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Generalized Scattering Coefficients

Generalizing the scattering coefficients at a multi-tube intersection (§C.12) by replacing the usual real tube wave impedance $ R_i=\rho c/A_i$ by the complex generalized wave impedance

$\displaystyle R(x)=-\frac{\rho c(x)}{1 + \frac{\mbox{ln}'A(x)}{\mbox{ln}'P(x)}}

from Eq.(C.173), or, as a special case, the conical-section wave impedance $ R_A^\pm (s)=[\rho c/A(x)]/[s/(s \pm 1/t_x)]$ from Eq.(C.172), we obtain the junction-pressure phasor [440]

$\displaystyle P_J = \left(G_J + \sum_{i=1}^N G_i^-\right)^{-1} \sum_{i=1}^N
\left(G_i^+ + G_i^- \right)P_i^+

where $ G_i^+
$ is the complex, frequency-dependent, incoming, acoustic admittance of the $ i$ th branch at the junction, $ G_i^-$ is the corresponding outgoing acoustic admittance, $ P_i^+$ is the incoming traveling pressure-wave phasor in branch $ i$ , $ P_i^- = P_J -
P_i^+$ is the outgoing wave, and $ G_J$ is the admittance of a load at the junction, such as a coupling to another simulation. For generality, the formula is given as it appears in the multivariable case.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University