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Chapters

Physical Audio Signal Processing
    Woodwinds
       Tonehole Modeling
          The Tonehole as a Two-Port Loaded Junction

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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 (H.87), in the two-port case with $ R_1=R_2=R_0$, and considering pressure waves rather than force waves, we have

$\displaystyle P_J(s)$ $\displaystyle =$ $\displaystyle \alpha P_1^{+}+ \alpha P_2^{+}, \quad \alpha = 2\Gamma _0/[G_J(s)+2\Gamma _0]$ (7.29)
$\displaystyle P_1^{-}(s)$ $\displaystyle =$ $\displaystyle P_J(s) - P_1^{+} = (\alpha-1)P_1^{+}+ \alpha P_2^{+}= \alpha(P_1^{+}+P_2^{+})-P_1^{+}$ (7.30)
$\displaystyle P_2^{-}(s)$ $\displaystyle =$ $\displaystyle P_J(s) - P_2^{+}= \alpha P_1^{+}+ (\alpha-1)P_2^{+} = \alpha(P_1^{+}+P_2^{+})-P_2^{+}$ (7.31)

Thus, the loaded two-port junction can be implemented in ``one-filter form'' as shown in Fig. 6.11 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 (6.24), 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. 6