Why does a narrower tube, for the same length, favour the upper register?
A few reasons. First consider organ pipes, which are often close to exact cylinders. Here, the frequency dependence of the end effect is important. This effect is greater for wide pipes, so in wide pipes (such as the flute rank) the higher resonances are more “out of tune” (ie less close to harmonic ratios of the first resonance). Thus the higher harmonics of the jet do not have a resonance to support their vibration, or to provide an impedance matcher to the radiation field.
It’s worth discussing these questions together. The reflection at the open end of a pipe depends on the ratio of the wavelength to the radius of the pipe. This effect is important for the high resonances, and thus for the high harmonics in the sound. Further, the end effects at both ends are functions of frequency. For the same wavelength, the reflection coefficient is higher for a narrow pipe than for a wide. The greater the reflection, the stronger the resonance (all else equal), and the smaller the band of frequencies for which it will occur. Consequently, the high resonances of a narrow pipe are also narrow: a narrower band of frequencies will cause resonance in a narrow pipe than in a wide pipe of the same length. The frequency dependence of the end effect is important. This effect is greater for wide pipes, so in wide pipes (such as the flute rank) the higher resonances are more “out of tune” (ie less close to harmonic ratios of the first resonance). Thus the higher harmonics of
