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

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

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]

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

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.