Lie analysis of the Webster horn equation with application to audio object recognition

The Webster horn equation describes the pressure wave in a duct of slowly varying cross section. We discuss symmetry reductions and exact solutions of the Webster horn equation using the classical Lie method of infinitesimals. The particular case of the exponential horn is examined and a complete set of reductions and solutions is formulated. The generation of a complete set of solutions using Lie analysis produces a set of group transformations. Particular attention is given to a new solution found, which contains an exponentially decaying Bessel function. The use of these group transformations as a tool for audio object recognition is also explored. Results indicate that the decaying Bessel function solution provides a particularly useful insight into exponential horn object recognition. Practical results are presented which indicate the group transformations offer an exciting new mechanism for identifying a specific audio object in a mixed audio scene.