Loaded Junctions

In discrete-time modeling of acoustic systems, it is often useful to attach waveguide junctions to external dynamic systems which act as a load. We speak in this case of a loaded junction [24]. The load is expressed in general by its complex admittance and can be considered a lumped circuit attached to the distributed waveguide network.

To derive the scattering matrix for the loaded parallel junction of $N$ lossless acoustic tubes, the Kirchhoff's node equation is reformulated so that the sum of velocities meeting at the junction equals the exit velocity (instead of zero). For the series junction of transversely vibrating strings, the sum of forces exerted by the strings on the junction is set equal to the force acting on the load (instead of zero).

The load admittance $\Gamma_{\rm L}$ is regarded as a lumped driving-point admittance [42], and the equation

$$U_{\rm L}(z) = \Gamma_{\rm L}(z) p_{\rm J}(z)$$ (65)

expresses the relation at the load.

For the general case of $N$ $m$-variable physical waveguides, the expression of the scattering matrix is that of (62), with

$$S=\left[{\mbox{\boldmath$1$}}^T\left({\sum_{i=1}^{N}{{\mbox{... ...}}\right){\mbox{\boldmath$1$}}+ \Gamma_{\rm L}\right]^{-1} \,.$$ (66)