Two-Port Series Adaptor for Force Waves
Figure F.6a illustrates a generic two-port description of the series adaptor.
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As discussed in §7.2, a series connection is characterized by a common velocity and forces which sum to zero at the junction:
![\begin{eqnarray*}
&& f_1(n) + f_2(n) = 0\\
&& v_1(n) = v_2(n) \isdef v_J(n)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4872.png)
The derivation can proceed exactly as for the parallel junction in
§F.2.1, but with force and velocity interchanged, i.e.,
, and with impedance and admittance interchanged,
i.e.,
. In this way, we may take the
dual of Eq.
(F.14) to get
![\begin{eqnarray*}
v^{-}_1 &=& -\rho v^{+}_1 + (1+\rho) v^{+}_2\\
v^{-}_2 &=& (1-\rho)v^{+}_1 + \rho v^{+}_2
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4875.png)
diagrammed in Fig.F.7. Converting back to force wave
variables via
and
, and noting
that
, we obtain, finally,
![\begin{eqnarray*}
f^{{-}}_1 &=& \rho f^{{+}}_1 - (1-\rho) f^{{+}}_2\\
f^{{-}}_2 &=& -(1+\rho)f^{{+}}_1 - \rho f^{{+}}_2
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4879.png)
as diagrammed in Fig.F.6b. The one-multiply form is now
![\begin{eqnarray*}
f^{{-}}_1 &=& -f^{{+}}_2 + \rho(f^{{+}}_1 + f^{{+}}_2)\\
f^{{-}}_2 &=& -f^{{+}}_1 - \rho(f^{{+}}_1 + f^{{+}}_2).
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4880.png)
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General Series Adaptor for Force Waves
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General Parallel Adaptor for Force Waves