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Mathematical Problems in Engineering
Volume 2011, Article ID 607196, 20 pages
http://dx.doi.org/10.1155/2011/607196
Research Article

On Step Approximation for Roseau's Analytical Solution of Water Waves

1Department of Marine Environmental Engineering, National Kaohsiung Marine University, Kaohsiung 811, Taiwan
2Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan 701, Taiwan

Received 24 November 2010; Accepted 3 March 2011

Academic Editor: Mohammad Younis

Copyright © 2011 Chia-Cheng Tsai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An indirect eigenfunction marching method (IEMM) is developed to provide step approximations for water wave problems. The bottom profile is in terms of successive flat shelves separated by abrupt steps. The marching conditions are represented by the horizontal velocities at the steps in the solution procedure. The approximated wave field can be obtained by solving a system of linear equations with unknown coefficients which represents the horizontal velocities under a proper basis. It is also demonstrated that this solution method can be exactly reduced to the transfer-matrix method (TM method) for a specific setting. The combined scattering effects of a series of steps can be described by a single two-by-two transfer matrix for connecting the far-field behaviors of both sides for this method. The solutions obtained by the IEMM are basically exact for water wave problems considering step-like bottoms. Numerical simulations were performed to validate the present and commonly used methods. Furthermore, it also shows that the solutions obtained by the IEMM converge very well to Roseau's analytical solutions for both mild and steep curved bottom profiles. The present method improves the converges of the TM method for solving water wave scattering over steep bathymetry.

1. Introduction

The scattering of linear water waves over arbitrary bed topography has been of interest for years. Analytic solutions are rare except for the cases of constant slope and Roseau's analytical solution [1]. Consequently, approximations are alternative for solving water waves. For example, Berkhoff [2] derived a mild-slope equation (MSE) by removing the vertical coordinate using the integration of depth function, and hence reducing the three-dimensional problem to a two-dimensional one. The mild-slope equation was later modified and extended with additional higher-order terms by several researches [36]. On the other hand, there are also other methods applicable for describing water wave problems, such as the integral equation method [7] and differential equation method in transformed domain [8] and so on.

For a two-dimensional problem, if the geometry of interest can be divided into separate regions with a constant fluid depth in each subdomain, the solution in each constant-depth subdomain is usually constructed in terms of eigenfunctions. The solutions are then matched at the vertical boundaries, resulting in a system of linear integral equations which must be truncated to a finite number of terms and solved numerically. Takano [9] used this direct eigenfunction marching method (DEMM) for solving the cases of waves travelling over an elevated sill and a fixed surface obstacle of normal wave incidence. Kirby and Dalrymple [10] extended the DEMM to the problems of waves passing a trench of oblique incidence. Moreover, Söylemez and Gören [11] applied the DEMM to problems of wave scattering over rectangle barriers.

On the other hand, Miles [12] innovated a variational formulation by approximating the horizontal velocity at each step using the corresponding propagating eigenfunction and incorporating with the wide-spacing approximation [13]. His method is formulated by a two-by-two scattering matrix for connecting the amplitudes of propagating modes in the two shelves near a step. Miles [12] used his variational formulation to study Newman's problem [14] of wave scattering by a step of infinite or arbitrarily finite depth. Mei and Black [15] applied the variational formulation to surface-wave scattering over rectangular obstacles. For waves propagating over a sequence of steps, Devillard et al. [13] generalized Miles' variational formulation by introducing a transfer matrix for representing the combined effects of all steps. For the case of an arbitrary bottom topography, O’Hare and Davies [16] approximated the smoothly varying bottom configuration using a series of shelves which are separated by abrupt steps and applied the prescribed method to find the desired transfer matrix of the problem. Furthermore, O'Hare and Davies [17] demonstrated that their solutions are comparable with those obtained using the MSE. However, their solutions converge to pseudosolutions for problems of steep bottom profiles.

In this study, an indirect eigenfunction marching method (IEMM) is developed to provide step approximations for describing wave propagating over an arbitrary bottom represented by several flat shelves separated by abrupt steps. In addition, it is demonstrated that the solution can be exactly reduced to the transfer-matrix (TM) method of Miles [12] and Devillard et al. [13] if the horizontal velocity at each step is approximated by the propagating eigenfunction and the wide-spacing approximation [13] is assumed. The accuracy of this new eigenfunction marching method is investigated to solve the problem of water wave scattering over steps. Furthermore, its applicability for wave propagating over an arbitrary bottom topography is examined by using Roseau's analytical solution [1]. Our results show that the solutions converge very well for both mild and steep curved beach profiles. This method circumvents the pseudoconverges for the TM method for solving problems of wave propagating over steep bottom profiles.

This paper is organized as follows: the wave problem is mathematically modeled in Section 2. Then, the indirect eigenfunction marching method is developed in Section 3. Its reduction to transfer-matrix method is given in Section 4. Some numerical experiments are carried out to validate the prescribed method in Section 5 and the conclusions are drawn in Section 6.

2. Wave Model

We consider the one-dimensional problem for a monochromatic wave propagating over an arbitrary bottom configuration with time dependence 𝑒𝐢𝜔𝑡, where 𝑡 is the time, 𝜔 the angular frequency and 𝐢 the unit of complex number. The sea bottom is represented by a succession of flat shelves as depicted in Figure 1. In the figure, there are 𝑀 shelves with depth 𝑚 in the interval of 𝑥𝑚1<𝑥<𝑥𝑚 for 𝑚=1,2,3,,𝑀 and 𝑀1 steps at 𝑥=𝑥𝑚 for 𝑚=1,2,3,,𝑀1. In order to make the formulation easier, it is assumed that 𝑥0= and 𝑥𝑀=, respectively. The coordinate (𝑥,𝑦) in Figure 1 is defined such that 𝑥 is the horizontal direction and 𝑦 is the vertical direction upwards positively from the still water level on 𝑦=0.

