Question

A soprano saxophone is approximately a conical tube 0.75 m long that is open at the wide end and closed at the apex. The wide end is about 8 cm in diameter. Find the frequency of the lowest note (in Hz) at 20 C. Neglect the end correction at the open end. The result you find should be higher than the actual lowest note of a soprano sax, which is A3-flat.

Answer 1

For a conical tube, if the width of the tube is changing slowly, the sound waves travel without reflection (like an open pipe).

If the change in area of the cross section is large compared to the wavelength of the sound, most of the sound is reflected by the surfaces.

For the case of a cone, the longer the wavelength of the sound, further out from the closed end it will reflect.

The resonant frequencies of a one end closed conical tube is exactly same as an open ended tube

For a cone of length L,

Where n = 1,2,3..

and v is the speed of sound in air.

At 20^oC, v = 343 m/s

The smallest frequency is for n=1

Here, L = 0.75 m

So,

This frequency is higher than the flat A note of frequency 110 Hz