Application of Hilbert space techniques and Green’s function methods to a realistic condenser microphone model

Abstract

A realistic model for a condenser microphone is theoretically investigated. The model consists of a rigid cylinder closed on the bottom by a membrane and on the top by a rigid plate. The cylinder is divided into two chambers by an electrode with a number of small holes. This coupled acoustic system is mathematically treated by an operator formalism. Each part of the system (outer space, membrane, electrode, two chambers) corresponds a linear operator in a Hilbert space. The interfaces (membrane, electrode with holes) correspond to state vectors in this Hilbert space. Applying Green’s integral formula leads to a system of equations in this space, corresponding conventionally to integral equations with Green’s functions as kernels. By approximating the operators of the two chambers by multiples of the identity operator (i.e., conventionally replacing the singular Green’s function by multiples of the Dirac delta function) it is possible to solve the problem analytically, taking into account exactly the pattern of the holes in the electrode, thus avoiding an approximation made in the preceding paper. The agreement with experiment is very satisfactory.