607196.fig.001
Figure 1: Schematic sketch of water wave scattering over a series of steps.

According to Airy's linear wave theory [18], the velocity potential is governed by Laplace equation given by2𝜙=0,(2.1) which is subject to the free surface condition𝜕𝜙𝜔𝜕𝑦2𝑔𝜙=0(2.2) and the bottom boundary condition: 𝜕𝜙𝜕𝑦=0on𝑦=𝑚,𝑥𝑚1<𝑥<𝑥𝑚for𝑚=1,2,3,,𝑀(2.3) and the condition for vertical walls𝜕𝜙𝜕𝑥=0on𝑥=𝑥𝑚,max𝑚,𝑚+1<𝑦<min𝑚,𝑚+1for𝑚=1,2,3,,𝑀1,(2.4) where 𝐾=𝜔2/𝑔 with 𝑔 being the acceleration of gravity. The following marching conditions are required at each step:𝜙𝑚=𝜙𝑚+1,on𝑥=𝑥𝑚,min𝑚,𝑚+1𝑈<𝑦<0for𝑚=1,2,3,,𝑀1,(2.5)𝑚(𝑦)=𝜕𝜙𝑚=𝜕𝑥𝜕𝜙𝑚+1𝜕𝑥on𝑥=𝑥𝑚,min𝑚,𝑚+1<𝑦<0for𝑚=1,2,3,,𝑀1.(2.6)

Furthermore, the following far-field conditions are required to make the solution unique:𝜙=cosh𝜅1𝑦+1cosh𝜅11𝑒𝐢𝜅1𝑥+𝑅𝑒𝐢𝜅1𝑥as𝑥,𝜙=cosh𝜅𝑀𝑦+𝑀cosh𝜅𝑀𝑀𝑇𝑒𝐢𝜅𝑀𝑥as𝑥,(2.7) where 𝑅 and 𝑇 are the reflection and transmission coefficients, respectively, and 𝜅𝑚 is the wavenumber in the interval of 𝑥𝑚1<𝑥<𝑥𝑚. The wavenumber 𝜅𝑚 is the positive root of the dispersion relation expressed by𝜅𝑚tanh𝜅𝑚𝑚=𝐾for𝑚=1,2,3,,𝑀.(2.8)

3. Indirect Eigenfunction Marching Method

According to the linear wave theory, a complete solution on the 𝑚th shelf can be constructed as follows:𝜙𝑚𝐴(𝑥,𝑦)=𝑚𝑒𝐢𝜅𝑚𝑥+𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝜒𝑚+(𝑦)𝑁𝑛=1𝐶𝑚,𝑛𝑒𝑘𝑚,𝑛(𝑥𝑥𝑚)+𝐷𝑚,𝑛𝑒𝑘𝑚,𝑛(𝑥𝑥𝑚1)𝜓𝑚,𝑛(𝑦).(3.1) This representation is arbitrarily well if 𝑁 is sufficiently large. In (3.1), the propagating eigenfunction 𝜒𝑚(𝑦) is written in the form:𝜒𝑚2(𝑦)=𝜅𝑚cosh𝜅𝑚𝑦+𝑚2𝜅𝑚𝑚+sinh2𝜅𝑚𝑚,(3.2) and the evanescent eigenfunction 𝜓𝑚,𝑛(𝑦) is expressed as𝜓𝑚,𝑛2(𝑦)=𝑘𝑚,𝑛cos𝑘𝑚,𝑛𝑦+𝑚2𝑘𝑚,𝑛𝑚+sin2𝑘𝑚,𝑛𝑚,(3.3) where 𝑘𝑚,𝑛 is the 𝑛th smallest positive root of 𝑘𝑚,𝑛tan𝑘𝑚,𝑛𝑚=𝐾.(3.4) Clearly, the following orthonormal relation can be found based on the Sturm-Liouville theory, that is,𝜒𝑚𝜒𝑚𝜒=1,𝑚𝜓𝑚,𝑙𝜓=0,𝑚,𝑛𝜓𝑚,𝑙=𝛿𝑛𝑙,(3.5) where 𝛿𝑛𝑙 is the Kronecker delta function and 𝐹𝐺 is defined by𝐹𝐺=0𝜆𝐹(𝑦)𝐺(𝑦)𝑑𝑦.(3.6) In (3.6), 𝐹 and 𝐺 are orthonormal eigenfunctions of 𝜒𝑚(𝑦) or 𝜓𝑚,𝑛(𝑦), and 𝜆 is the corresponding water depth of the eigenfunction 𝐹.

