- 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).
- Figure 5 illustrates a divergent conical bore truncated and rigidly terminated at a distance
*x*_{0}from its apex. - The pressure reflectance at
*x*_{0}is found from the boundary condition*U*(*x*_{0}) = 0, such that

where*s*is the Laplace transform variable and*c*is the speed of sound in air. - As
the reflectance approaches negative one as observed above.
- Using the bilinear transform, this expression can be discretized as follows

where

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.
- As
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.
**Figure 6:**The conical truncation reflectance: (top) Continuous-time and discrete-time filter phase responses; (bottom) Digital allpass filter coefficient value versus*x*_{0}. - The digital waveguide implementation of a closed-open conic section is shown in Fig. 7.
**Figure 7:**Digital waveguide implementation of a closed-open truncated conical section excited at*x*=*x*_{0}. - The cone truncation reflectance filter is represented by

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