A membrane occupies the plane z = 0 in a Cartesian coordinate system and is surrounded by a compressible fluid of wave speed c. The system is activated by a time–harmonic sound wave and the membrane is free to undergo transverse vibrations in response to the fluid loading, except for a small circular disc region (x2 + y2 < a2, z = 0) that is constrained to have zero displacement. This might represent the effect of a rivet or of a thin circular strut perpendicular to the plane of the membrane. By allowing for the incident and reflected waves for an infinite plane with no constraints, the problem is readily reduced to that in which there is no incident field, but with vibrations induced by a given time-harmonic normal velocity on the disc region. An asymptotic solution is sought in the small disc limit ϵ = ka → 0, where k = ω/c is the acoustic wave number and ω is the radian frequency. The inner and outer expansions are shown to involve gauge functions that are not simply powers of ϵ but which have the form ϵn (τ – lnϵ)−m, where τ is a constant.