Now, we are in a position to march the solutions on all shelves. Considering a specific step at 𝑥=𝑥𝑚 with 𝑚=1,2,3,,𝑀1, we apply the Galerkin method to (2.4) and (2.6) and use the orthonormal relation of (3.5) to obtain the following equations:0𝐻𝑚𝑈𝑚(𝑦)𝜒𝑚(𝑦)𝑑𝑦=𝐢𝜅𝑚𝐴𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚,(3.7)0𝐻𝑚𝑈𝑚(𝑦)𝜓𝑚,𝑛(𝑦)𝑑𝑦=𝑘𝑚,𝑛𝐶𝑚,𝑛𝐷𝑚,𝑛𝑒𝑘𝑚,𝑛(𝑥𝑚𝑥𝑚1)for𝑛=1,2,3,,𝑁,(3.8)0𝐻𝑚𝑈𝑚(𝑦)𝜒𝑚+1(𝑦)𝑑𝑦=𝐢𝜅𝑚+1𝐴𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚𝐵𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚,(3.9)0𝐻𝑚𝑈𝑚(𝑦)𝜓𝑚+1,𝑛(𝑦)𝑑𝑦=𝑘𝑚+1,𝑛𝐶𝑚+1,𝑛𝑒𝑘𝑚+1,𝑛(𝑥𝑚+1𝑥𝑚)𝐷𝑚+1,𝑛for𝑛=1,2,3,,𝑁,(3.10) where 𝐻𝑚 is the minimum value between 𝑚 and 𝑚+1, that is, 𝐻𝑚=min(𝑚,𝑚+1). Equations (3.7)~(3.10) are exactly equivalent to (2.4) and (2.6) when 𝑁 since 𝜒𝑚(𝑦) and 𝜓𝑚,𝑛(𝑦) have formed a complete basis. This has been shown by Miles [12].We make use of the orthonormal identity, (3.8) and (3.10), and are ready to obtain𝐶𝑚,𝑛=0𝐻𝑚𝑈𝑚(𝑦)𝜓𝑚,𝑛(𝑦)𝑑𝑦𝑒𝑘𝑚,𝑛(𝑥𝑚𝑥𝑚1)0𝐻𝑚1𝑈𝑚1(𝑦)𝜓𝑚,𝑛(𝑦)𝑑𝑦𝑘𝑚,𝑛1𝑒2𝑘𝑚,𝑛(𝑥𝑚𝑥𝑚1),𝐷𝑚,𝑛=𝑒𝑘𝑚,𝑛(𝑥𝑚𝑥𝑚1)0𝐻𝑚𝑈𝑚(𝑦)𝜓𝑚,𝑛(𝑦)𝑑𝑦0𝐻𝑚1𝑈𝑚1(𝑦)𝜓𝑚,𝑛(𝑦)𝑑𝑦𝑘𝑚,𝑛1𝑒2𝑘𝑚,𝑛(𝑥𝑚𝑥𝑚1),(3.11) for 𝑚=1,2,3,,𝑀.

