A Discrete Model for Tubular Acoustic Systems with Varying Cross Section - The Direct and Inverse Problems. Part 1: Theory

摘要

This paper presents two discrete lossy models for tubular acoustics systems, along with solutions to the inverse problem. The models discussed discretize the system into short cylindrical or conical segments; though these model are notnew, we point out various errors in the way they were dealt with in the past, and extend the mathematical treatment of the latter model, defining a "conical reflection coefficient" at the junction between two conical segments. The models are extended here to account also for viscous and thermal losses at the tube walls, using a method suggested by the authors in a previous paper. The cylindrical segment model is the basis for a filter termed a "lossy digital waveguide filter", which extends the con-: cept of the digital waveguide filter to account for such losses. For both models we present algorithms for solving the inverse problem. Though lossy inverse scattering is acknowledged to be a difficult problem in the general case, the properties of the systems examined here enable us to solve the lossy case for both models. The algorithms suggested here are based on the layer peeling algorithm used previously in seismic applications. Whereas the solution for the cylindrical segment model is based quite directly on this method, the solution for the conical segment model is more complicated and, to the best of our knowledge, is presented here for the first time. Finally, the simulation for a trumpet bell gives very good results. These results are shown to be superior to another method suggested in the literature. Simulation results also show that incorporating losses into the solution is important, without which a sizeable error results. A following paper will present results of the reconstruction algorithms presented here, applied to experimental data on brass wind instruments.