Scattering and trapping of wave energy by a submerged truncated paraboloidal shoal

Abstract

In this paper, we study the scattering and trapping of wave energy by a submerged truncated paraboloidal shoal. The analytical approach of Liu in 2004 for solving wave amplification around a circular island mounted on a shoal is extended in this study to investigate waves above a submerged shoal. In this approach, the mild-slope equation is first transformed into an explicit equation by using Hunt's 1979 Padé approximation to the wave dispersion equation. It is then solved analytically in terms of combined Fourier series and Taylor series. For incident waves varying from short to long waves, wave amplifications in the vicinity of the shoal are calculated using the newly derived analytical model and they agree excellently with the finite difference solutions of Lin in 2004. It is found that, for the particular shoal studied in this paper, there exists an energy focal point where the wave amplitude is the maximum. This point is the result of combined wave refraction, diffraction, and reflection. The point is always located along the x-axis and it is behind the shoal center for very short waves. As the increase of wave period, it moves upstream and it can come in front of the shoal center for very long waves when wave reflection is significant. For a particular incident wave, wave amplification increases and the focal point moves toward the shoal center with the reduction of the water depth above the shoal center. The maximum wave amplitude increases exponentially as the decrease of local water depth. It is also found that there exists the so-called "calm" areas for shallowly submerged shoal, which are located behind the focal point and the local amplitude can be as small as nearly zero. © 2007 ASCE.