On the other hand, the further application of the Galerkin method to the marching condition in (2.5) and using (3.11) can produce the following equations:𝐴𝑚𝑒𝐢𝜅𝑚𝑥𝑚+𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝜒𝑚𝜒𝑚𝐴𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚+𝐵𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚𝜒𝑚𝜒𝑚+1=𝑁𝑛=1𝜒𝑚𝜓𝑚+1,𝑛0𝐻𝑚+1𝑈𝑚+1(𝜂)𝜓𝑚+1,𝑛(𝜂)𝑑𝜂𝑘𝑚+1,𝑛sinh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚𝜒𝑚𝜓𝑚+1,𝑛0𝐻𝑚𝑈𝑚(𝜂)𝜓𝑚+1,𝑛(𝜂)𝑑𝜂𝑘𝑚+1,𝑛tanh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚𝜒𝑚𝜓𝑚,𝑛0𝐻𝑚𝑈𝑚(𝜂)𝜓𝑚,𝑛(𝜂)𝑑𝜂𝑘𝑚+1,𝑛tanh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚+𝜒𝑚𝜓𝑚,𝑛0𝐻𝑚1𝑈𝑚1(𝜂)𝜓𝑚,𝑛(𝜂)𝑑𝜂𝑘𝑚,𝑛sinh𝑘𝑚,𝑛𝑥𝑚𝑥𝑚1,𝐴(3.12)𝑚𝑒𝐢𝜅𝑚𝑥𝑚+𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝜓𝑚,𝑗𝜒𝑚𝐴𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚+𝐵𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚𝜓𝑚,𝑗𝜒𝑚+1=𝑁𝑛=1𝜓𝑚,𝑗𝜓𝑚+1,𝑛0𝐻𝑚+1𝑈𝑚+1(𝜂)𝜓𝑚+1,𝑛(𝜂)𝑑𝜂𝑘𝑚+1,𝑛sinh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚𝜓𝑚,𝑗𝜓𝑚+1,𝑛0𝐻𝑚𝑈𝑚(𝜂)𝜓𝑚+1,𝑛(𝜂)𝑑𝜂𝑘𝑚+1,𝑛tanh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚𝜓𝑚,𝑗𝜓𝑚,𝑛0𝐻𝑚𝑈𝑚(𝜂)𝜓𝑚,𝑛(𝜂)𝑑𝜂𝑘𝑚+1,𝑛tanh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚+𝜓𝑚,𝑗𝜓𝑚,𝑛0𝐻𝑚1𝑈𝑚1(𝜂)𝜓𝑚,𝑛(𝜂)𝑑𝜂𝑘𝑚,𝑛sinh𝑘𝑚,𝑛𝑥𝑚𝑥𝑚1(3.13) for 𝑗=1,2,3,,𝐽. In (3.12) and (3.13), 𝜒𝑚 and 𝜓𝑚,𝑙 are defined, respectively, by𝜒𝑚𝜒(𝑦)=𝑚(𝑦)if𝑚<𝑚+1,𝜒𝑚+1(𝑦)if𝑚+1<𝑚,𝜓𝑚,𝑙𝜓(𝑦)=𝑚,𝑙(𝑦)if𝑚<𝑚+1,𝜓𝑚+1,𝑙(𝑦)if𝑚+1<𝑚.(3.14) Note that (3.7), (3.9), (3.12), and (3.13) form a functional problem of 𝑈𝑚(𝑦). In order to solve this functional problem, the horizontal velocities in above steps should be represented by the complete basis, 𝜒𝑚 and 𝜓𝑚,𝑙, as follows:𝑈𝑚(𝑦)=𝑢𝑚,0𝜒𝑚(𝑦)+𝐿𝑙=1𝑢𝑚,𝑙𝜓𝑚,𝑙(𝑦),(3.15) where 𝑢𝑚,𝑙 are unknown coefficients to be determined. Then, substitution of (3.15) into (3.7), (3.9), (3.12), and (3.13) respectively, yield𝑢𝑚,0𝜒𝑚𝜒𝑚+𝐿𝑙=1𝑢𝑚,𝑙𝜓𝑚,𝑙𝜒𝑚=𝐢𝜅𝑚𝐴𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚,𝑢𝑚,0𝜒𝑚𝜒𝑚+1+𝐿𝑙=1𝑢𝑚,𝑙𝜓𝑚,𝑙𝜒𝑚+1=𝐢𝜅𝑚+1𝐴𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚𝐵𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚,𝐴𝑚𝑒𝐢𝜅𝑚𝑥𝑚+𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝜒𝑚𝜒𝑚𝐴𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚+𝐵𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚𝜒𝑚𝜒𝑚+1=𝑢𝑁𝑚+1,0𝑛=1𝜒𝑚+1𝜓𝑚+1,𝑛𝜒𝑚𝜓𝑚+1,𝑛𝑘𝑚+1,𝑛sinh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚+𝐿𝑙=1𝑢𝑁𝑚+1,𝑙𝑛=1𝜓𝑚+1,𝑙𝜓𝑚+1,𝑛𝜒𝑚𝜓𝑚+1,𝑛𝑘𝑚+1,𝑛sinh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚𝑢𝑁𝑚,0𝑛=1𝜒𝑚𝜓𝑚+1,𝑛𝜒𝑚𝜓𝑚+1,𝑛+𝜒𝑚𝜓𝑚,𝑛𝜒𝑚𝜓𝑚,𝑛𝑘𝑚+1,𝑛tanh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚𝐿𝑙=1𝑢𝑁𝑚,𝑙𝑛=1𝜓𝑚,𝑙𝜓𝑚+1,𝑛𝜒𝑚𝜓𝑚+1,𝑛+𝜓𝑚,𝑙𝜓𝑚,𝑛𝜒𝑚𝜓𝑚,𝑛𝑘𝑚+1,𝑛tanh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚+𝑢𝑁𝑚1,0𝑛=1𝜒𝑚1𝜓𝑚,𝑛𝜒𝑚𝜓𝑚,𝑛𝑘𝑚,𝑛sinh𝑘𝑚,𝑛𝑥𝑚𝑥𝑚1+𝐿𝑙=1𝑢𝑁𝑚1,𝑙𝑛=1𝜓𝑚1,𝑙𝜓𝑚,𝑛𝜒𝑚𝜓𝑚,𝑛𝑘𝑚,𝑛sinh𝑘𝑚,𝑛𝑥𝑚𝑥𝑚1,𝐴𝑚𝑒𝐢𝜅𝑚𝑥𝑚+𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝜓𝑚,𝑗𝜒𝑚𝐴𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚+𝐵𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚𝜓𝑚,𝑗𝜒𝑚+1=𝑢𝑁𝑚+1,0𝑛=1𝜒𝑚+1𝜓𝑚+1,𝑛𝜓𝑚,𝑗𝜓𝑚+1,𝑛𝑘𝑚+1,𝑛sinh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚+𝐿𝑙=1𝑢𝑁𝑚+1,𝑙𝑛=1𝜓𝑚+1,𝑙𝜓𝑚+1,𝑛𝜓𝑚,𝑗𝜓𝑚+1,𝑛𝑘𝑚+1,𝑛sinh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚𝑢𝑁𝑚,0𝑛=1𝜒𝑚𝜓𝑚+1,𝑛𝜓𝑚,𝑗𝜓𝑚+1,𝑛+𝜒𝑚𝜓𝑚,𝑛𝜓𝑚,𝑗𝜓𝑚,𝑛𝑘𝑚+1,𝑛tanh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚𝐿𝑙=1𝑢𝑁𝑚,𝑙𝑛=1𝜓𝑚,𝑙𝜓𝑚+1,𝑛𝜓𝑚,𝑗𝜓𝑚+1,𝑛+𝜓𝑚,𝑙𝜓𝑚,𝑛𝜓𝑚,𝑗𝜓𝑚,𝑛𝑘𝑚+1,𝑛tanh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚+𝑢𝑁𝑚1,0𝑛=1𝜒𝑚1𝜓𝑚,𝑛𝜓𝑚,𝑗𝜓𝑚,𝑛𝑘𝑚,𝑛sinh𝑘𝑚,𝑛𝑥𝑚𝑥𝑚1+𝐿𝑙=1𝑢𝑁𝑚1,𝑙𝑛=1𝜓𝑚1,𝑙𝜓𝑚,𝑛𝜓𝑚,𝑗𝜓𝑚,𝑛𝑘𝑚,𝑛sinh𝑘𝑚,𝑛𝑥𝑚𝑥𝑚1(3.16) for 𝑗=1,2,3,,𝐽.

