The scattering properties of a membrane constrained on a small circular region

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 activate d by a time-harmonic sound wave and the membrane is free to undergo transve rse vibrations in response to the fluid loading, except for a small circula r disc region (x(2) + y(2) < a(2), z = 0) that is constrained to have zero displacement. This might represent the effect of a rivet or of a thin circu lar strut perpendicular to the plane of the membrane. By allowing for the i ncident 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 di sc region. An asymptotic solution is sought in the small disc limit epsilon = ka --> 0, where k = omega/c is the acoustic wavenumber and omega is the radian frequency. The inner and outer expansions are shown to involve gauge functions that are not simply powers of epsilon but which have the form ep silon(n)(tau - ln epsilon)(-m), where tau is a constant.