The domain exterior to an island in a channel with topography is not simply connected and so the circulation about the island is indeterminate. Rhines shows that, in a rotating flow with a rigid lid, requiring the pressure to be continuous forces the circulation to be constant in time. It is shown here that the constant-circulation continuous-pressure solution also conserves energy and, moreover, the circulation associated with scattering of incident shelf waves is identically zero. The scattering problem is then well posed and a method is given for constructing the scattered field at arbitrary frequencies.
The problem simplifies greatly in the low frequency limit and an explicit solution for waves scattered by a thin barrier follows by decomposing the motion into propagating modes and a geostrophic current following Hsieh and Buchwald. Explicit values are given for the round-island flux and the distribution of scattered wave energy for a long, thin island in the center of a channel. It is shown that with increasing island length the round-island flux decreases rapidly from the value determined by requiring the volume flux to be continuous at the leading edge of the island towards a solution with little flux between the island and a coastal boundary, as in Wilkin and Chapman.