Waveguide Network for the Euler-Bernoulli System

It is possible to design a waveguide network which simulates the behavior of equation (5.1), but there are some extra features we must add which were not necessary in the case of the transmission line. In addition, the overloaded symbols for the wave variables become even more overloaded, due to the fact that we can no longer interleave the two dependent variables spatially, and are faced with a double set of wave variables at every grid point. (This can be remedied with recourse to other more involved difference methods, but we will not pursue this subject here.) The structure of interest is shown in Figure 5.1.

This is still a (1+1)D waveguide network, like that which simulates the (1+1)D transmission line equations, but we have drawn the junctions which calculate and separately; it should be kept in mind that they operate at the same spatial locations. As before, we use grey/white coloring of junctions to signify operation at different time steps. Here we have interpreted (which we will identify with of difference scheme (5.5), and thus with ) as aWe note that the wave variables at the series scattering junction at location are indicated by a tilde, to distinguish them from those at the parallel junction at the same location, even though the two sets of variables are calculated at alternate time steps. As for the (1+1)D transmission line, we index wave variables and immittances at the left and right ports of any junction by and respectively, and the same such quantities associated with any self-loop are subscripted with . We also have new waveguides connecting parallel and series junctions at the same grid point; immittances and wave variables are subscripted with a in this case. With reference to Figure 5.2, we can define the junction admittance at the parallel junction, and the junction impedance at the series junction to be

It should be clear that this waveguide network is really a pair of coupled (1+1)D transmission lines; the coupling is via the waveguide connecting the series and parallel junctions at the same grid location (the vertical waveguide in Figure 5.1).

(This equivalence can easily be derived through the manipulation of

We now trace the signal flow in the network to show that it does indeed solve the Euler Bernoulli system. Beginning from a series junction at grid point , we have:

which is identical to (5.5b) if we replace by and by , and if we have

Beginning from the series junction, we arrive at a similar requirement for , namely

As in the case of the transmission line, three families of waveguide networks are distinguishable: