In order to implement the equivalent circuit of a conical waveguide using digital waveguide techniques, it is necessary to express the lumped impedance elements of Fig. 14 in terms of traveling-wave parameters and then convert these expressions to discrete-time filters.
The impedance of the input inertance, given in terms of a Laplace transform, is
, where is the mass density of air, Ao is the area of the spherical wavefront at the waveguide input, and s is Laplace transform frequency variable.
The effective impedance at the waveguide input is determined as the parallel combination of Mo and an input load impedance. If the input is rigidly terminated, the input load is infinite and the pressure-wave reflectance is given by:
is a real, locally defined characteristic impedance parameter.
The reflectance filter, discretized with the bilinear transform, is
and is the bilinear transform constant that controls frequency warping.
This first-order allpass filter accurately accounts for the phase delay experienced by pressure traveling-wave components reflecting from a rigid input termination in a conical waveguide.
The output inertance, Me, tends to be less significant than that at the input, particularly in the presence of an open-end load impedance. In general, a single output reflectance filter can be designed based on the parallel combination of Me and an appropriate open-end impedance characterization.
Figure 18 shows a truncated conic structure and the corresponding digital waveguide block diagram, using input and output reflectance filters as discussed above.
A closed-open conic structure (top) and its digital waveguide block diagram (bottom).
The goal here is to model a conical bore instrument system by attaching a simple, memory-less, non-linear excitation mechanism to a conical air column representation.
The traditional reed function/air column coupling, however, is derived for an input cylindrical section using a real wave impedance. It is not a simple process to re-derive the reed function using the complex wave impedance of a conic frustum. Even if we ignored this complication, direct coupling of the reed function to the allpass inertance element at the input to the conical ``circuit'' would produce a delay-free loop in the digital waveguide implementation. These constraints lead to the modeling approach discussed first in this section.