## Diameter and Taper Discontinuities

A frequency-domain, transfer-matrix method for modeling piece-wise conical sections, as previously presented for cylindrical sections, is possible. The matrix elements are a bit more complicated, but the approach is well documented and validated and can include accurate characterizations of thermoviscous losses. In this section, however, we focus on the time-domain, traveling-wave approach for modeling combinations of conical sections.

• At the boundary of two discontinuous and lossless conical sections, Fig. 10, the abrupt change in diameter and rate of taper will cause scattering of traveling wave components.

• Assuming continuity of pressure and conservation of volume flow at the boundary,
 (29)

and
 (30)

where Yc1 is the characteristic admittance for section 1 at the boundary looking in the positive x direction.

• The characteristic admittance for spherical waves in a cone is given by
 (31)

where applies to traveling-wave components propagating away from the cone apex in the positive x direction and to traveling-wave components propagating toward the cone apex in the negative x direction.

• Solving for at the junction,
 (32)

• The frequency-dependent scattering coefficient that relates to is
 = = (33)

where B is the ratio of wave front surface areas A1/A2 at the boundary and is given by
 (34)

(Martínez and Agulló, 1988; Gilbert et al., 1990).

• This reflectance is given a negative superscript to indicate scattering in the negative x direction.

• The parameters x1 and x2 are measured from the (imaginary) apices of cones 1 and 2, respectively, to the discontinuity.

• Similarly, the expression for at the junction is
 (35)

and the reflectance that relates to is
 = = (36)

• This reflectance is given a positive superscript to indicate scattering in the positive x direction.

• The scattering equations can then be expressed in terms of and as
 = (37) = (38)

• It is possible to define transmittances that indicate scattering through the junction as and

• These expressions are equally valid when either acoustic section is cylindrical, rather than conical. Replacing cone 1 by a cylindrical section, and A1 is given by the cylinder's cross-sectional area at the discontinuity.

• Whereas the junction scattering coefficients for cylindrical bore diameter discontinuities were real and constant, these expressions are frequency-dependent and must be transformed to discrete-time filters for time-domain implementation.

• Figure 11(a) illustrates the general scattering junction implementation for diameter and taper discontinuities in conical bores.

• Because and are different, the one-multiply form of the scattering junction implementation is not possible.

• However, if the wavefront surface areas are approximated by cross-sectional areas at the discontinuity, which is reasonable only for small changes in taper rate and cross-section, the reflectances and become identical for a discontinuity of taper only. In this case, a one-multiply scattering junction implementation is possible, as shown in Fig. 11(b) (Välimäki, 1995).

• Strictly speaking, however, the propagating wave fronts in cones are spherical and thus and will never be identical, even for a simple taper discontinuity.

• Equation (33) can be transformed to the time domain, resulting in the reflection function
 (39)

where is the Dirac impulse and is the Heaviside unit step function (Martínez and Agulló, 1988).

• An appropriate discrete-time filter is found by making the bilinear transform frequency variable substitution in Eq. (33) with the result
 (40)

where is as given in Eq. (34), is the bilinear transform constant, and
 (41)

is the first-order filter pole location in the z-plane.

• Unfortunately, is unstable for negative which occurs any time cone 2 has a lower rate of taper than cone 1.

• Equivalently, has no causal inverse Fourier transform for negative

• This corresponds to a growing exponential in the reflection function r-(t).

• The reflectance is a junction characteristic that is blind'' to the boundary conditions upstream or downstream from it. Physically realistic boundary conditions in such cases, however, will always limit the time duration over which the growing exponential can exist.

• Experimental measurements of reflection functions due to discontinuities have verified this general behavior (Agulló et al., 1995).

• The problem here in terms of digital waveguide modeling, is that the growing exponentials become unstable digital filters in the discrete-time domain.