Journal of Applied Mathematics

Volume 2015, Article ID 256084, 11 pages

http://dx.doi.org/10.1155/2015/256084

## Shear Current Effects on Monochromatic Water Waves Crossing Trenches

^{1}Department of Civil and Environmental Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 133-791, Republic of Korea^{2}Civil and Environment Division, Hyundai Engineering and Construction, Gye-dong, Jongno-gu, Seoul 110-920, Republic of Korea

Received 28 October 2014; Accepted 6 January 2015

Academic Editor: Michael Meylan

Copyright © 2015 Jun-Whan Lee 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

The reflection coefficients of monochromatic water waves over trenches with shear current are estimated analytically. The diffraction of waves by an abrupt depth change and shear current is formulated by the matched eigenfunction expansion method. The proper number of steps and evanescent modes are proposed by a series of convergence tests. The accuracy of the predicted reflection coefficients is checked by estimating the wave energy. Reflection and transmission characteristics are studied for various shear current conditions. The different combinations of strength, width of shear current, and incident wave angle with constant water depth topography are examined. The optimal figure of the trench with shear current is obtained by estimating the reflection coefficients for various sloped transitions. The resonant reflection of the water waves is found by multiarrayed optimal trenches and the interaction of sinusoidally varying topography with shear current.

#### 1. Introduction

When water waves propagate over the nonuniform bottom topography and shear current field, they undergo many physical phenomenon including refraction and partial reflection. It is well known that the nearshore and coastal environments with the stability of coastal structures are significantly affected by water waves [1]. Therefore, the prediction of wave transformation is very important for coastal engineers to design counter facilities that can protect coastal structures from wave attacks. A number of wave-scattering theoretical models treating bathymetry and shear current or both have been developed.

A widely studied wave diffraction problem is abrupt depth changes. Specifically, the analytic method based on the potential flow theory has been used for decades because of its simple but absolutely accurate characteristics. By employing the well-known semianalytic matched eigenfunction expansion method to the potential flow theory, breakwaters and trenches can be represented by a series of steps. One of the earliest studies solving the scattering problem of normal incident wave over discontinuous bathymetry was given by Takano [2]. The linear integral equations were set up by the matching boundary conditions and were solved numerically for a truncated series. Miles [3] used a plane-wave and variational approximations and solved a scattering matrix for the case of discontinuity in depth. Kirby and Dalrymple [4] extended the previous studies to involve the obliquely incident water waves and asymmetric geometry of a rectangular trench. By formulating the scattering problem in terms of Schwinger’s type integrals, Lassiter III [5] has solved the wave diffraction due to a trench containing two different fluids. Bender and Dean [6] have considered scattering of normally incident waves over a trench with sloping sides by using three different methods and concluded that the step method is valid for an arbitrary water depth, while other methods are limited to the shallow water region. Recently, several kinds of techniques for solving the scattering problem, such as Galerkin method, the first and second kinds of Bessel functions, perturbation method, have been employed [7–9]. Also, several kinds of conditions that cause the scattering problem, such as porous plates, floating plate, and cylinders, have been studied [10–12].

Horizontally sheared currents such as those formed at river mouths, tidal inlets, and tidal races and around coastal structures are also one of the main factors of wave transformation [13]. However, even within the linear theory framework, the analysis of the water waves across a shear current is remarkably intractable. Therefore, many approximate methods based on mild-slope, mild-shear equations, and vortex sheets have been developed. The case of incident waves on a single vortex sheet representing a shear current was studied by Evans [14] using a Galerkin approximation and was extended to a two vortex sheets problem based on conservation of wave action and vertical averaging by Smith [15]. Smith [16] also presented a variety of approximate solutions to the wave diffraction problem caused by a uniform shear current and depth change. Kirby et al. [17] developed a matched eigenfunction expansion model to study the effect of shear current flowing along a trench in finite water depth, and the model was verified with a boundary-integral-equation method. Liu et al. [18] applied the eigenfunction expansion method to the scattering problem of obliquely incident wave groups over a trench with a shear current. An arbitrary number of discontinuities of shear current were used by McKee [19] to allow realistic representation of natural shear current by gradual transitions in water of a constant depth. The results were compared with the mild-shear equation model [20] and the extended mild-shear equation model [21].

One of the most interesting topics in the wave scattering problem is Bragg resonant reflection. When the bottom topography contains the patches of periodic undulations and the wavelength of the bottom ripple is one-half of the wavelength of the incident wave, the so called Bragg conditions, waves are scattered and amplified due to the resonant effect [22]. Dalrymple and Kirby [23] used a boundary integral equation method and calculated the wavenumber ratio for the obliquely incident wave that could yield the resonant reflection. Guazzelli et al. [24] compared the experimental measurements with numerical predictions obtained from the potential theory of linear waves. McKee [25] investigated the Bragg reflection by sinusoidally varying the shear current using a mild-shear equation and compared the result with the modulation theory. The model based on the eigenfunction expansion method was developed by Cho and Lee [26] and the obtained results were compared with the laboratory measurements of singly and doubly sinusoidally varying topographies. For a practical counter facility’s design using the Bragg reflection concept, Kirby and Anton [27] and Hsu et al. [28] investigated periodically spaced rectified sinusoidal and rectangular breakwaters, respectively. Although multiarrayed breakwaters show good performance in wave protection, these are not suitable for the region where vessel seaway is required. For this problem, Kim et al. [29] studied the reflection by multiarrayed trenches and suggested the optimal figure of trench which reflects the incident wave mostly. However, the proper number of steps and evanescent modes are not proposed, and the effect of shear current is not considered.

The aim of the present study is to extend Cho and Lee [30] by including shear current effects and to investigate the role of shear current for several cases. In this study, the semianalytic matched eigenfunction expansion technique is used to calculate the reflection coefficients. Preceding the investigation of shear current effects, after explaining the methodology briefly for completeness, the proper number of steps and evanescent modes are suggested to fully converge. Next, the significance of shear current in the wave diffraction problem is checked by adding several shear currents. The investigation is carried out for finding the main factors of the maximum reflection coefficient and its location, as well as finding the optimal figure of trench. Case studies for ideal conditions are conducted to identify the resonant reflection.

#### 2. Eigenfunction Expansion Method

The diffraction of monochromatic waves by an abrupt depth change and shear current is formulated by the eigenfunction expansion method. As shown in Figure 1, the variation of the bottom topography is limited to the -axis direction and shear current occurs only in the -axis direction. The origin of the coordinate system is placed at the beginning point where water depth or shear current differs from the incident condition. The regions where the shear current and the bottom topography change are represented by a finite number of steps; thus, the parameters are constant at each region. The relative strength of the shear current is represented by the Froude number as where is the shear current velocity, is the gravity acceleration, and is the water depth. By dividing the domain into several steps and assuming that the shear current is small enough so that there would be no possibility of turbulence flow, we could extend the scope of analysis to various topographies with nonhomogeneous shear current [15, 31].