Water-wave scattering by thick vertical barriers

This paper is concerned with two-dimensional scattering of a normally incident surface wave train on an obstacle in the form of a thick vertical barrier of rectangular cross section in water of uniform finite depth. Four different geometrical configurations of the barrier are considered. The barrier may be surface-piercing and partially immersed, or bottom-standing and submerged, or in the form of a submerged rectangular block not extending down to the bottom, or in the form of a thick vertical wall with a submerged gap. Appropriate multi-term Galerkin approximations involving ultraspherical Gegenbauer polynomials are used for solving the integral equations arising in the mathematical analysis. Very accurate numerical estimates for the reflection coefficient for each configuration of the barrier are then obtained. The reflection coefficient is depicted graphically against the wave number for each configuration. It is observed that the reflection coefficient depends significantly on the thickness for a wide range of values of the wave number, and as such, thickness plays a significant role in the modelling of efficient breakwaters.