Physics, Modeling Instruction, Educational Technology, and other stuff I find interesting

# Longitudinal Standing Waves in a Spring

Posted on April 19, 2012

In the spring of 2011 I had a pretty amazing student teacher, Mat. One of the cool things he did was to pillage his college lab equipment (my alma mater as well, St. Olaf College) to bring in a function generator that drives a speaker, producing standing waves on a string. I found out towards the end of the year that Vernier makes equipement that, though not as high end as what he brought in, does the job for about \$300 (I needed a power amplifier and speaker accessory, as I already had a Labquest and Loggerpro, either of which can act as a function generator). I had taught waves for years, but lacked this equipment which, through his example, I found made standing waves much more tangible for students.

So this year I started standing waves just as Mat did, by producing standing waves on a string, shown below, and deriving the pattern that yields that the nth harmonic is n times the fundamental frequency (n=positive integers), or

$\dpi{100} \bg_white f_{n}=nf_1$

3rd harmonic of a standing wave on a string

A node for a standing longitudinal wave on a spring

Mat had found a cool video which showed standing waves on a spring, so naturally, I wanted to replicate this for myself. I managed to do so without too much effort, and it’s pretty sweet. Nodes show up as single coils that don’t move, and both nodes and antinodes can be identified by sticking a small piece of paper in the spring. Antinodes move the paper up and down by a couple of centimeters, while at the nodes the paper remains stationary. This is better shown in the video I made to show the apparatus and how it works, linked here as well as embedded at the bottom of the post.

I figured that since the spring was fixed at one end but driven at the other that it would act like a node anti-node (NA) standing wave, similar to a pipe closed at one end and open at the other. This follows the same pattern as above except that n can only be odd; that is, only the odd harmonics are produced.  Additionally, I was not certain without theory how the fundamental frequencies of the two would relate. Though I could find nodes (which was awesome!), I failed for a while to come up with a nice pattern for the resonant frequencies that mimicked NA.

A node formed near the end of the spring

Then springs started falling off, which was weird and only in some resonant cases. Next I noticed that while most of the time the nodes were not evenly spaced (as would be expected with an antinode at the driver), occasionally I found a node right at the driver, as shown at right, which I figured meant that was part of the NN pattern.

I started wondering what kind of patterns I may actually be producing and if there was theory to back them up. Could I have a node in some cases at the driver, while in other cases have an antinode? How would the fundamentals of each pattern be related?

I had looked around for a while to find a way to calculate the theoretical fundamental frequency for waves on a spring. After much searching I finally found this article, which states that the speed v of a longitudinal wave on a spring of mass  ms with spring constant k and length L is

$v=L\sqrt{\frac{k}{m_{s}}}\text{ \ \ (eq 1)}$

All references I found to this equation state that it could then be used with the following equation to find the resonant frequencies of longitudinal waves on a spring;

$f_{n}=\frac{nv}{2L} \text{ for }n=1,2,3...\text{ \ \ (eqn 2)}$

which means that they assume that there is a node at both ends of the spring, such that it behaves like a standing wave on a string. Substituting equation 1 into equation 2 yeilds

$f_{n}=\frac{nv}{2L}=\frac{n\left[L\sqrt{\frac{k}{m_{s}}}\right]}{2L} \Rightarrow f_{n}=\frac{n}{2}\sqrt{\frac{k}{m_{s}}}\text{ for }n=1,2,3...\text{ \ \ (eqn 3)}$

Thus once the value of $f_{1}=\frac{1}{2}\sqrt{\frac{k}{m_{s}}}$  is established, the frequency of the nth harmonic for NN standing waves is simply

$f_{n}=nf_{1}\text{ for } n=1,2,3... \text{ \ (eqn 4)}$

This would be great, except that it didn’t fit the values that I was getting! Additionally, I noticed that I would get other standing waves in-between the ones that I thought were the harmonics I was looking for. So then I took the equation for the harmonic frequencies for a standing wave with a node at one end and an antinode at the other,

$f_{n}=\frac{nv}{4L} \text{ for }n=1,3,5...\text{ \ \ (eqn 5)}$

and substituted equation 1;

$f_{n}=\frac{nv}{4L}=\frac{n\left[L\sqrt{\frac{k}{m_{s}}}\right]}{4L} \Rightarrow f_{n}=\frac{n}{4}\sqrt{\frac{k}{m_{s}}}\text{ for }n=1,3,5...\text{ \ \ (eqn 6)}$

For my spring with mass 11.6 grams and experimentally determined spring constant 1.14 N/m, I should get a fundamental at 5 Hz for NN standing waves and 2.5 Hz for NA standing waves. The fundamentals are almost impossible, in my experience, to see due to the lack of nodes (the NN pattern should have a node at the bottom, but I had trouble getting a passible node for the fundamental), and thus I started confirming with n=2. My predictions and data are shown below;

As you can see, the node-antinode predictions work out quite well, but the node-node predictions are systematically low. I suspect this has to do with the fact that the node at the bottom of the spring is not all the way at the bottom of the spring, therefore the length of the spring needs to be adjusted in a similar way that produces end effects in pipes (Document link) changes the effective length of a pipe that is open at the end. I have to find some time to investigate this more formerly, however.

One interesting aspect of spring standing waves is that the frequencies that generate standing waves do not depend on the length of the spring. This can be shown by changing the length of the spring and observing that the standing wave remains. Note that the above mentioned end effect would take place because if the L term in equation 1 (which would be the total length of the spring) did not completely cancel out with the L term in equation 2 (where L is the effective length of the standing wave, node to node).

One obvious application of the dual pattern is organs. Organs utilize both open-open (node-node) and open-closed (node-antinode) pipes, and the relationships between their frequencies can easily be demonstrated (and calculated, though as an analogy since v is calculated differently with a spring) using this apparatus. Plus it’s really cool.

One last fun fact; It turns out that node-node AND node-antinode patterns can also be produced on a string with transverse waves; more on that in a later post. For now, check out the video of the longitudinal standing waves on a spring, below.

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