Two-dimensional sound field reproduction based on Mathieu function expansion

Abstract

In this study, a two-dimensional sound field reproduction method based on Mathieu function expansion (MFE) is proposed. Mathieu functions are orthogonal solutions of the Helmholtz equation in an elliptical coordinate system. A MFE, which is similar to the conventional circular harmonic expansion, is applied to the sound field. The MFE-based method introduces elliptical properties such that the listening area is an ellipse. Three methods are described: an analytical sound field reproduction method for elliptical loudspeaker arrays, a method applying transformation (i.e., stretch, rotation, and translation) to the listening area, and a numerical approach for arbitrarily shaped arrays. A suitable truncation order for the MFE is also derived. The performance of all methods is tested in computer simulations with examples.