One-Multiply Form for the Mass-String Scattering Junction

The above form of the dynamic scattering junction is analogous to the Kelly-Lochbaum (KL) scattering junction (§C.8.4). The one-multiply form of the KL junction is well known (§C.8.5) [299]. The same basic derivation works for our problem as well to yield a one-filter form for the string-mass scattering junction.

We cannot simply substitute our reflectances and transmittances for the reflection and transmission coefficients in the KL result because our problem is a series connection of two identical string sections loaded by a junction mass, while the KL junction is derived for the parallel connection of two acoustic tube sections with no load at the junction; these differences lead to different signs as well as introducing the Laplace variable $s$ .

For force waves, we get the following:

$$\begin{eqnarray*} F^{-}_1 &=& \hat{\rho}_f F^{+}_1 + (1-\hat{\rho}_f) F^{-}_2 \;=\; F^{-}_2 + \hat{\rho}_f\cdot(F^{+}_1-F^{-}_2)\\ [5pt] F^{+}_2 &=& (1-\hat{\rho}_f) F^{+}_1 + \hat{\rho}_f F^{-}_2 \;=\; F^{+}_1 - \hat{\rho}_f\cdot(F^{+}_1-F^{-}_2) \end{eqnarray*}$$

The one-filter form follows from the observation that $\hat{\rho}_f\cdot(F^{+}_1-F^{-}_2)$ appears in both computations, and therefore need only be implemented once:

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

This structure is diagrammed in Fig.9.20.

Figure 9.20: Continuous-time force-wave simulation diagram, in one-filter form, for an ideal string with a point mass attached.

Again, the above results follow immediately from the more general formulation of §C.12.