Analysis of sound attenuation in elliptical chamber mufflers by using Green's functions

The present work aims at finding the transmission loss of an elliptical expansion chamber, the inlet and outlet of which are located at arbitrary locations of the chamber, i.e. the side wall or on the face of the muffler. The analysis is based on the Green's function solution for an elliptical cavity with homogeneous boundary conditions. Solving field problems with elliptical geometries require the computation of Mathieu and modified Mathieu functions. These are the eigenfunctions of the wave equation in elliptical coordinates and their computations pose a considerable challenge. In our present study, we have tried to develop a formulation for finding the transmission loss using the properties of the Mathieu and the modified Mathieu functions. The Green's function is found by considering the boundary to be rigid walls with homogeneous boundary conditions. The inlet and outlet are assumed to be uniform velocity piston sources. The velocity potential inside the muffler is found by adding the individual potentials arising from the inlet and outlet pistons. The pressure in the chamber is obtained from the velocity potential through the linear momentum equation. The pressure at the inlet and at the outlet is approximated by the averaging the acoustic pressure over the piston area. The four-pole parameter is derived from the average pressure values and hence the transmission loss is calculated. The results are compared to those available in literature. It is shown that the results obtained from the present work agree well with those reported in literature.