It is known that there are two infinite sets of discrete frequencies for which deep water waves may be either totally reflected or totally transmitted by a pair of surface-piercing barriers of equal length. Here the problem is further investigated using an accurate method based on eigenfunction expansions. It is found that for sufficiently closely spaced barriers each zero of reflection is accompanied by a zero of transmission at a nearby frequency. As the barrier spacing is increased the two lowest frequency zeros of transmission coalesce and are lost. This loss of zeros of transmission in pairs continues as the barrier spacing is further increased.
The effects of different barrier lengths and finite depth are also considered. For barriers of different lengths the zeros of reflection no longer occur while the existence of the zeros of transmission is unaffected.