- In general, abrupt diameter discontinuities rarely occur in vocal tract or wind-instrument air column profiles.
- However, these discontinuities will be encountered in the approximation of complex air column shapes using cylindrical pipe sections.
- At the boundary of two discontinuous and lossless cylindrical sections, Fig. 11, there will be a change of characteristic impedance which results in partial reflection and transfer of traveling-wave components.
- Assuming continuity of pressure and conservation of volume flow at the boundary,

*p*_{1}^{+}+*p*_{1}^{-}=*p*_{2}^{+}+*p*_{2}^{-}(41)

and

(42) *Z*_{c1}is the characteristic impedance of cylindrical section 1. - Because the characteristic wave impedance of a cylindrical pipe is real, these expressions apply to both time- and frequency-domain wave variables.
- Solving for
*p*_{1}^{-}and*p*_{2}^{+}at the junction,

where is the reflectance for the junction of cylinders 1 and 2. is given by

= (45) = (46)

where*A*_{1}is the cross-sectional area of section 1. - The relationships of Eqs. (43) and (44) are referred to as scattering equations.
- The scattering equations are implemented by the structure shown in Fig. 12a, which was first derived for an acoustic tube model used in speech synthesis (Kelly and Lochbaum, 1962).
**Figure 12:**(a) The Kelly-Lochbaum scattering junction for diameter discontinuities in cylindrical bores; (b) The one-multiply scattering junction [after (Markel and Gray, 1976)]. - Equations (43) and (44) can also be written in the form

where

- In this way, the Kelly-Lochbaum scattering junction can be implemented with a single multiply, as shown in Fig. 12b (Markel and Gray, 1976).
- Smith (1987) points out that junction passivity is guaranteed for
- The digital waveguide implementation of lossless wave propagation in two discontinuous cylindrical sections is shown in Fig. 13.
- In this way, any combination of co-axial cylindrical sections can be modeled using only digital delay lines and one-multiply scattering junctions.
- Thermoviscous losses can be implemented with digital filters designed in accordance with previous theoretical analyses.
- Because these models are linear and time-invariant, these loss characteristics can be commuted with an open-end filter to maximize efficiency.

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