A procedure for constructing explicitly solvable models of small inhomogeneities in boundary-contact acoustic problems is presented. The procedure is based on the theory of self-adjoint extensions of symmetric operators and enables the diffraction problem to be reduced to two simpler problems. The first problem is for a totally rigid plate and the second is for an isolated plate. In a number of cases the asymptotic analysis of these problems enables one to construct a model for inhomogeneity in the original boundary-contact problem. This procedure is used to investigate the diffraction of a plane acoustic wave at a plate with a circular aperture of small radius. The problem of diffraction of a plane wave by the aperture in a completely rigid plate and the problem of diffraction of a bending wave by the aperture in an isolated plate can be solved by separation of variables in ellipsoidal and polar coordinates, respectively. The asymptotic behaviour of the field for the original problem in the far zone is obtained.