The problem of surface water wave scattering by two thin nearly vertical barriers submerged in deep water from the same depth below the mean free surface and extending infinitely downwards is investigated here assuming linear theory, where configurations of the two barriers are described by the same shape function. By employing a simplified perturbational analysis together with appropriate applications of Green's integral theorem, first-order corrections to the reflection and transmission

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... cients are obtained. As in the case of a single nearly vertical barrier, the first-order correction to the transmission coefficient is found to vanish identically, while the correction for the reflection coefficient is obtained in terms of a number of definite integrals involving the shape function describing the two barriers. The result for a single barrier is recovered when two barriers are merged into a single barrier. 2000 Mathematics subject classification: primary 76B07; secondary 76B15. of Levine and Rodemich [6] and Jarvis [4] employ the Schwartz-Christoffel transformation of complex variable theory. Evans and Morris [3] reinvestigated Levine and Rodemich's problem [6] of two equal surface-piercing barriers and obtained approximate expressions for the reflection and transmission coefficients which are simpler to compute numerically than Levine and Rodemich's explicit results [6] involving definite integrals whose integrands are complicated functions of elliptic integrals. For two unequal surface-piercing barriers in finite depth water, Mclver [12] used the method of matched eigenfunction expansions to obtain very accurate numerical estimates for the reflection and transmission coefficients. A substantial amount of research relating to water wave scattering problems involving thin vertical barriers has been carried out during the last six decades (see Mandal and Chakrabarti [8]). Problems involving thin curved barriers or inclined straight plane barriers have also been studied mostly by using hypersingular integral equation formulations (see, for example, Parsons and Martin [14] and [15], Midya et al. [13], Kanoria and Mandal [5], Mandal and Gayen (Chowdhury) [9]) which essentially involve somewhat heavy numerical computations in obtaining numerical estimates for the reflection and transmission coefficients. A thin plate in the form of a circular arc symmetric about the vertical through its centre was investigated by Mclver and Urka [11] by two methods, one based on the method of matched series expansions and the other based on Schwinger variational approximation. If the thin barrier differs slightly from the vertical position and in general is of a curved nature described by a shape function, the corresponding scattering problem has no explicit solution. An integral equation formulation of the problem is always possible, whose explicit solution is almost impossible to obtain. For the case of a nearly vertical surface-piercing thin barrier, Shaw [17] employed a perturbational approach to obtain first-order corrections to the reflection and transmission coefficients R\,T X in terms of some definite integrals involving the shape function, which are however somewhat difficult to evaluate. He used a physical argument to prove that T\ vanishes identically and then used this result to simplify R\. Soon afterwards Mandal and Chakrabarti [7] devised a simplified perturbation method employed for the governing partial differential equation, the boundary and other conditions, describing the original problem. This procedure reduced to first order, and the original problem became two problems involving a vertical barrier. The solution of the first problem is well known in the literature. Without solving the second problem, the first-order corrections R\, T\ were obtained by simple applications of Green's integral theorem. That T x vanishes identically was shown in a straightforward manner, and the result for ^i in the form obtained by Shaw [17] for the partially immersed barrier was also obtained without much effort. The complementary problem of a nearly vertical barrier submerged in deep water and extending infinitely downwards was also investigated by Mandal and Chakrabarti [7] by the same procedure. The case of a nearly vertical thin plate was use, available at https://www.cambridge.org/core/terms. https://doi.