Conical Bores

To a first approximation, a truncated cone can be regarded as a cylinder open on both ends [4]. To make a more precise model, the phase shift between traveling pressure and velocity needs to be taken into account, or, equivalently, the imaginary part of the wave impedance needs to be modeled [5]. Digital waveguide models have been derived for conical-bore instruments [53,62,70], and limited simulations have been successful. However, there is a surprising result in the theory: When a conical tube suddenly decreases in taper angle, such as when crossing from a diverging conical segment into a converging one, the impulse response of the junction actually contains growing exponentials [2]. This means that, at such a junction, the straightforward waveguide model must use unstable reflection and transmission filters! It seems highly inappropriate to use unstable filters to model a passive physical bore, and numerically it is highly inadvisable without elaborate schemes to reset the growing round-off noise inside the filters. However, so far, no efficient solution has been found. One expensive solution is to replace the unstable IIR junction filters by large FIR filters which explicitly implement truncated growing exponentials; this solution is based on the observation that in realistic (finite-length) bore geometries, the growing exponentials are always ultimately canceled by reflections from the terminations.