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) . 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 .
For force waves, we get the following:
The one-filter form follows from the observation that appears in both computations, and therefore need only be implemented once:
This structure is diagrammed in Fig.9.20.
Again, the above results follow immediately from the more general formulation of §C.12.