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Summary of Mass-String Scattering Junction

In summary, we have characterized the mass on the string in terms of its reflectance and transmittance from either string. For force waves, we have outgoing waves given by

F^{-}_1(s) &=& \hat{\rho}_f(s) F^{+}_1(s) + \hat{\tau}_f(s) F^{-}_2(s)\\ [5pt]
F^{+}_2(s) &=& \hat{\tau}_f(s) F^{+}_1(s) + \hat{\rho}_f(s) F^{-}_2(s)


$\displaystyle \left[\begin{array}{c} F^{+}_2 \\ [2pt] F^{-}_1 \end{array}\right] \eqsp \left[\begin{array}{cc} \hat{\tau}_f(s) & \hat{\rho}_f(s) \\ [2pt] \hat{\rho}_f(s) & \hat{\tau}_f(s) \end{array}\right] \left[\begin{array}{c} F^{+}_1 \\ [2pt] F^{-}_2 \end{array}\right]

in terms of the incoming waves $ F^{+}_1$ and $ F^{-}_2$ , the force reflectance $ \hat{\rho}_f(s)=ms/(ms+2R)$ , and the force transmittance $ \hat{\tau}_f(s)=1-\hat{\rho}_f(s)=2R/(ms+2R)$ . We may say that the mass creates a dynamic scattering junction on the string. (If there were no dependency on $ s$ , such as when a dashpot is affixed to the string, we would simply call it a scattering junction.) The above form of the dynamic scattering junction is analogous to the Kelly-Lochbaum scattering junctionC.8.4). The general relation $ \hat{\tau}_f = 1+\hat{\rho}_f$ can be used to simplify the Kelly-Lochbaum form to a one-filter scattering junction analogous to the one-multiply scattering junctionC.8.5):

F^{-}_1 &=& \hat{\rho}_f F^{+}_1 + (1+\hat{\rho}_f) F^{-}_2 \;=\; F^{-}_2 + \hat{\rho}_f\cdot(F^{+}_1+F^{-}_2)\\ [5pt]
F^{+}_2 &=& (1+\hat{\rho}_f) F^{+}_1 + \hat{\rho}_f F^{-}_2 \;=\; F^{+}_1 + \hat{\rho}_f\cdot(F^{+}_1+F^{-}_2)

The one-filter form follows from the observation that $ \hat{\rho}_f\cdot(F^{+}_1+F^{-}_2)$ appears in both computations, and therefore need only be implemented once:

F^{+}&\isdef & \hat{\rho}_f\cdot(F^{+}_1+F^{-}_2)\\ [5pt]
F^{-}_1 &=& F^{-}_2 + F^{+}\\ [5pt]
F^{+}_2 &=& F^{+}_1 + F^{+}

This structure is diagrammed in Fig.9.20.

Figure 9.20: Continuous-time force-wave simulation diagram, in one-filter form, for an ideal string with a point mass attached.

Again, the above results follow immediately from the more general formulation of §C.12.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2020-01-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University