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

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

or

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

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.)