To obtain a complete solution in the present model, it is desirable to compare the far-field behaviors (2.7) with (3.1) for obtaining the following relations:𝐴1=2𝜅11+sinh2𝜅112𝜅1cosh𝜅11,𝐵𝑀𝐵=0,(3.17)1=𝑅2𝜅11+sinh2𝜅112𝜅1cosh𝜅11,𝐴𝑀=𝑇2𝜅𝑀𝑀+sinh2𝜅𝑀𝑀2𝜅𝑀cosh𝜅𝑀𝑀.(3.18)

It should be noted that the model defined in Section 2 is analytically converted to the problem of finding 𝑢𝑚,𝑙 and 𝐴𝑚 and 𝐵𝑚 such that (3.16) are satisfied. This conversion is arbitrarily well if all of 𝑁, 𝐿, and 𝐽 are sufficiently large.

To obtain a numerical solution to the prescribed problem, the numbers 𝑁, 𝐿, and 𝐽 should be truncated to finite values. In the model, there are (𝐿+3)(𝑀1) unknowns of 𝑢𝑚,𝑙 and 𝐴𝑚 and 𝐵𝑚, except that 𝐴1 and 𝐵𝑀 have been defined in (3.17). Therefore, 𝐽=𝐿 should be set to obtain (𝐿+3)(𝑀1) equations as given in (3.16). After the coefficients 𝐴𝑚 and 𝐵𝑚 are solved, (3.18) can thus be used to obtain the reflection and transmission coefficients of the water wave problem. These complete the procedure of numerical solutions in the IEMM. This method has the advantage to solve the velocity 𝑈𝑚 in a direct manner instead of solving the velocity potential. If it is required to solve the velocity potential, (3.1), (3.11) can be implemented to get the solution 𝜙𝑚.

4. Transfer-Matrix Method

Equation (3.16) is readily reduced to the TM method of Miles [12] and Devillard et al. [13] using the wide-spacing approximation [13] subject to 𝐽=𝐿=0, defined assinh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚0,tanh𝑘𝑚+1,𝑛𝑥𝑚+1𝑥𝑚1.(4.1) In (4.1), we assumed that the evanescent eigenfunctions originating at one step are negligible when they reach the next step. Based on these assumption, (3.16) can be simplified in the following expressions:𝑢𝑚,0𝜒𝑚𝜒𝑚=𝐢𝜅𝑚𝐴𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚,𝑢𝑚,0𝜒𝑚𝜒𝑚+1=𝐢𝜅𝑚+1𝐴𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚𝐵𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚,𝐴𝑚𝑒𝐢𝜅𝑚𝑥𝑚+𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝜒𝑚𝜒𝑚𝐴𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚+𝐵𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚𝜒𝑚𝜒𝑚+1=𝑢𝑚,0𝑋𝑚,𝑁,(4.2) where𝑋𝑚,𝑁=𝑁𝑛=1𝜒𝑚𝜓𝑚+1,𝑛𝜒𝑚𝜓𝑚+1,𝑛+𝜒𝑚𝜓𝑚,𝑛𝜒𝑚𝜓𝑚,𝑛𝑘𝑚+1,𝑛.(4.3) Following Devillard et al. [13], (4.2) can solved independently for every single step. Considering a specific step at 𝑥=𝑥𝑚, suppose that the following two quantities on the left shelf are known as follows:Ψ𝑚=𝐴𝑚𝑒𝐢𝜅𝑚𝑥𝑚+𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚,Ω𝑚1=𝜅𝑚𝑑Ψ𝑚𝑑𝑥𝑚=𝐢𝐴𝑚𝑒𝐢𝜅𝑚𝑥𝑚+𝐢𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚.(4.4) Details of these definitions are referred to [19]. Using (4.4), we can replace 𝐴𝑚 and 𝐵𝑚 by Ψ𝑚 and Ω𝑚 in (4.2). The resultant equations read𝑢𝑚,0𝜒𝑚𝜒𝑚+𝜅𝑚Ω𝑚𝑢=0,𝑚,0𝜒𝑚𝜒𝑚+1𝜅𝑚+1Ψ𝑚+1sin𝜅𝑚+1𝑥𝑚+1𝑥𝑚Ω𝑚+1cos𝜅𝑚+1𝑥𝑚+1𝑥𝑚Ψ=0,𝑚𝜒𝑚𝜒𝑚Ψ𝑚+1cos𝜅𝑚+1𝑥𝑚+1𝑥𝑚+Ω𝑚+1sin𝜅𝑚+1𝑥𝑚+1𝑥𝜒𝑚𝜒𝑚+1+𝑢𝑚,0𝑋𝑚,𝑁=0.(4.5) Equations (4.5) are sufficient to obtain the unknowns Ψ𝑚+1 and Ω𝑚+1 on the right shelf in which 𝑢𝑚,0 have been cancelled as follows:Ψ𝑚+1Ω𝑚+1=cos𝜅𝑚+1𝑥𝑚+1𝑥𝑚sin𝜅𝑚+1𝑥𝑚+1𝑥𝑚sin𝜅𝑚+1𝑥𝑚+1𝑥𝑚cos𝜅𝑚+1𝑥𝑚+1𝑥𝑚×𝜒𝑚𝜒𝑚𝜒𝑚𝜒𝑚+1𝜅𝑚𝑋𝑚,𝑁𝜒𝑚𝜒𝑚𝜒𝑚𝜒𝑚+10𝜅𝑚𝜒𝑚𝜒𝑚+1𝜅𝑚+1𝜒𝑚𝜒𝑚Ψ𝑚Ω𝑚.(4.6) Equation (4.6) is the transfer matrix method of Devillard et al. [13] based on the variational formulation of Miles [12].Clearly, the combined effect of the series of steps can be achieved by simply applying matrix multiplications to the transfer matrices of all the steps. We also notice that (4.6) can be further reduced to the plane-wave approximation of Lamb [20] by assuming 𝑋𝑚,𝑁=0. This reduction has been addressed by Miles [12].

