The applicability of the Moebius transformation to mechanical and acoustic impedance modifications

Abstract

The Moebius transformation maps straight lines or circles in one complex domain into straight lines or circles in another complex plane. This paper will demonstrate that the acoustic response will trace a circle in the complex plane for straight line or circular modifications to mechanical or acoustical impedance. This is due to the fact that the equations relating the acoustic response to the modification are in a form consistent with the Moebius transformation. This is demonstrated for series and parallel mechanical and acoustic impedances. The principles presented in this paper for the case of mechanical impedance are in essence equivalent to what has been termed the generalized Vincent circle by other authors. This paper shows that the principle is valid for multiple excitation problems while also showing the applicability of the principle to acoustic impedance. The key qualifications for applying the Moebius transformation to mechanical and acoustic impedances is that the problem should be linear, and the impedance modification should be in one coordinate direction and at one position. The principle is shown to be valuable for understanding the effect of acoustic impedance modifications in waveguides. The principle is illustrated for several examples including a point excited plate, a construction cab, and an acoustic waveguide.