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)

(30) *Y*_{c1}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) *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

where*B*is the ratio of wave front surface areas*A*_{1}/*A*_{2}at the boundary and is given by

(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
*x*_{1}and*x*_{2}are measured from the (imaginary) apices of cones 1 and 2, respectively, to the discontinuity. - Similarly, the expression for
at the junction is

(35)

= = (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
*A*_{1}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.
**Figure 11:**(a) The scattering junction for diameter and taper discontinuities in conical bores; (b) The one-multiply scattering junction for a taper discontinuity only [after (Välimäki, 1995)]. - 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

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)

(41) *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.

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