For a science project I chose to investigate the relationship between the resonance frequency of cups and the amount of water filled in it. I filled the cup with 10 grams of water each time, and used a pencil to hit the cup. After trying a lot of combinations, I found that there is a linear relationship between $1/f^2$ and $m^3$, where $f$ is the resonance frequency and $m$ is the mass of water added. Can anyone explain this relationship? I searched all over the internet, but I couldn't find anything.

I used cups that are cylindrical, so the mass can be converted into water level and the relationship maybe between $1/f^2$ and $h^3$ where $h$ is the water level.

These are my research that I think are most relevant:

How and why does the height of water in a glass affect it's resonance? (resonant frequency)

Understanding how the amount of water affects frequency emitted by wine glass


Maybe the cups can be modeled as a spring-mass model? The guy in the first link said, "resonant frequency always looks like the square root of some kind of restoring force divided by some kind of inertia"

Something like this:

  • $\begingroup$ What are the cups made of, out of curiosity? If hitting them with a pencil is enough to produce a clear tone, I would expect them to be made of glass (in which case "glasses" would be the more appropriate term). $\endgroup$ – probably_someone May 24 at 16:44
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    $\begingroup$ Also, how sure are you that $1/f^2\propto m^3$ is the actual best-fit relationship? For example, have you checked to see that a plot of $m^4$ on the $x$-axis doesn't look just as good? I would recommend running a (constant+power-law) fit on the data, just to be sure. $\endgroup$ – probably_someone May 24 at 16:54
  • $\begingroup$ The first cup is made of glass, and the second is made of porcelain. I will try more fit, but this one has r-squared values of 0.99x, so I think it is the best-fit relationship. $\endgroup$ – David305 May 25 at 1:10
  • $\begingroup$ Oh my god I found this paper and the formula derived in it matches with my results! However, I don't understand the derivation. Can anyone explain it in simple terms? $\endgroup$ – David305 May 26 at 12:08

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