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Mass Transmittance from String to String

Referring to Fig.9.15, the velocity transmittance from string 1 to string 2 may be defined as

$\displaystyle \hat{\tau}_v(s)
\eqsp \frac{V^{+}_2(s)}{V^{+}_1(s)}.

By physical symmetry, we expect the transmittance to be the same in the opposite direction: $ \hat{\tau}_v(s) = V^{-}_1(s)/V^{-}_2(s)$ . Assuming the incoming wave $ V^{-}_2$ on string 2 is zero, we have $ V^{+}_2=V$ , which we found in Eq.(9.16):

$\displaystyle V \eqsp \frac{2R}{ms+2R}V^{+}_1

Thus, the mass transmittance for velocity waves is

$\displaystyle \zbox {\hat{\tau}_v(s) \eqsp \frac{2R}{ms+2R} \eqsp 1-\hat{\rho}_f(s) \eqsp 1+\hat{\rho}_v(s)}

We see that $ m\to\infty$ corresponds to $ \hat{\tau}_v(s)\to 0$ , as befits a rigid termination. As $ m\to0$ , the transmittance becomes 1 and the mass has no effect, as desired.

We can now refine the picture of our scattering junction Fig.9.17 to obtain the form shown in Fig.9.18.

Figure 9.18: Velocity-wave scattering junction for a mass $ m$ (impedance $ ms$ ) attached to an ideal string having wave impedance $ R$ .

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University