- 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

where - 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 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*=*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

Equation (18) can be rewritten in the form

(19) - 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
*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.

©2004-2016 McGill University. All Rights Reserved. Maintained by Gary P. Scavone. |