Lossless Scattering
The delay-line inputs (outgoing traveling waves) are computed by
multiplying the delay-line outputs (incoming traveling waves) by the
feedback matrix (scattering matrix)
. By
defining
,
, we obtain the more
usual DWN notation
where
![$ \mathbf{p}^+$](http://www.dsprelated.com/josimages_new/pasp/img4056.png)
![$ n$](http://www.dsprelated.com/josimages_new/pasp/img146.png)
![$ \mathbf{p}^-$](http://www.dsprelated.com/josimages_new/pasp/img4057.png)
![$ n$](http://www.dsprelated.com/josimages_new/pasp/img146.png)
![$ \mathbf{A}$](http://www.dsprelated.com/josimages_new/pasp/img569.png)
The junction of physical waveguides determines the structure of the
matrix
according to the basic principles of physics.
Considering the parallel junction of lossless acoustic tubes, each
having characteristic admittance
, the continuity of pressure and
conservation of volume velocity at the junction give us the following
scattering matrix for the pressure waves [433]:
where
![]() |
(C.121) |
Equation (C.121) can be derived by first writing the volume velocity at the
![$ j$](http://www.dsprelated.com/josimages_new/pasp/img664.png)
![$ v_j = (p_j^+ - p_j^-)\Gamma_j$](http://www.dsprelated.com/josimages_new/pasp/img4061.png)
![$\displaystyle p = 2 \sum_{i=1}^{N}\Gamma_{i} p_i^+ / \Gamma_J
$](http://www.dsprelated.com/josimages_new/pasp/img4062.png)
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Normalized Scattering
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Diffuse Reflections in the Waveguide Mesh