Truncated Cones

• Conical bores are always truncated to some extent to allow excitation at their small end. Assuming the mouth of the cone at $x=L$ is ideally open, so that $Z_{L} = 0,$ the reflectance $C^{-}/C^{+}$ becomes $-e^{-2jkL}$ and the input admittance of a truncated cone reduces to
    $$Y_{in}(x_{0}) = \frac{A(x_{0})}{j \rho c} \left[ \cot(k\gamma) + \frac{1}{k x_{0}} \right],$$ (16)
  where $\gamma = L - x_{0}.$

• If the small end of the cone, at $x=x_{0},$ is assumed ideally open, the resonance frequencies of the open-open (o-o) conic frustum are at the infinities of Eq. (16), which are given for $n = 1, 2, \ldots$ by
    $$f^{(\mbox{o-o})} = \frac{n c}{2 (L-x_{0})} = \frac{n c}{2 \gamma}.$$ (17)

• Thus, the higher natural frequencies of the open-open conic frustum are also related to the fundamental by integer multiple ratios.

• If the input end of the cone at $x=x_{0}$ is assumed ideally closed, which is nearly the case for reed-driven conical woodwind instruments, the resonance frequencies are found at the zeros of the input admittance.

• In this case, the partials of the closed-open conic frustum do not occur at exact integer multiples of the fundamental frequency, but are generally more widely spread apart depending on the magnitude of $x_{0}.$

• The natural frequencies of the truncated cone closed at its small end are found by solving the transcendental equation
    $$\tan(k\gamma) = -k x_{0}.$$ (18)
  Equation (18) can be rewritten in the form
    $$\tan\left(\pi \frac{f}{f_{0}}\right) = -\frac{\beta}{1 - \beta}\left(\pi \frac{f}{f_{0}}\right),$$ (19)
  where $\beta = x_{0}/L$ and $f_{0} = c/(2 \gamma)$ is the fundamental frequency for the open-open conic frustum of length $\gamma$ (Ayers et al., 1985).

  Figure 3: Partial frequency ratios, relative to $f_{0},$ for a closed-open conic frustum.

  

• Figure 3 illustrates the partial frequency ratios for a closed-open conic frustum, relative to the fundamental frequency of an open-open conic frustum of the same length, for $\beta=0$ (complete cone) to $\beta=1$ (closed-open cylinder).

• The dotted lines indicate exact integer relationships above the first partial and serve only to make more apparent the stretching of the partial ratios.

• The perfect harmonicity of the partials of an open-open conic frustum are distorted when a single-reed excitation mechanism is placed at one end.

• The fundamental frequency of the conic section is most affected by truncation and closure, as the inertance term in the admittance is largest for low frequencies.

• In terms of the fundamental wavelength of an open-open conic frustum of the same length, the inertance contributes a positive length correction that increases with truncation $x_{0}$ but decreases with frequency. Viewed in terms of the normal modes of a closed-open cylinder, however, the inertance contributes a negative length correction that is inversely proportional to truncation $x_{0}$ and frequency.