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
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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Bowing as Periodic Plucking
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Tonehole Filter Design