#### 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

*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).

**Next Section:**

Bowing as Periodic Plucking

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Tonehole Filter Design