## Reflection at a Rigid Boundary

• 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 x0 from its apex.

• The pressure reflectance at x0 is found from the boundary condition U(x0) = 0, such that
 U+(x0) + U-(x0) = 0 = 0 = = (23)

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
 (24)

where
 (25)

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.

• The digital waveguide implementation of a closed-open conic section is shown in Fig. 7.

• The cone truncation reflectance filter is represented by