The problem of oblique wave scattering by a rectangular submarine trench is investigated assuming a linearized theory of water waves. Due to the geometrical symmetry of the rectangular trench about the central line $x=0$, the boundary value problem is split into two separate problems involving the symmetric and antisymmetric potential functions. A multi-term Galerkin approximation involving ultra-spherical Gegenbauer polynomials is employed to solve the first-kind integral equations arising in the mathematical analysis of the problem. The reflection and transmission coefficients are computed numerically for various values of different parameters and different angles of incidence of the wave train. The coefficients are depicted graphically against the wave number for different situations. Some curves for these coefficients available in the literature and obtained by different methods are recovered.

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 coefficients 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.