Longitudinal Waves in Air (Experimentally determining the speed of sound)

[Von Neumann]

Problem:

In my physics class, we conducted an experiment involving a column of air set vibrating by a tuning fork of a known frequency f held at the upper end. The wave travels from the source to a fixed end (namely the water in the lower end of the tube) & reflected back to the source.

Assuming it takes a half-integral number of periods,

(n-1/2)T=(n-1/2)/f

for the wave to return to the source, then the compressional wave sent from the tuning fork in its downward motion and reflected by the water will arrive back at the fork just in time to aid its upward motion. Since there is no phase change in the compressional wave reflected off of the water, the condition for resonance is

(n-1/2)/f=2l/v

where n is a positive integer, and l is the length of the column of air, and v is the speed of the wave (the speed of sound). Solving for v,

v=2lf/(n-1/2)

We then compare the results of the experimentally determine v to the theoretically calculated v using the formula

v=\sqrt{\frac{(\gamma)RT}{M}}

Where \gamma is the thermodynamic constant for air, R is the universal gas constant, T is the absolute temperature, and M is the average molecular weight of air.

However, an alternate formula for v in terms of the distance l' from the first node to the bottom of the air column is

v=2l'f/(n-1)

This formula is given with no explanation, and I am wondering if it is mathematically equivalent to the other experimental formula for v. It is obvious that this particular formula will not yield a result when n=1.

 

[mfb]

The last formula corresponds to two open (or two closed) ends.

 

[Von Neumann]

The last formula corresponds to two open (or two closed) ends.

If you wouldn't mind, can you elaborate on why this is the case?

 

[mfb]

With a closed end, you get pressure differences a both sides, so you should match a wave of high pressure from one end, traveling 2l through the pipe, with the next (or next to next, or ...) time of high pressure. This gives an integer multiple of wavelengths in 2l.
You can drop the -1 if you like.

 

[Von Neumann]

If, as you claim, the equation v=2l'f/(n-1) correlates to the situation involving a pipe with 2 open or 2 closed ends, then it certainly has no relevance in this experiment having to do with a pipe with 1 open end. Correct?

 

[Von Neumann]

Anyone have any HELPFUL suggestions?

 

[mfb]

If, as you claim, the equation v=2l'f/(n-1) correlates to the situation involving a pipe with 2 open or 2 closed ends, then it certainly has no relevance in this experiment having to do with a pipe with 1 open end. Correct?

Correct

Anyone have any HELPFUL suggestions?

I am not sure how to interpret that post...