#### The Tonehole as a Two-Port Loaded Junction

It seems reasonable to expect that the tonehole should be representable as a load along a waveguide bore model, thus creating a loaded two-port junction with two identical bore ports on either side of the tonehole. From the relations for the loaded parallel junction (C.101), in the two-port case with , and considering pressure waves rather than force waves, we have   (10.63)   (10.64)   (10.65)

Thus, the loaded two-port junction can be implemented in one-filter form'' as shown in Fig. 9.48 with ( ) and  Comparing with (9.58), we see that the simplified Keefe tonehole model with the negative series inertance removed ( ), is equivalent to a loaded two-port waveguide junction with , i.e., the parallel load impedance is simply the shunt impedance in the tonehole model. Each series impedance in the split-T model of Fig. 9.43 can be modeled as a series waveguide junction with a load of . To see this, set the transmission matrix parameters in (9.55) to the values , , and from (9.51) to get      (10.66)

where is the alpha parameter for a series loaded waveguide junction involving two impedance waveguides joined in series with each other and with a load impedance of , as can be seen from (C.99). To obtain exactly the loaded series scattering relations (C.100), we first switch to the more general convention in which the '' superscript denotes waves traveling into a junction of any number of waveguides. This exchanges '' with '' at port 2 to yield      (10.67)

Next we convert pressure to velocity using and to obtain      (10.68)

Finally, we toggle the reference direction of port 2 (the current'' arrow for on port 2 in Fig. 9.43) so that velocity is positive flowing into the junction on both ports (which is the convention used to derive (C.100) and which is typically followed in circuit theory). This amounts to negating , giving      (10.69)

where . This is then the canonical form (C.100).
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Tonehole Filter Design