Water wave scattering by two surface-piercing and one submerged thin vertical barriers

The problem of water wave scattering by three thin vertical barriers present in infinitely deep water is investigated assuming linear theory. Out of the three, two outer barriers are partially immersed and the inner one is fully submerged and extends infinitely downwards. Havelock’s expansion of water wave potential along with inversion formulae is employed to reduce the problem into a set of linear first-kind integral equations which are solved approximately by using single-term Galerkin approximation technique. Very accurate numerical estimates for the reflection and transmission coefficients are then obtained. The numerical results obtained for various arrangements of the three vertical barriers are depicted graphically in several figures against the wavenumber. These figures exhibit that the reflection coefficient vanishes at discrete wavenumbers only when the two outer barriers are identical. Few known results of a single submerged wall with a gap, single fully submerged barrier extending infinitely downwards, and two barriers partially immersed up to the same depth in deep water are recovered as special cases. This establishes the correctness of the method employed here.