A new application of an improved DRBEM model for water wave propagation over a frictional uneven bottom Ali Reza Soltankoohi a,n , Behrouz Gatmiri a,b , Asadollah Noorzad a a School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran b Ecole Nationale des Ponts et Chaussees, Paris, France article info Article history: Received 11 March 2011 Accepted 7 October 2011 Available online 14 December 2011 Keywords: Water wave diffraction and refraction Bottom friction Extended mild-slope equation Dual reciprocity boundary element method Axi-symmetric pit abstract This paper presents a numerical scheme to approximate water wave diffraction, refraction and frictional dissipation over an axi-symmetric pit. Based on an improved extended mild-slope equation (EMSE) including bottom friction effect, as the elliptic governing differential equation, dual reciprocity boundary element method (DRBEM) is employed to model water wave propagation over an axi-symmetric pit. To the authors’ knowledge, this is the first application of DRBEM for water wave scattering over a pit. In order to promote accuracy of the model, not only effects of the bottom curvature and the slope-squared terms which are neglected in the mild-slope equation (MSE), are considered, but also effect of the bottom friction is measured by the improved EMSE. Numerical results are compared with existing analytical or numerical solutions or with experimental data by several examples. Through these numerical experiments reliability and efficiency of present DRBEM model for determining the total wave field over an uneven bottom is approved. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction Combined diffraction and refraction of water waves which illustrates the total wave field around obstacles and over variable bed topographies is of great interest to coastal and ocean engineers, either in design procedures or in the wave hydrody- namics and sediment transportation studies. The mild-slope equation (MSE), which was firstly derived by Berkhoff [1], is a powerful mathematical model to simulate the scattering of surface gravity water waves on moderately varying water depth. It is capable to consider simultaneous effects of diffraction, refrac- tion, reflection, and shoaling of simple harmonic water waves. In fact, the MSE is a depth-integrated form of the well-known Laplace equation under the assumption of moderately water depth varia- tion. Thus, as a great advantage of the MSE rather than the Laplace potential equation for simulating water wave field in large areas, the dimension of the problem can be reduced by one. Ito and Tanimoto [2] also derived an equation describing the propagation of short waves which is similar in form to the long- wave equation. Smith and Sprinks [3] proposed a formal deriva- tive of the MSE which is a time-dependent equation. The MSE as a general equation is valid for all water depths, i.e. shallow, intermediate and deep waters. In shallow water depth, it reduces to the long wave (shallow water wave) equation whereas in deep or constant water depth, it reduces to the Helmholtz equation. Due to its advantage in reducing a three-dimensional problem to a two-dimensional one and its applicability to a wide range of wave frequencies from short to long water waves, many studies have been conducted in which the applicability of the MSE to various coastal and offshore engineering problems is investi- gated [4–9]. As we know, the MSE as well as the Laplace equation is an elliptic partial differential equation which needs to prescribe boundary conditions on all boundaries and solves as a boundary value problem. This can be computationally tedious and expen- sive, in particular for large and complicated domains. The diffi- culties involved in the solution of an elliptic equation prompted the development by Radder [10] of the parabolic approximation to the MSE. But the reflected wave field is assumed negligible in his parabolic approximation. Without the loss of the reflected wave, Copeland [11] expressed the MSE in the form of a pair of first-order equations, which constitute a hyperbolic system. Many numerical studies have been reported to solve water wave scattering using the elliptical form of the MSE. Among numerical solutions that have been given since are those by Berkhoff [1], Bettess and Zienkiewicz [4], Houston [5], Tsay and Liu [6], Li and Anastasiou [7], Zhu [8], and Zhu et al. [9]. All of them used the finite element method with the exception of Li and Anastasiou [7] who Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements 0955-7997/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2011.10.001 n Corresponding author. E-mail addresses: asoltankoohi@ut.ac.ir (A.R. Soltankoohi), behrouz.gatmiri@andra.fr (B. Gatmiri), noorzad@ut.ac.ir (A. Noorzad). Engineering Analysis with Boundary Elements 36 (2012) 537–550