Properties of Conic Frusta

• It is generally agreed that harmonically aligned air column mode ratios better support a stable “regime of oscillation” via mode cooperation (Worman, 1971).

• It is thus expected that a synthesis model incorporating a nonlinear excitation mechanism will likewise benefit from such mode cooperation in attempting to produce a robust, stable oscillatory regime.

• Ayers et al. (1985) presents an exploration of the properties of conic frusta. Of particular note, the mode ratios for truncated closed-open frusta of length $L$ are shown to vary with respect to the parameter $\beta$ as depicted in Fig. 15.

  Figure 15: Mode frequencies for a closed-open conic frustum normalized by the fundamental frequency of an open pipe of the same length as the frustum. The dotted curves indicate integer relationships to the first mode.

  

• $\beta$ can be defined in terms of the ratio of input to output end radii, $r_{o} / r_{e}$, or in terms of the ratio $x_{o} / (x_{o} + L)$, where $x_{o}$ is the length of the missing, truncated apical section (see Fig. 13).

• A complete cone is given by $\beta=0$, while a closed-open cylinder is given by $\beta=1$.

• It should be obvious from Fig. 15 that any closed-open truncated conical section will have inharmonic mode ratios and that the extent of this inharmonicity is dependent on the dimensions of the frustum.

• Attempts at robust synthesis using a model that correctly simulates the behavior of a truncated conical section may in turn be hindered.

• Benade (1976) reports that the effects of truncation can be reduced by utilizing a reed/mouthpiece cavity with an equivalent volume equal to that of the missing conical section.

• This constraint is based on a lumped characterization of the reed/mouthpiece cavity, which is only appropriate for low-frequency modes whose wavelengths are large in comparison to the dimensions of the cavity.

• Higher-frequency modes are less likely to benefit from such a change because they are more directly affected by changes in waveguide shape.

• Figure 16 plots the mode ratios for a cylinder-cone compound horn designed so that the cylindrical section volume is equal to the truncated conic section volume.

  Figure 16: Mode frequencies for a closed-open, cylinder-cone compound horn in which the cylindrical section volume is equivalent to the missing conic section volume. The dotted curves indicate integer relationships to the first mode.

  

• For $\beta=1$, the structure is of infinite length and all its modes converge to zero. In comparison with Fig. 15, the compound horn displays nearly harmonic mode ratios out to values of $\beta$ in the range 0.2–0.3.

• Another property of conic frusta can be directly attributed to the input inertance element, $M_{o}$, in the equivalent circuit (Fig. 14). The inertance, whose magnitude varies with the parameter $\beta$, tends to “shunt” low-frequency wave components, thus imposing a “high-pass” characteristic on the resulting air column mode structure.

• For longer conic sections, the lowest modes can be significantly attenuated, which in turn destabilizes oscillatory regimes dependent on these modes. This behavior is often apparent in the lowest notes of saxophones, which tend to be difficult to control under soft playing conditions.

• Figure 17 shows an example conic section input impedance in which this effect is demonstrated. The smooth curve indicates the combined influence of the conicity inertance and the open-end load impedance.

  Figure 17: An example conic frustum input impedance.