Mass Transmittance

Force transmittance is similarly derived:

$$\hat{\tau}_f(s) \isdef \frac{F}{F^{+}} = \frac{F^{+}+F^{-}}{F^{+}} = 1 + \frac{F^{-}}{F^{+}} = 1+\hat{\rho}(s)$$

as is velocity transmittance:

$$\hat{\tau}_v(s) \isdef \frac{V}{V^{+}} = \frac{V^{+}+V^{-}}{V^{+}} = 1 + \frac{V^{-}}{V^{+}} = 1-\hat{\rho}(s)$$

For the mass-on-string problem:

$$\begin{eqnarray*} \hat{\tau}_f(s) &=& 1+\hat{\rho}(s) = 1 + \frac{ms}{ms+2R} = 2... ...s) &=& 1-\hat{\rho}(s) = 1 - \frac{ms}{ms+2R} = \frac{2R}{ms+2R} \end{eqnarray*}$$

Limiting Behavior:

• For $m=0$, both transmission filters become 1, as expected
• For $m=\infty$, $\hat{\tau}_v(s)\to0$, which makes good physical sense (infinite mass = rigid termination)
• For $m=\infty$, $\hat{\tau}_f(s)\to 2$!
• Recall that signal power is force times velocity