A Stable Acoustic Impedance Model of the Clarinet Using Digital Waveguides

Digital waveguide (DW) modeling techniques are typically associated with a traveling-wave decomposition of wave variables and a “reflection function” approach to simulating acoustic systems. As well, it is often assumed that inputs and outputs to/from these systems must be formulated in terms of traveling-wave variables. In this paper, we provide a tutorial review of DW modeling of acoustic structures to show that they can easily accommodate physical input and output variables. Under certain constraints, these formulations reduce to simple “Schroeder reverb-like” computational structures. We also present a stable single-reed filter model that allows an explicit solution at the reed / air column junction. A clarinet-like system is created by combining the reed filter with a DW impedance model of a cylindrical air column. 1. REFLECTION FUNCTION CALCULATIONS The use of digital waveguides (DW) to model wave propagation within cylindrical air columns has been well documented [1, 2]. A DW structure like that diagrammed in Fig. 1 can be used to compute the time-domain pressure reflection function, rp(t), of a uniform pipe. The reflection function is defined as the pressure response at the input of an air column caused by the introduction there of a pressure impulse, assuming no reflections at the input end (an anechoic input termination). The digital filter RL models the frequency-dependent reflectance of the load impedance connected to the far end of the pipe. It can be expressed as RL(f) = » ZL(f)− Zc ZL(f) + Zc – , (1) where Zc is the real characteristic wave impedance of the pipe, ZL(f) is the frequency-dependent load impedance, and f is frequency in Hertz. A closed end is reasonably modeled by a load impedance ZL = ∞, in which case RL = 1. This indicates that pressure traveling-waves reflect from a rigid boundary in a cylindrical pipe with unity gain and no phase shift. For an open end condition, an analytic solution for RL(f) has been reported by [3]. A second-order digital filter is sufficient to achieve a good fit to that analytic result for most pipe dimensions of musical interest [4]. The structure of Fig. 1 is a time-domain computational model of one-dimensional traveling-wave propagation along the length of the air column, together with wave reflection from the load impedance at the far end. The frequency-domain counterpart to RL z−M z−M