Acoustic Modelling of a Convex Pipe Adapted for Digital Waveguide Simulation
This work deals with the physical modelling of acoustic pipes for real-time simulation, using the “Digital Waveguide Network” approach and the horn equation. With this approach, a piece of pipe is represented by a two-port system with a loop which involves two delays for wave propagation, and some subsystems without internal delay. A well-known form of this system is the “Kelly-Lochbaum” framework, which allows the reduction of the computation complexity. We focus this work on the simulation of pipes with a convex profile. But, using the “Kelly-Lochbaum” framework with the horn equation, two problems occur: first, even if the outputs are bound, some substates have their values which diverge; second, there is an infinite number of such substates. The system is then unstable and cannot be simulated as such. The solution of this problem is obtained with two steps. First, we show that there is a simple standard form compatible with the “Waveguide” approach, for which there is an infinite number of solutions which preserve the input/output relations. Second, we look for one solution which guarantees the stability of the system and which makes easier the approximation in order to get a low-cost simulation.