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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 $ R_1=R_2=R_0$ , and considering pressure waves rather than force waves, we have

$\displaystyle P_J(s)$ $\displaystyle =$ $\displaystyle \alpha P_1^{+}+ \alpha P_2^{+}, \quad \alpha = 2\Gamma _0/[G_J(s)+2\Gamma _0]$ (10.63)
$\displaystyle P_1^{-}(s)$ $\displaystyle =$ $\displaystyle P_J(s) - P_1^{+}
= (\alpha-1)P_1^{+}+ \alpha P_2^{+}= \alpha(P_1^{+}+P_2^{+})-P_1^{+}$ (10.64)
$\displaystyle P_2^{-}(s)$ $\displaystyle =$ $\displaystyle P_J(s) - P_2^{+}= \alpha P_1^{+}+ (\alpha-1)P_2^{+}
= \alpha(P_1^{+}+P_2^{+})-P_2^{+}$ (10.65)

Thus, the loaded two-port junction can be implemented in ``one-filter form'' as shown in Fig. 9.48 with $ A(\omega)=1$ ( $ L(\omega)=0$ ) and

$\displaystyle T(\omega)=\alpha = \frac{2\Gamma _0}{2\Gamma _0+ G_J(s)} = \frac{2R_J(s)}{2R_J(s)+R_0}

Comparing with (9.58), we see that the simplified Keefe tonehole model with the negative series inertance removed ($ R_a=0$ ), is equivalent to a loaded two-port waveguide junction with $ R_J=R_s$ , i.e., the parallel load impedance is simply the shunt impedance in the tonehole model.

Each series impedance $ R_a/2$ in the split-T model of Fig. 9.43 can be modeled as a series waveguide junction with a load of $ R_a/2$ . To see this, set the transmission matrix parameters in (9.55) to the values $ T_{11} = T_{22} = 1$ , $ T_{12} = R_a/2$ , and $ T_{21}=0$ from (9.51) to get

$\displaystyle P_1^-$ $\displaystyle =$ $\displaystyle (1-\alpha) P_1^+ + \alpha P_2^-$  
$\displaystyle P_2^+$ $\displaystyle =$ $\displaystyle \alpha P_1^+ + (1-\alpha) P_2^-$ (10.66)

where $ \alpha = 2R_0/(2R_0+R_a/2)$ is the alpha parameter for a series loaded waveguide junction involving two impedance $ R_0$ waveguides joined in series with each other and with a load impedance of $ R_a/2$ , 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
$\displaystyle P_1^-$ $\displaystyle =$ $\displaystyle (1-\alpha) P_1^+ + \alpha P_2^+$  
$\displaystyle P_2^-$ $\displaystyle =$ $\displaystyle \alpha P_1^+ + (1-\alpha) P_2^+$ (10.67)

Next we convert pressure to velocity using $ P_i^+ = R_0U_i^+$ and $ P_i^- = -R_0U_i^-$ to obtain
$\displaystyle U_1^-$ $\displaystyle =$ $\displaystyle (\alpha-1) U_1^+ - \alpha U_2^+$  
$\displaystyle U_2^-$ $\displaystyle =$ $\displaystyle -\alpha U_1^+ + (\alpha-1) U_2^+$ (10.68)

Finally, we toggle the reference direction of port 2 (the ``current'' arrow for $ u_2$ 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 $ U_2^{\pm}$ , giving
$\displaystyle U_1^-$ $\displaystyle =$ $\displaystyle U_J - U_1^+$  
$\displaystyle U_2^-$ $\displaystyle =$ $\displaystyle U_J - U_2^+$ (10.69)

where $ U_J \isdef (\alpha U_1^+ + \alpha U_2^+)$ . This is then the canonical form (C.100).

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University