Scattering of gravity waves by a porous rectangular barrier on a seabed*

Qing-rui MENG （孟慶瑞）， Dong-qiang LU （盧東強）

Shanghai Institute of Applied Mathematics and Mechanics， Shanghai University， Shanghai 200072， China

Shanghai Key Laboratory of Mechanics in Energy Engineering， Shanghai University， Shanghai 200072， China，

E-mail: mqr.grad@gmail.com

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Scattering of gravity waves by a porous rectangular barrier on a seabed*

Qing-rui MENG （孟慶瑞）， Dong-qiang LU （盧東強）

Shanghai Institute of Applied Mathematics and Mechanics， Shanghai University， Shanghai 200072， China

Shanghai Key Laboratory of Mechanics in Energy Engineering， Shanghai University， Shanghai 200072， China，

E-mail: mqr.grad@gmail.com

Within the frame of linear potential flow theory， the impact of a porous rectangular barrier on a seabed on the dynamic characteristics of gravity waves is investigated. The porous barrier can be regarded as an abstract representative such as a seabed plant， a wave breaker， inhomogeneous topography， and trussed supporting of ocean engineering platform， etc. In the process of mathematical modeling， the method of matched eigenfunction expansions is employed for analysis， where a newly defined form of inner product is introduced to improve the simplicity of derivation. Under this definition， the inner product is automatically orthogonal， which will provide great simplification to obtain the expansion coefficients. Once the wave numbers for the fluid region and the barrier region are obtained， the reflection and transmission coefficients of the wave motion can readily be calculated.

gravity waves， dispersion relation， permeability， orthogonality

Introduction

Considering the impact of a porous medium on wave motions is one of the meaningful and significant research issues in hydrodynamics. Many types of constructs and topographies in the ocean environment，such as wave breaker， permeable seabed， water plants zone， etc.， can be modeled by porous media. The permeability of relevant objects is usually formulated in two different ways. One is to neglect the thickness of the porous medium by considering the changes of velocity field on its surface to carry out appropriate boundary conditions， e.g.， Martha et al.［1］and Mohapatra［2］. Another way is to regard the porous medium as a special region where the wave motion is described by a potential function after equivalent averaging. For example， Das and Bora［3］studied the case of wave propagating across a porous medium occupying the whole depth of the fluid and afterwards reflected by a vertical rigid wall. Recently， Metallinos et al.［4］investigated wave scattering by a porous trapezoidal structure submerged on the seabed of shallow water region by a computational method.

In this Letter， an improved semi-analytical approach is proposed to obtain the reflection and transmission coefficients of free-surface gravity wave scattering by a porous rectangular barrier fixed on the fluid bottom. The method of matched eigenfunction expansions is employed in the derivation. A newly defined form of inner products with orthogonality for the vertical eigenfunctions in the porous barrier region is found to deal with the matching relations between different fluid regions， by which the number of simultaneous equations is reduced by half.

The physical model of the related problem is given under a two-dimensional Cartesian coordinate systemplaced on the surface of a homogeneous fluid with thez axis upwards vertically， as shown in Fig.1. Free-surface gravity waves income fromonto the porous rectangular barrier symmetrically mounted on the seabed with respect to thez axis. The width and the height of the barrier areandrespectively， and the distance from the top of the barrier to the free surface isbe the depth of the fluid. Obviously

Fig.1 Schematic diagram of free surface gravity waves incoming onto a porous rectangular barrier mounted on seabed

With the assumptions that the fluid is ideal and incompressible and the wave motion is irrotational and time-harmonic， the velocity field of the fluid can be described within the potential flow theory. Letbe the velocity potential function， whereis the time. The fluid inside the porous barrier still satisfies the conservation law of mass， such that the wave motion inside can be described by the potential flow theory as well. By regardingas a piecewise function for the whole fluid domain， the part inside the barrier is actually generated by the equivalent averaging of the relevant fluid motion. Letbe the frequency of the incident waves. We separate the time variable ofvia taking the form of， such that， the spatial part of the potential function， should satisfy

On the impermeable seabed， the boundary condition reads

According to the linearized Bernoulli equation， the combined boundary condition on the free surface is

Associating Eqs.（4）-（7）， the matching relations can be derived as

For a given incident frequency， one can find two real rootsand infinite numbers of pure imaginary rootsfrom Eq.（11）. Forin the region， Eq.（14） will exhibit the similar root situation as well， while for a non-zero linear friction factor， all of the real and imaginary roots will diverge from the real and imaginary axes and change to complex ones. Hereinafter let2，L）denote the complex roots of regionshould be the most propagative wave mode varying from the original real root while the rest ofj must be related to the original pure imaginary roots.

Based on the vertical eigenfunctions and far-field conditions， the potential functions for the regionscan be expanded according to relevant wave numbers as follows:

In order to obtain the unknown coefficients， some proper approach should be applied to the matching relations. In this letter， the inner product method is employed for derivation. It is found that the set of vertical eigenfunctions for the regionhas the property of orthogonality by taking the definition of

Fig.2 Reflection coefficientsversus incident wave numbersat different linear friction factors

Fig.3 Transmission coefficientsversus incident wave numbersat different linear friction factors

It is shown that with the increase of the incident wave number， the reflection coefficientexhibits a fluctuant profile and the transmission coefficientwill decrease firstly and then increase. For the impact of the linear friction factorwill be strengthened whilewill be weakened by a larger

In conclusion， the reflection and transmission coefficients of free-surface gravity wave scattering by a porous rectangular barrier mounted on seabed is calculated by an inner product method proposed in this letter. The correctness of the method can be validated by checking the energy conservation relationat f =0.

References

［1］ MARTHA S. C.， BORA S. N. and CHAKRABARTI A. Oblique water-wave scattering by small undulation on a porous sea-bed［J］. Applied Ocean Research， 2007， 29（1）: 86-90.

［2］ MOHAPATRA S. The interaction of oblique flexural gravity incident waves with a small bottom deformation on a porous ocean-bed: Green’s function approach［J］. Journal of Marine Science and Application， 2016， 15（2）: 112-122.

［3］ DAS S.， BORA S. N. Wave damping by a vertical porous structure placed near and away from a rigid vertical wall［J］. Geophysical and Astrophysical Fluid Dynamics，2014， 108（2）: 147-167.

［4］ METALLINOS A. S.， REPOUSIS E. G. and MEMOS C. D. Wave propagation over a submerged porous breakwater with steep slopes［J］. Ocean Engineering， 2016， 111: 424-438.

［5］ SOLLITT C. K.， CROSS R. H. Wave transmission through permeable breakwaters［J］. Coastal Engineering Proceedings， 1972， 1（13）: 1827-1846.

May 20， 2015， Revised May 27， 2015）

* Project supported by the National Basic Research Development Program of China （973 Program， Grant No. 2014CB046203）， the National Natural Science Foundation of China （Grant No. 11472166） and the Natural Science Foundation of Shanghai （Grant No. 14ZR1416200）.

Biography: Qing-rui MENG （1989-）， Male， Master Candidate

Dong-qiang LU，

E-mail: dqlu@shu.edu.cn