Question

A hollow tube of length L open at both ends as shown, is held in midair. A tuning fork with a frequency f_o vibrates at one end of the tube and causes the air in the tube to vibrate at its fundamental frequency. Express your answers in terms of L and f_o

a. Determine the wavelength of the sound.

b. Determine the speed of sound in the air inside the tube.

c. Determine the next higher frequency at which this air column would resonate.

The tube is submerged in a large, graduated cylinder filled with water. The tube is slowly raised out of the water and the same tuning fork, vibrating with frequency f_o, is held a fixed distance from the top of the tube.

d. Determine the height h of the tube above the water when the air column resonates for the first time. Express your answer in terms of L.

Answer 1

(a) The fundamental in a open–open pipe looks like this and is ½ of a wavelength of the wave. Since this ½ wavelength fits in the length L, the total wavelength would have to be 2L.

(b) Simply use v = f λ → v = 2Lf_o

(c) Harmonics are multiples of the fundamental, so the next frequency is 2f_o

(d) This is the same tuning fork so it is the same wavelength and waveform but the bottom is now closed so the wave looks like this.

The tube we had initially, fit ½ of a wavelength inside, and since its the same wavelength wave, again ½ of the wavelength of this wave would fit in length L and it would look like this.

This is impossible for a standing wave in an open–closed tube, and its also not the fundamental anyway so we have to change the length to make it look like the fundamental, Shown below. To do this, we make the length half of what it used to be.