Power law transfer matrix and the acoustic impedance of Gabriel's Horn

Abstract

One dimensional acoustic waveguides such as tubes and musical instruments can be mathematically characterized by a 2 × 2 transfer matrix which relates the pressure and flow at the inlet to the outlet of the waveguide. Matrix elements are a function of the geometry of the waveguide, frequency, and the speed of sound in the fluid medium. Transfer matrices are a mathematically convenient representation of waveguides because multiple waveguides may be easily combined, and certain characteristics such as sound transmission loss and impedance can be derived from them. Impedance maxima are frequencies where a small excitation of the volume flow results in high sound pressure. Brass instruments are usually played near these resonant frequencies. A derivation of the terms of a waveguide transfer matrix is presented and analytical solutions are shown for an assortment of geometries including waveguides with cross sectional areas varying according to a power law. This information is then used to calculate the normalized impedance of a well defined but non-physical instrument, Gabriel's Horn, which has infinite length, infinite surface area, yet finite volume. The impedance of Gabriel's Horn is compared to that of cylinders of equivalent length and input area.