A regular wave train travels along the surface of a body of slightly compressible fluid of infinite depth or constant depth, and is scattered by totally immersed obstacles. The scattering geometry may be either two-or three-dimensional, but is taken to be compact in the sense that the ratio ∈, of characteristic width against depth, is small. An asymptotic solution is found in the limit ∈ ↑ 0 using the method of mathod expansions, in which a locally incompressible inner approximation is matched with an outer solution that corresponds to point- or line-source singularities. A general treatment is developed and explicit results are given for the surface waves scattered by either a pair of circular cylinders or by a prolate spheroid at incidence.