Returning to the ``input-output'' waveguide defined in §4.2.1 and §4.2.2, we now must deal with connecting bidirectional delay lines; this is done in the same way as in the wave digital filtering framework, namely through the use of Kirchoff's Laws, which conserve instantaneous power at a connection. The resulting equations relating input to output waves at such a connection or scattering junction are identical to the adaptor equations for wave digital filters already mentioned in §2.3.5. For completeness sake, we will re-derive the scattering equations for a series connection of bidirectional delay lines, of impedances
,
.
At such a series connection, we must have
Thus, we have
![]() |
![]() |
![]() |
![]() |
![]() |
The scattering equations for a dual parallel connection are similar under the replacement of and
by
and
,
by
and
by the junction admittance, defined by
![]() |
A waveguide's immittance is placed at the port at which it is connected to the junction, and the junction quantity to be calculated from incoming waves appears at the center of the junction. Sometimes, if there is no room in the figure, we will indicate the immittance of a waveguide by an overbrace (see, e.g., Figure 4.8). In the case of electrical variables, a junction current
Instantaneous power is preserved at the scattering junction (here again, as in the WDF case, the scattering junction is no more than a wave variable implementation of Kirchoff's Laws, which preserve power by definition). The power-normalization strategy employed in the wave digital filter setting can also be used here as well, and gives rise to the same orthogonality property of the scattering junction in either the series or parallel case (see §2.3.5). Power-normalized waves can be used in order to construct time-varying passive waveguide networks [165], though for time-invariant problems, the use of such power-normalized quantities involves more arithmetic operations.