# Standing waves in a resonance tube

I am doing an experiment about standing waves in a resonance tube. I use a bucket of water, a waterproof tube (open at both ends), and a frequency generator app. I have two set-ups, A and B:

A. Setting the frequency to be constant and dipping the tube into the bucket of water until I hear resonance that signifies standing waves. I record the effective length of tube at which this occurs, from the part where the tube touches the water to the other end. Then, I compute the corresponding wavelengths.

B. Setting the length to be constant but with increasing frequency. I record the frequency at which the standing waves occur and compute the wavelengths.

Using the formula for the speed of sound $$v =$$ wavelength x frequency, I have solved the experimental speed of sound. Now, using the room temperature to solve for the theoretical speed of sound, I found out that the Set B resulted to less deviation from this theoretical value compared to the case of Set A. Is this expected in general? What do you think are the main sources of error?

• Welcome JD Maximo. An interesting question, but you might want to provide a bit more experimental detail. For example, how big a deviation are you talking about? Did you monitor temperature continuously, or just once? What was the uncertainty in the length and frequency measurement? Oct 10, 2022 at 3:15

You are using the same apparatus to either, 1 detect a maximum loudness whilst altering the length or, 2 detecting a maximum loudness whilst altering the frequency.

From your post it would appear that method 2 allows you to estimate the maximum loudness with greater precision which is perhaps no surprise as I think that I would find it easier to twiddle a knob to change the frequency in "comfort" and estimate the frequency for maximum loudness as oppose to moving a tube up and down whilst keeping it vertical and setting the tube at estimated maximum loudness.

I do not know how you analysed your data but you as you have not mentioned an end correction, $$e$$, for the tube which is a correction for the fact the the displacement node (pressure node) does no occur exactly at the open end of the tube.
This means that for the fundamental resonance position $$L+e= \lambda/4$$ where $$L$$ is the physical length of the tube and $$\lambda$$ the wavelength of the sound.

$$\dfrac {\lambda}{4} = \dfrac cf \Rightarrow L = \dfrac {4c}{f}-e$$ where $$f$$ is the frequency of the sound and $$c$$ the speed of sound.
So a graph of $$L$$ against $$\dfrac 1f$$ has a gradient of $$4c$$ and an intercept on the $$L$$ axis of $$-e$$.

Note also that if you do not account for the end correction it produces a smaller error in the estimate of the wavelength the larger the wavelength (the smaller the frequecncy) is.

It's hard to comment on a specific experimental setup, but I am not surprised by your result. In B, the machine can control the value of the frequency much more precisely than you can manually submerge the tube into the water to a certain length in A.

One thing to try calculating is, if your length measurement is off by say $$\mathrm{0.1 \, cm, 0.5 \, cm, \,or\, 1 \, cm}$$, what % change does that produce in the speed of sound result?

Another thing to assess is whether your Method A results seem to be consistently too high, too low, or randomly scattered around the true value.