5. Numerical Results

The validity of the present approximation is examined by three cases of waves propagating over a step, a rectangle obstacle, and a trench. All the IEMM, DEMM, TM method, and plane-wave approximations are applied to solve the prescribed wave-scattering problems. Here, the IEMM is introduced in Section 3. On the other hand, the TM method and plane-wave approximation are described in Section 4. For completeness, the DEMM of Takano [9] is briefly reviewed in the Appendix for solving water wave scattering over a series of steps since all previous studies [911] do not consider this general configuration. In all of these cases, the reflection coefficients are tabulated and compared with the results in the literatures.

If the derived methods were verified well, they can be applied to the problem of water wave scattering over Roseau's curved profile. This result demonstrates the applicability of our method for solving water wave scattering by an arbitrary bottom topography.

5.1. Water Wave Scattering over a Step

We first consider the problem of water wave scattering over a step as depicted in Figure 2. Miles [12] and Evans and Linton [19] have solved the problem using the variational formulation and the intermediate mapping technique, respectively. Table 1 presents the reflection coefficients obtained by the IEMM, TM method, plane-wave approximation, and DEMM and a comparison with the solutions of Evans and Linton [19]. In Table 1, we note that the convergence of the reflection coefficients obtained by the IEMM is significant. Furthermore, it is very interesting to observe that the solutions of the IEMM and DEMM are equal to each other up to the sixth decimal places. However, only (𝐿+3)(𝑀1) linear equations are needed to be solved in the IEMM and this number is quite smaller than that of (2𝑁+2)(𝑀1) in the DEMM. For some specific settings described in Section 4, our codes can reproduce the solutions of the TM method and plane-wave approximation. It is shown that they are in good agreement with the results listed in [19] to the third decimal places. This indicates that the solutions of IEMM or DEMM should converge to the analytical solution if no round-off errors are cumulated.

tab1
Table 1: Reflection by a single step and comparison with the solution of Miles [12].
607196.fig.002
Figure 2: Schematic sketch of water wave scattering over a step.

Figure 3 gives the reflection and transmission coefficients against 𝑘 for various depths of the downstream shelf 𝜀. This result provides a very good example for the comparison of other methods.

fig3
Figure 3: Reflection and transmission coefficients of water wave scattering over a step. (a) up steps; (b) down steps.
5.2. Water Wave Scattering over a Rectangle Obstacle

Then, we consider the water wave scattering over a rectangle obstacle as shown in Figure 4. Here, 𝐻/=2 is assumed in this example. This problem has also been solved by Mei and Black [15]. They solved the problem using Miles' variational formulation without considering the wide-spacing approximation. Their formulation is in exact agreement with our formulation of IEMM except the different definition of the coordinate system. In Table 2, the reflection coefficients for various widths of the rectangle obstacle are obtained by the IEMM, TM method, plane-wave approximation, and DEMM. A comparison with the solution of Mei and Black [15] is also presented. A good agreement between the IEMM and DEMM can also be observed. Note that the reflection coefficients obtained by the IEMM converge perfectly to the fifth decimal places.

tab2
Table 2: Reflection by a rectangle obstacle and comparison with the solution of Mei and Black [15].
607196.fig.004
Figure 4: Schematic sketch of water wave scattering over a rectangle obstacle.

Figure 5 describes the comparison of the reflection coefficients against 𝑘𝐻 obtained by the TM method, IEMM, and Mei and Black's method [15]. Good agreement between the later two methods can be observed. The reflection and transmission coefficients against 𝑘𝐻 are given in Figure 6. Full transmission for some specific 𝑘𝐻 can be observed as addressed in [15]. These specific 𝑘𝐻 values are tabulated in Table 3.

tab3
Table 3: 𝑘𝐻 values for full transmission by a rectangle obstacle.
607196.fig.005
Figure 5: Comparison of reflection coefficients of water wave scattering over a rectangle obstacle.
607196.fig.006
Figure 6: Reflection and transmission coefficients of water wave scattering over a rectangle obstacle.
5.3. Water Wave Scattering over a Trench

The last typical example of water wave scattering over a trench is defined in Figure 7. Here, 𝑏=10 is assumed since Kirby and Dalrymple [10] have solved this problem by the DEMM. Table 4 gives the reflection coefficients corresponding to different trench depths obtained by all the methods mentioned above. The first three decimals of the reflection coefficients obtained by our implementation are the same as those obtained by Kirby and Dalrymple [10]. Furthermore, the significant convergence of the DEMM can also be observed. In the table, only a few reflection coefficients obtained by the IEMM are addressed since they are the same with the DEMM up to the sixth decimals. Furthermore, the reflection and transmission coefficients against 𝑘 are given in Figure 8. In the figure, the full transmission can also be reached and its corresponding 𝑘 values are addressed in Table 5.

tab4
Table 4: Reflection by a trench and comparison with the solution of Kirby and Dalrymple [10].
tab5
Table 5: 𝑘 values for full transmission by a trench.
607196.fig.007
Figure 7: Schematic sketch of water wave scattering over a trench.
fig8
Figure 8: (a) Reflection and (b) transmission coefficients of water wave scattering over a trench.
5.4. Roseau's Explicit Solution

Finally, we apply our method of step approximation to water wave scattering over Roseau's curved profiles as depicted in Figure 9. The bottom profiles are written in the form𝛽𝑥=ln𝜇+𝜀12ln1+2𝜇cos𝛽+𝜇2,(5.1) where𝜇=tan𝛽(𝑦+1)1𝜀sin𝛽tan𝛽(𝑦+1)1𝜀cos𝛽1.(5.2) Equation (5.1) defines a step of depths from 1 to 𝜀 with 𝛽(0,𝜋/2) being the steepness of the step. In the present investigation, three typical cases of mild (𝛽=0.1), middle (𝛽=1.0), and steep (𝛽=1.0) slopes are all considered. These values are selected according to Evans and Linton [19]. The analytical reflection coefficient was derived by Roseau [1] and can be expressed as||𝑅||=||||sinh𝜋𝛽1𝑘𝜀𝑘+sinh𝜋𝛽1𝑘+𝜀𝑘+||||,(5.3) where the plus and minus signs follow from the requirements of progressive and transmission waves. Their wavenumbers are defined by𝐾=𝑘tanh𝑘=𝑘+tanh𝜀𝑘+.(5.4)

607196.fig.009
Figure 9: Roseau's curved profiles.

