Connections between potentials describing wave scattering by complementary arrangements of vertical barriers

In this paper connections are made between the solutions of two water wave scattering problems, namely the diffraction of oblique waves by a thin vertical barrier with gaps and the complementary problem where the barriers are interchanged with the gaps. It is shown that the potential everywhere for the barrier problem is expressible in terms of the potential for the gap problem and a connection potential also associated with gaps in barriers. As a result the reflection coefficients are also shown to be connected. The theory is illustrated in two ways. First, by analytically deriving Ursell's (Q. J. Mech. Appl. Math. 1 (1947)) explicit result for a surface-piercing barrier in infinite depth from Dean's (Proc. Camb. Phil. Soc. 41 (1945)) explicit result for a submerged barrier in infinite depth. Secondly, numerical results for complementary arrangements of barriers and gaps in finite depth and under oblique wave incidence are presented.This article is dedicated to the memory of Professor Fritz Ursell who died in 2012.