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
.

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.

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