Table 6 gives the reflection and transmission coefficients obtained by the IEMM and TM method and its comparison with the results of Evans and Linton [19] and the analytical solutions obtained by (5.3). In the numerical calculations, we assume 𝑁=30 since it is sufficient for most of our studies. In the table, it can be seen that more steps (bigger 𝑀) generally produce better results. The best accuracy of all the three cases are basically up to four decimals. For mild-slope case, the results obtained by the IEMM and TM method are basically the same because the wide-spacing assumption is reasonable. Furthermore, the increases of 𝐿 have little help to the numerical results. On the other hand, for middle-slope case, the results of TM method basically converge to another value due to the fact that the wide-spacing approximation is not valid in this slope. Finally, for the steep-slope case the solutions corresponding to 𝐿=0 are not accurate enough. This implies that the value of 𝐿 should be increased to improve the result.

tab6
Table 6: Comparison with the explicit solution of Roseau and the method of Evans and Linton [19] (most accurate solutions are bold).

This study demonstrates the applicability of the IEMM for solving problems of water wave scattering over smoothly varying bottom profiles for both mild and steep slopes. At the same time, it is clear that the DEMM is not suitable to this example since its system matrix is too large. In addition, the IEMM could avoid the pseudoconvergence of the TM method.

6. Discussions and Conclusion

An indirect eigenfunction marching method (IEMM) is derived to solve problems of water wave scattering over a series of steps. In the solution procedure, the solution is represented by the horizontal velocity above the steps and a system of linear equations is resulted by using the Galerkin method. Furthermore, it is demonstrated that the IEMM can be exactly reduced to the transfer-matrix (TM) method when the wide-spacing approximation is applied.

Numerical methods were carried out to validate the applicability for the proposed IEMM and TM method and comparisons were made with the direct eigenfunction marching method (DEMM) and other available results in the literatures. The numerical results demonstrated that the IEMM has the same accuracy when compared with the DEMM for problems of water wave scattering over a step, a rectangle obstacle, and a trench. However, the IEMM has a smaller dimension of matrix to save computer time. For water wave scattering caused by Roseau's curved profile, the IEMM also provides accurate reflection and transmission coefficients up to four decimals whilst the DEMM fails to work due to a very large system matrix. Furthermore, the IEMM can avoid the pseudoconvergence of the TM method for problems of steep-slope bottom profiles.

Appendix

The direct eigenfunction marching method of Takano [9] is governed by the following four equations:𝐢𝜅𝑚𝐴𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝜒𝑚𝜒𝑚+𝑁𝑛=1𝑘𝑚,𝑛𝐶𝑚,𝑛𝐷𝑚,𝑛𝑒𝑘𝑚,𝑛(𝑥𝑚𝑥𝑚1)𝜓𝑚,𝑛𝜒𝑚=𝐢𝜅𝑚+1𝐴𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚𝐵𝑚𝑒𝐢𝜅𝑚+1𝑥𝑚𝜒𝑚+1𝜒𝑚+𝑁𝑛=1𝑘𝑚+1,𝑛𝐶𝑚+1,𝑛𝑒𝑘𝑚+1,𝑛(𝑥𝑚+1𝑥𝑚)𝐷𝑚+1,𝑛𝜓𝑚+1,𝑛𝜒𝑚,(A.1)𝐢𝜅𝑚𝐴𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝜒𝑚𝜓𝑚,𝑙+𝑁𝑛=1𝑘𝑚,𝑛𝐶𝑚,𝑛𝐷𝑚,𝑛𝑒𝑘𝑚,𝑛(𝑥𝑚𝑥𝑚1)𝜓𝑚,𝑛𝜓𝑚,𝑙=𝐢𝜅𝑚+1𝐴𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚𝐵𝑚𝑒𝐢𝜅𝑚+1𝑥𝑚𝜒𝑚+1𝜓𝑚,𝑙+𝑁𝑛=1𝑘𝑚+1,𝑛𝐶𝑚+1,𝑛𝑒𝑘𝑚+1,𝑛(𝑥𝑚+1𝑥𝑚)𝐷𝑚+1,𝑛𝜓𝑚+1,𝑛𝜓𝑚,𝑙𝐴for𝑙=1,2,3,,𝑁,(A.2)𝑚𝑒𝐢𝜅𝑚𝑥𝑚+𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝜒𝑚𝜒𝑚+𝑁𝑛=1𝐶𝑚,𝑛+𝐷𝑚,𝑛𝑒𝑘𝑚,𝑛(𝑥𝑚𝑥𝑚1)𝜒𝑚𝜓𝑚,𝑛=𝐴𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚+𝐵𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚𝜒𝑚𝜒𝑚+1+𝑁𝑛=1𝐶𝑚+1,𝑛𝑒𝑘𝑚+1,𝑛(𝑥𝑚+1𝑥𝑚)+𝐷𝑚+1,𝑛𝜒𝑚𝜓𝑚+1,𝑛,𝐴(A.3)𝑚𝑒𝐢𝜅𝑚𝑥𝑚+𝐵𝑚𝑒𝐢𝜅𝑚𝑥𝑚𝜓𝑚,𝑙𝜒𝑚+𝑁𝑛=1𝐶𝑚,𝑛+𝐷𝑚,𝑛𝑒𝑘𝑚,𝑛(𝑥𝑚𝑥𝑚1)𝜓𝑚,𝑙𝜓𝑚,𝑛=𝐴𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚+𝐵𝑚+1𝑒𝐢𝜅𝑚+1𝑥𝑚𝜓𝑚,𝑙𝜒𝑚+1+𝑁𝑛=1𝐶𝑚+1,𝑛𝑒𝑘𝑚+1,𝑛(𝑥𝑚+1𝑥𝑚)+𝐷𝑚+1,𝑛𝜓𝑚,𝑙𝜓𝑚+1,𝑛for𝑙=1,2,3,,𝑁,(A.4) where𝜒𝑚𝜒(𝑦)=𝑚+1(𝑦)if𝑚<𝑚+1,𝜒𝑚(𝑦)if𝑚+1<𝑚,𝜓𝑚,𝑙𝜓(𝑦)=𝑚+1,𝑙(𝑦)if𝑚<𝑚+1,𝜓𝑚,𝑙(𝑦)if𝑚+1<𝑚.(A.5) In (A.1)~(A.4), there are (𝑀1)(2𝑁+2) unknowns, 𝐴2,,𝐴𝑀, 𝐵1,,𝐵𝑀1, 𝐷2,,𝐷𝑀, 𝐶1,,𝐶𝑀1, and can be solved by the above (𝑀1)(2𝑁+2) equations.

