In an earlier section, it was observed that pressure and volume flow traveling-wave components grow increasingly out of phase as they approach the cone apex. This behavior is defined by the complex characteristic impedance of conical air columns.
The boundary condition imposed by a rigid termination requires that volume flow normal to the boundary equal zero.
Where spherical volume velocity traveling-wave components reflect with an inversion, so that their sum is zero, spherical pressure traveling-wave components reflect with a phase angle
In the case of a complete cone,
and pressure traveling-wave components reflect from the cone apex with a phase shift (or an inversion).
Divergent conical section rigidly terminated by a spherical cap at x0.
Figure 5 illustrates a divergent conical bore truncated and rigidly terminated at a distance x0 from its apex.
The pressure reflectance at x0 is found from the boundary condition U(x0) = 0, such that
U+(x0) + U-(x0)
where s is the Laplace transform variable and c is the speed of sound in air.
the reflectance approaches negative one as observed above.
Using the bilinear transform, this expression can be discretized as follows
and is the bilinear transform constant that controls frequency warping.
Equation (24) is a first-order discrete-time allpass filter. This filter structure implements the
frequency-dependent phase delay experienced by pressure traveling-wave components reflecting from a rigid termination in a conical waveguide.
The upper plot of Fig. 6 illustrates the continuous-time phase response of the reflectance given by Eq. (23) and the discrete-time phase response of the digital filter of Eq. (24).
The lower plot of Fig. 6 indicates the allpass filter coefficient value as a function of cone truncation.
the filter becomes unstable because
and the allpass pole falls on the unit circle. Thus, the truncation filter cannot be used to simulate a complete cone, but in this case the reflectance is simply negative one anyway.
The conical truncation reflectance: (top) Continuous-time and discrete-time filter phase responses; (bottom) Digital allpass filter coefficient value versus x0.
The digital waveguide implementation of a closed-open conic section is shown in Fig. 7.
Digital waveguide implementation of a closed-open truncated conical section excited at x=x0.
The cone truncation reflectance filter is represented by