NPG Nonlinear Processes in Geophysics NPG Nonlin. Processes Geophys. 1607-7946 Copernicus Publications Göttingen, Germany 10.5194/npg-20-1023-2013 Modelling and simulation of waves in three-layer porous media Pudjaprasetya S. R. 1 Mathematics Department, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung, 40132, Indonesia 25 11 2013 20 6 1023 1030 Copyright: © 2013 S. R. Pudjaprasetya 2013 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://npg.copernicus.org/articles/20/1023/2013/npg-20-1023-2013.html The full text article is available as a PDF file from https://npg.copernicus.org/articles/20/1023/2013/npg-20-1023-2013.pdf

The propagation of gravity waves in an emerged three-layer porous medium is considered in this paper. Based on the assumption that the flow can be described by Darcy's Law, an asymptotic theory is developed for small-amplitude long waves. This leads to a weakly nonlinear Boussinesq-type diffusion equation for the wave height, with coefficients dependent on the conductivities and depths of each layer. In the limit of equal conductivities of all layers, the equation reduces to the single-layer result recorded in the literature. The model equations are numerically integrated in the case of an incident monochromatic wave hitting the layers. The results exhibit dissipation and also a downstream net height rise at infinity. Wave transmission coefficient in three-layer porous media with conductivity of mangrove is discussed. Numerically, propagation of an initial solitary wave through a porous medium shows the emergence of wave reflection and transmission that both evolve as permanent waves. Additionally we examine the impact of a solitary gravity wave on a porous medium breakwater.

 References Chwang, A. T. and Chan, A. T.: Interaction between porous media and wave motion, Annu. Rev. Fluid Mech., 30, 53–84, 1998. Dahdouh-Guebas, F.: Mangrove forests and tsunami protection, 2006 McGraw-Hill Yearbook of Science & Technology, McGraw-Hill Professional, New York, USA, 187–191, 2006. Fernando, H. J. S., Samarawickrama, S. P., Balasubramanian, S., Hettiarachchi, S. S. L., and Voropayev, S.: Effects of porous barriers such as coral reefs on coastal wave propagation, Journal of Hydro-environment Research, 1, 187–194, 2008. Kathiresan, K. and Rajendran, N.: Comments on "Coastal mangrove forests mitigated tsunami", Estuar Coast. Shelf S., 67, 539–541, 2006. Lynett, P. J., Liu, P. L. F., Losada, I. J., and Vidal, C.: Solitary wave interaction with porous breakwaters, J. Waterw. Port C.-ASCE, 314–322, 2000. Madsen, O. S.: Wave transmission through porous structures, J. Waterway Div.-ASCE, 100, 169–188, 1974. Masda, Y., Magi, M., Kogo, M., and Nguyen Hong, P.: Mangrove as a coastal protection from waves in the Tong King delta, Vietnam, Mangroves and Salt Marshes, 1, 127–135, 1997. Massel, S. R., Furukawa, K., and Brinkman, R. M.: Surface wave propagation in mangrove forests, Fluid Dyn. Res., 24, 219–249, 1999. Nielsen, P.: Tidal dynamics of the water table in beaches, Water Resour. Res., 26, 2127–2134, 1990. Lin, P. and Anuja Karunarathna, S. A. S.: Numerical Study of Solitary Wave Interaction with Porous Breakwaters, J Waterw. Port C.-ASCE, 133, 352–363, <a href="http://dx.doi.org/10.1061/(ASCE)0733-950X(2007)133:5(352)">https://doi.org/10.1061/(ASCE)0733-950X(2007)133:5(352)</a>, 2007. Liu, P. L. F. and Jiangang Wen, J.: Nonlinear diffusive surface waves in porous media, J. Fluid Mech., 347, 119–139, 1997. Liu, P. L. F., Lin, P., Chang, K. A., and Sakakiyama, T.: Numerical modeling of wave interaction with porous structures, J. Waterw. Port C.-ASCE, 125, 322–330, <a href="http://dx.doi.org/10.1061/(ASCE)0733-950X(1999)125:6(322)">https://doi.org/10.1061/(ASCE)0733-950X(1999)125:6(322)</a>, 1999. Liu, W. C., Hsu, M.H., and Wang, C.F.: Modeling of Flow Resistance in Mangrove Swamp at Mouth of Tidal Keelung River, Taiwan, J. Waterw. Port C.-ASCE, 129, 86–92, <a href="http://dx.doi.org/10.1061/(ASCE)0733-950X(2003)129:2(86)">https://doi.org/10.1061/(ASCE)0733-950X(2003)129:2(86)</a>, 2003. Scarlatos, P. D. and Singh, V. P.: Long wave transmission through porous breakwaters, Coast. Eng., 11, 141–157, 1987. Sollitt, C. K. and Cross, R. H.: Long-wave transmission through porous breakwaters, in: Proc. 13th Coastal Eng. Conf., ASCE, New York, USA, 3, 1827–1846, 1972. Sulisz, W.: Wave reflection and transmission at permeable breakwaters of arbitrary cross-section, Coast. Eng., 9, 371–386, 1985. Susilo, A. and Ridd, P. V.: The bulk hydraulic conductivity of mangrove soil perforated with animal burrows, Wetl. Ecol. Manag., 13, 123–133, 2005. Teh, S. Y., Koh, H. L., Liu, P. L. F., Ismail, A. I. Md., and Lee, H. L.: Analytical and numerical simulation of tsunami mitigation by mangroves in Penang, Malaysia, J. Asian Earth Sci., 36, 38–46, 2009. van Gent, M. R. A.: Wave interaction with permeable coastal structure, Ph.D. thesis, Delft University, Delft, the Netherlands, 1995. Vo-Luong, H. P. and Massel, S.: Energy dissipation in non-uniform mangrove forests of arbitrary depth, J. Marine Syst., 74, 603–662, 2008. Wang, K. H. and Li, W.: Modeling propagation of nonlinear shallow-water waves past a porous barrier, in: Proc. 15th Eng. Mech. Div. Conf. of the American Soc. of Civil Engineers, Columbia University, New York, USA, 2–5 June 2002. Whitham, G. B.: Linear and Nonlinear Waves, John Wiley & Sons, Inc., New York, USA, 1973.