Wave Scattering by a Mass on a String

We have derived the reflectance and transmittance of a mass $m$ as seen from either string at impedance $R$. We can now derive the complete scattering relations:

For force waves, the outgoing waves are

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

where the incoming waves are $F^{+}_1$ and $F^{-}_2$, and

$$\begin{eqnarray*} \hat{\rho}(s)&=&\frac{ms}{ms+2R} \quad \mbox{(force reflectanc... ...{\rho}(s)=\frac{2(ms+R)}{ms+2R}\quad \mbox{(force transmittance)}\end{eqnarray*}$$

We may say that the mass creates a dynamic scattering junction on the string