Wave Motion 43 (2005) 167–175 Water wave scattering by two partially immersed nearly vertical barriers Soumen De, Rupanwita Gayen, B.N. Mandal ∗ Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India Received 14 June 2005; received in revised form 25 July 2005; accepted 21 September 2005 Available online 20 October 2005 Abstract This paper provides a mathematical investigation of the problem of scattering of surface water waves by two surface-piercing barriers that are almost vertical and are described by the same shape function in the context of linear theory by employing a simplified perturbational analysis. Green’s identity is used to express the perturbation to the quantities of interest—the reflection and transmission coefficients—in terms of the solution to the unperturbed system. As in the case of single nearly vertical barrier, here also the perturbed transmission coefficient vanishes identically while the perturbed reflection coefficient is obtained in terms of a number of definite integrals involving shape function. When the two barriers are merged into a single barrier, the known result for a single barrier is recovered. © 2005 Elsevier B.V. All rights reserved. Keywords: Water wave scattering; Linear theory; Nearly vertical barriers; Perturbation analysis; Reflection and transmission coefficients 1. Introduction Within the framework of linearised theory of water waves, the problems involving scattering of normally incident surface water wave trains in deep water by thin fixed plane vertical barriers, admit of exact solutions (cf.Ursel [1], Evans [2], Porter [3] for a single barrier, Levine and Rodemich [4], Jarvis [5] for two equal parallel barriers). A substantial amount of research work related to water wave scattering problems involving the thin vertical barriers has been carried out during the last six decades (see Mandal and Chakrabarti [6]). Problems involving thin curved barriers or inclined straight plane barriers do not admit of exact solutions but can be studied by some approximate methods. These have been studied by using hypersingular integral equation formulations (cf. Parson and Martin [7], [8], Midya et al [9], Kanoria and Mandal [10], Mandal and Gayen(Chowdhury) [11]). The hypersingular integral equation arising in each problem has been solved approximately and numerical estimates for the reflection and transmission coefficients are then obtained. When the barriers are slightly curved, which we call nearly vertical barriers, it is also not possible to find exact solutions, rather perturbed forms of the solution can be found. A problem associated with a surface-piercing nearly vertical single barrier was first handled by Shaw [12] using perturbational analysis which involved the solution of a singular integral equation. Evans [13], in a short note, gave an idea for computation of the wave amplitude produced by small oscillations of a partially immersed flexible plate which involved an application of ∗ Corresponding author. Tel.: +91 33 2575 3034; fax: +91 33 2577 3026. E-mail address: biren@isical.ac.in (B.N. Mandal) 0165-2125/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2005.09.001