An Extended Analytic Solution of Combined Refraction andDiffraction of Long Waves Propagating over Circular Island

Yao‐Lung Kuo ,Yu-Hsuan Huang,Todd Hsu ,Chia‐Cheng Tsai

doi:10.1155/2012/742705

Abstract

An analytic solution of long waves scattering by a cylindrical island mounted on a permeable circular shoal was obtained by solving the linear long wave equation (LWE). The solution is in terms of the Bessel function expressed by complex variables. The present solution is suitable for arbitrary bottom configurations described by a power function with two independent parameters. For the case of the paraboloidal shoal, there exists a singular point (α = 2) which can be removed using Frobenius series, where α is a real constant. The present solution is reduced to Yu and Zhang’s (2003) solution for impermeable circular shoal. The numerical results show some special features of the combined effect of wave refraction and diffraction caused by a porous circular island. The effect of key parameters of the island dimension, the shoal slope, and permeability on wave scattering was discussed based on the analytic solution.

Highlights

The combined effect of wave refraction and diffraction is caused by the varying bottom topography or the presence of islands and structures
Numerical models are commonly used in practical applications, it is still limited by the problems of accuracy and numerical stability
An analytic solution of the long-wave equation for waves propagating over a porous circular island was derived

Summary

Introduction

The combined effect of wave refraction and diffraction is caused by the varying bottom topography or the presence of islands and structures. Journal of Applied Mathematics needed to be verified by comparing with measured data or analytic solutions In some cases, it takes time and expensive cost for laboratory experiments because of the complexity in completion of the model and facilities setup. A typical example of the analytic study of combined refraction and diffraction is the case when the long-wave approximation is applied to the MSE in one dimension or axisymmetric bathymetry in two dimensions. The present investigation is to extend Yu and Zhang’s 18 analytic solution to long wave propagation over a porous circular island with arbitrary topography. The combined effect of refraction and diffraction of long wave motion as well as key features of the intensity of wave ray focus behind the shoal is presented. The effect of key parameters on wave scattering is intensively discussed based on the analytic solution

Theoretical Formulation

Results and Discussions

Conclusions