Mass Transmittance from String to String

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

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

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):

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

Thus, the mass transmittance for velocity waves is

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

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 $R$ .