Cylinder with Conical Cap

Consider a cylindrical acoustic tube adjoined to a converging conical cap, as depicted in Figure C.50a. We may consider the cylinder to be either open or closed on the left side, but everywhere else it is closed. Since such a physical system is obviously passive, an interesting test of acoustic theory is to check whether theory predicts passivity in this case.

Figure C.50: Cylindrical acoustic tube with a conical cap. a) Physical outline. b) Continuous-time waveguide formulation.

It is well known that a growing exponential appears at the junction of two conical waveguides when the waves in one conical taper angle reflect from a section with a smaller (or more negative) taper angle [7,302,8,161,9]. The most natural way to model a growing exponential in discrete time is to use an unstable one-pole filter [510]. Since unstable filters do not normally correspond to passive systems, we might at first expect passivity to not be predicted. However, it turns out that all unstable poles are ultimately canceled, and the model is stable after all, as we will see. Unfortunately, as is well known in the field of automatic control, it is not practical to attempt to cancel an unstable pole in a real system, even when it is digital. This is because round-off errors will grow exponentially in the unstable feedback loop and eventually dominate the output.

The need for an unstable filter to model reflection and transmission at a converging conical junction has precluded the use of a straightforward recursive filter model [409]. Using special ``truncated infinite impulse response'' (TIIR) digital filters [544], an unstable recursive filter model can in fact be used in practice [532]. All that is then required is that the infinite-precision system be passive, and this is what we will show in the special case of Fig.C.50.