Asymptotic Expansion and Numerical Solution for Gravity Waves in a Porous Media

Abstract

In this paper, we study wave interaction with a porous structure. A nonlinear diffusive wave equation is used to describe gravity surface wave propagation in a porous media. We solve the equations using asymptotic expansion method and numerically using a staggered finite volume method. We then derive the dispersion relation that holds for gravity waves inside a porous structure. This dispersion relation explains the diffusive mechanism of wave amplitude inside the porous structure. Analysis of the dispersion relation shows that amplitude reduction depends on porous medium parameters such as porosity, friction coefficient, length of the structure, and wave frequency. To validate our numerical scheme, we compare the wave reduction amplitude from the numerical result with the asymptotic solution. A good agreement of the comparison is observed. Furthermore, the numerical model is employed to investigate the effectiveness of porous media in dissipating wave energy of an incoming wave. The results from this paper can be used to determine the optimum dimension of the porous medium so that the incoming wave can be reduced as much as possible to protect shoreline. References H.J.S. Fernando, S.P. Samarawickrama, S. Balasubramanian, S.S.L. Hettiarachchi, and S. Voropayev. Effects of porous barriers such as coral reefs on coastal wave propagation. Journal of Hydro-environment Research, 1(3):187 – 194, 2008. Pengzhi Lin and S. A. Karunarathna. Numerical study of solitary wave interaction with porous breakwaters. Journal of Waterway, Port, Coastal, and Ocean Engineering, 133(5):352–363, 2007. Philip L.-F. Liu, Pengzhi Lin, Kuang-An Chang, and Tsutomu Sakakiyama. Numerical modeling of wave interaction with porous structures. Journal of Waterway, Port, Coastal, and Ocean Engineering, 125(6):322–330, 1999. Patrick J. Lynett, Philip L.-F. Liu, Inigo J. Losada, and Cesar Vidal. Solitary wave interaction with porous breakwaters. Journal of Waterway, Port, Coastal, and Ocean Engineering, 126(6):314–322, 2000. O.S. Madsen. Wave transmission through porous structures. Journal of Waterways, Harbors, Coastal Engineering, 100:169–188, 1974. I. Magdalena, S. R. Pudjaprasetya, and L. H. Wiryanto. Wave interaction with an emerged porous media. Advances in Applied Mathematics and Mechanics, 6(5):680–692, 06 2015. S.R. Pudjaprasetya and I. Magdalena. Momentum conservative scheme for dam break and wave run up simulations. East Asia Journal on Applied Mathematics, 4(2):152–165, 2014. Panagiotis D. Scarlatos and Vijay P. Singh. Long-wave transmission through porous breakwaters. Coastal Engineering, 11(2):141 – 157, 1987. C.K. Sollitt and R.H. Cross. Long-wave transmission through porous breakwaters. Proc. 13th Coastal Eng. Conf. 3, pages 1827–1846, 1972. W. Sulisz. Wave reflection and transmission at permeable breakwaters of arbitrary cross-section. Coastal Engineering, 9:371–386, 1985. M. R. A. Van Gent. Wave interaction with permeable coastal structures. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 33(6):277A, 1996.