An analytical solution of the mild-slope equation for waves around a circular island on a paraboloidal shoal

Abstract

In this paper, we develop an analytical technique in terms of series expansions to solve the mild-slope equation on an axi-symmetric topography. This technique is applied to study the combined refraction and diffraction of plane monochromatic waves by a circular cylindrical island mounted on a paraboloidal shoal. By using the direct solution for the wave dispersion equation by Hunt [J. Waterw., Port, Coast., Ocean Div. Proc ASCE 4 (1979) 457], the mild-slope equation becomes explicit and it is then solved in terms of combined Fourier series and Taylor series. It is found that, to calculate the wave elevation along a perimeter with a specific radius, more terms in the Taylor series and angular modes in the Fourier series are needed for shorter waves. On the other hand, for the same incident wave, the outer the solutions are sought, more angular modes are needed to obtain the converged result of Fourier series. The comparison with the analytical solution based on the linear shallow-water equation by Homma [Geophys. Mag. 21 (1950) 199] is made for long wave incidence and excellent agreements are obtained. For long waves and waves in intermediate water depth, comparisons are made with other numerical results of the mild-slope equation and an equally good quality of agreement is achieved.

Introduction

Due to the undoubted practical importance in understanding the phenomenon of wave refraction and diffraction, there have been many research activities in this area including experimental measurements, numerical modelling and theoretical analysis. Analytical solutions can be more favorable than experimental data when they are used to verify numerical solutions as the accuracy of analytical solution is only limited by its own assumption. However, analytical solutions are in general only obtainable for simple bottom geometry and particular type of wave (i.e., long wave or short wave). Examples include the solution of the Helmholtz equation for a cylindrical island standing in the middle of an open ocean with constant water depth (MacCamy and Fuchs, 1954), the solution of the linear shallow-water equation for a circular cylindrical island mounted on a paraboloidal shoal (Homma, 1950), solutions of the linear shallow-water equation for a conical island (Zhang and Zhu, 1994) and the solution of the linear shallow-water equation for a circular cylindrical island mounted on a conical shoal (Zhu and Zhang, 1996). Based on the linear shallow-water equation, Yu and Zhang (2003) derived a uniform form of the solutions for several axi-symmetrical bottom geometries. Kânoǧlu and Synolakis (1998) also developed an analytical method to solve the linear shallow-water equation for simple piecewise linear topographies. However, all these analytical techniques fail when one has to deal with a more complicated geometry or start from a more general equation, i.e., the mild-slope equation.

The mild-slope equation was originally proposed by Berkhoff, 1972, Berkhoff, 1976. With the assumption that the bottom slope is mild and thus the mean water depth variation is moderate, he demonstrated that a three-dimensional problem can be well approximated by this two-dimensional equation. Ever since it was derived, the mild-slope equation has proved to be a very useful model for a wide range of water wave problems. In recent years, various extensions of the mild-slope equation have been made, such as Chamberlain and Porter (1995), Porter and Staziker (1995), Chandrasekera and Cheung (1997), Suh et al. (1997), Miles and Chamberlain (1998), Li and Fleming (1999) and Agnon and Pelinovsky (2001).

Because of its ability to describe combined wave refraction and diffraction, the mild-slope equation has become a popular basis of numerical models for calculating surface waves on slowly varying water depth. For example, Jonsson et al. (1976) solved the mild-slope equation using a collocation method for Homma's (1950) island. Following Chen and Mei's (1974) numerical work in solving the Helmholtz equation in a constant depth region, Bettess and Zienkiewicz (1977) and Houston (1981) developed a hybrid method to solve the mild-slope equation for wave scattering by Homma's (1950) island, in which an infinite computational domain is divided into two: an outer region where infinite elements (Bettess and Zienkiewicz, 1977) or eigenfunctions (Houston, 1981) can be adopted and an inner region where finite element or finite difference techniques can be used to obtain solutions. Most of the subsequent work differs only on the treatment of the outer domain. Li (1994a) proposed a generalized conjugate gradient method with a fast convergence rate. Li (1994b) used the so-called Aternating Direction Implicit (ADI) method to solve the evolution mild-slope equation and the numerical scheme is unconditional stable. Employing Nardini and Brebbia's (1983) dual reciprocity boundary element method (DRBEM), Zhu (1993) developed a DRBEM model and then Zhu et al. (2000) and Liu (2001) further extended it to the general DRBEM (GDRBEM), which greatly improved the numerical efficiency in terms of both computational time and computer memory required in comparison with their hybrid counterparts.

However, due to the insufficient field data for the problem of combined wave refraction and diffraction around an island and the lack of analytical solutions to the original mild-slope equation for general wave conditions, numerical solutions mentioned above were only verified with the results from other numerical models and with available analytical solutions of the shallow-water equation. In case that numerical results from different numerical methods did not match with each other, it was difficult to judge which one was correct. For shorter waves, the situation becomes even worse since almost all the known numerical models are incapable of dealing with the increasing demand of computational resources. In fact, for Homma's (1950) island, numerical results with incident waves periods less than 120 s had rarely been reported except that of Zhu et al. (2000) and Liu (2001) who presented two GDRBEM results of T=90 s and T=60 s. However, these two solutions remain to be verified. Recently, Lin (2004) developed a finite difference model to investigate the same problem. Obviously, the pursuit of analytical solutions of the mild-slope equation in the whole range from long waves to short waves is of great significance and is, naturally, of great difficulty.

It is well-known that the principal difficulty in solving the mild-slope equation analytically comes from the fact that its coefficients are related by the linear dispersion equation which is implicit for waves in intermediate water depth. Thus, the governing equation with its coefficients being transcendental functions in such a complicated manner seems impossible to be solved exactly. Hence, we turn to search for some approximate analytical solutions in this study. In the past several decades, many attempts were made to provide explicit approximate solution to wave dispersion relationship. The early works include Hunt (1979), Venezian (1980) and Nielsen (1982). A good review article was written by Fenton and Mckee (1990) on these early approximate direct solutions. The recent works include Newman (1990), Chamberlain and Porter (1999a), Guo (2002) and You (2003). Recently, an approximate solution of the dispersion relation in conjunction with an explicit power series solution was used by Ehrenmark and Williams (2001) to determine the local solution of the mild-slope equation at the shoreline in a two-dimensional context.

In this paper, by using Hunt's (1979) approximate direct solution of the implicit dispersion equation, the mild-slope equation is first transformed into an approximate form with its coefficients being explicitly expressed. An analytical solution in terms of combined Fourier series and Taylor series is then sought for a circular cylindrical island mounted on a paraboloidal shoal. The solution is compared to Homma's (1950) analytical solutions based on the linear shallow-water equation, Liu's (2001) GDRBEM solutions based on the mild-slope equation and Lin's (2004) finite difference solution based on the time-dependent mild-slope equation for waves in shallow and intermediate water depths.