Wavelength of a wave in a closed tube

[warfreak131]

Homework Statement

In a lab that we performed, we had a closed tube with piezo electric transducers on both ends, and they were attached to a frequency generator. We found the frequencies that caused resonance, and the lab wants us to calculate the velocity of the wave using data. The lab says that there are antinode pressure waves at both ends, and therefore, the length of the tube is equal to one wavelength at the lowest frequency.

But at the lowest frequency, wouldn't one antinode to the next correspond to half a wavelength? The two anti nodes in question would be one antinode with max positive displacement to one with the max negative displacement.

 

[ehild]

You are right, there is half wavelength distance between the antinodes. But it could be that those transducers generated in-phase pressure changes, then there was one wavelength between the ends.

ehild

 

[warfreak131]

so should i go with my gut and say half wavelength or trust the manual to be right?

also, assuming its a full wavelength, it asks us to find the velocity at higher resonant frequencies. so would the number of waves inside the tube increase by 1/2 or by a whole when another resonant frequency is reached?

 

[ehild]

I am confused about the set-up of the experiment. Was that tube exited at one end and closed at the other end where the pressure was measured? The pressure can have antinode at a closed end, but than the length of the tube is L=((2n+1)/4) λ with n=0, 1, 2, 3, ... If the tube was either closed or open at both ends it was a half wave between the ends in case of the fundamental frequency. In case of the other resonant modes, the length of the tube is integer multiple of the half wavelength: L=n(λ/2).

See http://en.wikipedia.org/wiki/Acoustic_resonance#Cylindersehild

 

[rwisz]

I would like to clarify that in a closed tube, the wavelength of a wave is equal to twice the length of the tube. This is because in a closed tube, there are two points of maximum displacement (antinodes) at each end. The distance between these two points is equal to half a wavelength. Therefore, in order to calculate the wavelength of the wave, we would need to multiply the length of the tube by 2.

Furthermore, the frequency at which resonance occurs in a closed tube is determined by the length of the tube and the speed of the wave. Therefore, we can use the formula v = fλ to calculate the velocity of the wave, where v is the velocity, f is the frequency, and λ is the wavelength.

In this case, since the length of the tube is equal to one wavelength at the lowest frequency, we can simply use the lowest frequency and the length of the tube to calculate the velocity. However, it is important to note that the wavelength will change at different frequencies, so it is best to take multiple measurements and use an average to calculate the velocity accurately.

In conclusion, the wavelength of a wave in a closed tube is equal to twice the length of the tube and can be used to calculate the velocity of the wave using the formula v = fλ. It is important to take multiple measurements and use an average to ensure accurate results.