  Citation: Magdalena, I.; Firdaus, K. Numerical Study for Unsteady Waves Generated by Flow over a Permeable Wavy Bed. Fluids 2022, 7, 9. https://doi.org/10.3390/fluids 7010009 Academic Editors: Alberto Alberello, Richard Manasseh and Laura A. Miller Received: 14 August 2021 Accepted: 21 December 2021 Published: 27 December 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). fluids Article Numerical Study for Unsteady Waves Generated by Flow over a Permeable Wavy Bed Ikha Magdalena 1,2, * ,† and Kemal Firdaus 1,† 1 Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Bandung 40132, Indonesia; kemalfds@gmail.com 2 Center for Marine and Coastal Development, Bandung Institute of Technology, Bandung 40132, Indonesia * Correspondence: ikha.magdalena@math.itb.ac.id; Tel.: +62-22-2502545 (ext. 116) † These authors contributed equally to this work. Abstract: In this paper, we formulate a numerical model to study unsteady waves generated by fluid flow over a permeable wavy bed. The model is derived from boundary value problems using potential theory. We solve the model numerically using a finite difference method. As a result, we found that the flow over a porous layer generates wave disturbed by bumps on the porous layer. The simulation also showed that the wave profile shifts from the permeable bed. The results of this study can be incorporated into the design of submerged artificial and natural breakwaters. Keywords: permeable wavy bed; Darcy’s law; potential theory; Boussinesq equation; finite difference 1. Introduction Most coastal protection structures consist of permeable beds or porous media that cause the waves passing through them to be reflected or dissipated. A structure is declared efficient if it is able to maximize dissipation and minimize reflections. Some examples of the use of porous mediums in coastal defense structures include submerged porous breakwaters, which can be formed by natural and artificial coral reefs. Besides their beauty and utility as marine habitats, coral reefs are also able to reduce beach erosion [1–3]. Several studies on flows over a porous layer can be found in the literature. For instance, the numerical scheme of Liu et al. [4] was validated using data from simple experiments on liquid flows through different porous mediums types. Carotenuto and Minale [5] experi- mentally investigated the velocity profile of a fluid undergoing simple shear above a porous medium. Kim, Cho, and Choi [6] analyzed the experimental result with theoretical results calculated using Darcy’s Law and Forchheimer’s equation. However, experiments are costly. Consequently, recent studies have focused on the use of mathematical modeling to in- vestigate flows over a porous media. Examples of researchers who have studied flows over a porous layer using Navier–Stokes equations include Liu et al. [4], who constructed a nu- merical scheme based on Navier–Stokes equations, Bruneau and Mortazavi [7] who build a numerical model based on Navier–Stokes equations, and Cimolin and Discacciati [8], who compared the Navier–Stokes/Forchheimer, Navier–Stokes/Darcy, and penalization models. Models for flow over permeable beds have also been formulated with the shallow water equations (SWEs) as their base. These are known for their simplicity. For example, Mag- dalena et al. [9] studied wave interaction with submerged porous mediums, Wiryanto [10] analyzed wave propagation over a submerged breakwater for monochromatic and solitary waves, and Wiryanto [11] studied fluid disturbances caused by the uneven surface of a porous medium. There are trade-offs involved when choosing between these approaches. Models based on Navier–Stokes equations are more computationally expensive [12] and SWEs are less precise for solving problems with high complexity [13]. In this paper, we derive a model for unsteady waves generated by a flow over a per- meable wavy bed based on potential theory and Darcy’s law. Potential theory is then used to build a model similar to a Boussinesq-type equation. This model was chosen because Fluids 2022, 7, 9. https://doi.org/10.3390/fluids7010009 https://www.mdpi.com/journal/fluids