude and bottom slope is used to calculate the wave field around an island. This is of circular cylindrical shape, and is situated on a paraboloidal shoal in an ocean of constant depth (Figure 1). The sides of the island are assumed fully reflecting. The incident waves are plane, periodic, and of small amplitude. Periods up to 30 min are investigated, and the Coriolis force is neglected. The wave equation is solved analytically, and a great number of numerical computations are carried through. The total wave field over the shoal is presented for two discrete periods in the upper end of the tsunami frequency range. The amplitudes at the middle of the front face of the island, and at the middle of the lee face, are given as functions of the wave period, and the existence of “resonance”; periods is thus demonstrated. Comparison with solutions to the linearized long‐wave equation is made, and the validity range of the shallow water theory is estimated. The geometrical optics (i.e., refraction) approach is shown to give extremely inaccurate amplitudes using intermediate depth theory (or shallow water theory for that matter). Finally, the wave field around a circular island in an ocean of constant depth is illustrated.
Notes
Present address: Laboratory of Applied Mathematical Physics, Technical University of Denmark, Building 303, DK‐2800 Lyngsby.