Mass Transmittance from String to String

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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: $\hat{\tau}_v(s) = \frac{V^{-}_1(s)}{V^{-}_2(s)}$ . 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

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

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.