Acknowledgment

The support of National Science Council, Taiwan under the Grant of NSC 99-2221-E-022-007 is gratefully acknowledged.

References

  1. M. Roseau, Asymptotic Wave Theory, vol. 2 of North-Holland Series in Applied Mathematics and Mechanics, North-Holland, Amsterdam, The Netherlands, 1976. View at Zentralblatt MATH
  2. J. C. W. Berkhoff, “Computation of combined refraction-diffraction,” in Proceedings of the 13th International Conference on Coastal Engineering, pp. 471–490, 1972.
  3. P. G. Chamberlain and D. Porter, “The modified mild-slope equation,” Journal of Fluid Mechanics, vol. 291, pp. 393–407, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. S. R. Massel, “Extended refraction-diffraction equation for surface waves,” Coastal Engineering, vol. 19, no. 1-2, pp. 97–126, 1993. View at Google Scholar
  5. D. Porter, “The mild-slope equations,” Journal of Fluid Mechanics, vol. 494, pp. 51–63, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. D. Porter and D. J. Staziker, “Extensions of the mild-slope equation,” Journal of Fluid Mechanics, vol. 300, pp. 367–382, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. R. Porter and D. Porter, “Water wave scattering by a step of arbitrary profile,” Journal of Fluid Mechanics, vol. 411, pp. 131–164, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. D. Porter and R. Porter, “Approximations to wave scattering by an ice sheet of variable thickness over undulating bed topography,” Journal of Fluid Mechanics, vol. 509, pp. 145–179, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. K. Takano, “Effets d'un obstacle parallélépipédique sur la propagation de la houle,” La Houille Blanche, vol. 15, pp. 247–267, 1960. View at Google Scholar
  10. J. T. Kirby and R. A. Dalrymple, “Propagation of obliquely incident water waves over a trench,” Journal of Fluid Mechanics, vol. 133, pp. 47–63, 1983. View at Google Scholar
  11. M. Söylemez and Ö. Gören, “Diffraction of oblique waves by thick rectangular barriers,” Applied Ocean Research, vol. 25, no. 6, pp. 345–353, 2003. View at Publisher · View at Google Scholar
  12. J. W. Miles, “Surface-wave scattering matrix for a shelf,” Journal of Fluid Mechanics, vol. 28, pp. 755–767, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. P. Devillard, F. Dunlop, and B. Souillard, “Localization of gravity waves on a channel with a random bottom,” Journal of Fluid Mechanics, vol. 186, pp. 521–538, 1988. View at Google Scholar
  14. J. N. Newman, “Propagation of water waves over an infinite step,” Journal of Fluid Mechanics, vol. 23, pp. 399–415, 1965. View at Publisher · View at Google Scholar
  15. C. C. Mei and J. L. Black, “Scattering of surface waves by rectangular obstacles in waters of finite depth,” Journal of Fluid Mechanics, vol. 38, no. 3, pp. 499–511, 1969. View at Google Scholar
  16. T. J. O'Hare and A. G. Davies, “A new model for surface wave propagation over undulating topography,” Coastal Engineering, vol. 18, no. 3-4, pp. 251–266, 1992. View at Google Scholar
  17. T. J. O'Hare and A. G. Davies, “A comparison of two models for surface-wave propagation over rapidly varying topography,” Applied Ocean Research, vol. 15, no. 1, pp. 1–11, 1993. View at Google Scholar
  18. G. B. Airy, “On tides and waves,” Encyclopedia Metropolitan, pp. 241–396, 1845. View at Google Scholar
  19. D. V. Evans and C. M. Linton, “On step approximations for water-wave problems,” Journal of Fluid Mechanics, vol. 278, pp. 229–249, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. H. Lamb, Hydrodynamics, C. J. Clay and Sons, London, UK, 2008.