## 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 ZL = 0, the reflectance C-/C+ becomes -e-2jkL and the input admittance of a truncated cone reduces to
 (16)

where

• If the small end of the cone, at x=x0, 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 by
 (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=x0 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 x0.

• The natural frequencies of the truncated cone closed at its small end are found by solving the transcendental equation
 (18)

Equation (18) can be rewritten in the form
 (19)

where and is the fundamental frequency for the open-open conic frustum of length (Ayers et al., 1985).

• 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 (complete cone) to (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 x0 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 x0 and frequency.