Analytical solution of a condenser microphone model as an example of the mathematical treatment of coupled acoustic systems

Abstract

In this work an analytical model for a condenser microphone with a pattern of holes in the electrode is solved approximately by applying an operator method of quantum mechanics. Essential is the replacement of the operators corresponding to the Green’s functions of the cylinder in the fundamental integral equations of the system by products of identity operators times an appropriately chosen constant. The second approximation is the averaging of the holes and by replacing the effect of the holes in the electrode by a product of the identity operator times the part of the area of the small holes in the electrode compared to the whole area of the electrode. The last approximation is the replacement of the displacement of the fluid in the small holes by their mean value and the assumption of small holes compared to the total area of the electrode. By a summation method over zeros of Bessel functions it is possible to do infinite summations exactly and to provide an analytical solution of the average displacement of the membrane as a function of the frequency of the incoming plane wave. The approximations are justified by the good agreement with experiment.