Water wave scattering by multiple thin vertical barriers

A study of obliquely incident water wave scattering by two, three and four unequal partially immersed vertical barriers in water of uniform finite depth has been carried out in this paper employing Havelock’s expansion of water wave potential. A formulation involving integral equations in terms of either horizontal component of velocities across the gap below each barrier or difference of potentials across each barrier are obtained using the Havelock’s inversion formulae. A multi-term Galerkin approximation technique with Chebychev’s polynomials (multiplied by appropriate weights) as basis functions is adapted to solve these integral equations and to compute the reflection and transmission coefficients numerically. The numerical results are depicted graphically against the wavenumber in several figures for various arrangements of the vertical barriers. From these figures zeros of reflection coefficient are observed when the vertical barriers are immersed upto equal depths below the mean free surface. However, this observation is not true always for non-identical vertical barriers. For two non-identical partially immersed barriers reflection coefficient never vanishes whereas for three non-identical partially immersed barriers reflection coefficient vanishes at discrete frequencies if the two outer barriers have equal lengths of submergence. For four partially immersed barriers arranged symmetrically about a vertical line, zeros of reflection coefficient are always observed. The known results of a single barrier are recovered as special cases so as to establish the correctness of the present method.