Bottle Marimba calculations? How does pressurized air in a bottle change the sound of the note being played (more air = higher note) but what is the explanation behind why this works and possibly also the calculations to achieve each note.
I'm making an instrument for a high school physics project and I was wondering how the density of air changes the perceived note and the explanation behind it. I've managed a few possible theories but none that I was fully confident in. I'll reference a video which inspired my idea, adding more pressurized air to the bottle changes the pitch but I'm not exactly sure how this could work, does anyone have any answers? Thanks for the help.
https://www.youtube.com/watch?v=koIC_MAB3MU&ab_channel=JetKyeChong
 A: The bottles are essentially balloons under tension. Kuo, Hunt and Lister calculated the frequency of a spherical shell to be the following "simple" formula:
$$
\omega^2_n = \frac{A\left(1+\frac{N}{1+\nu}\right) +\tau N + 2 
\pm\sqrt{\left(A\left(1+\frac{N}{1+\nu}\right)+\tau N +2\right)^2 - 4AN\left(\tau- 1+\frac{\tau N+2}{1+\nu}\right)}
}{2\rho R^2A(1-\nu)/E}
$$
and $\tau=T(1-\nu)/(Eh)$, $A=1+((\rho_e/\rho)/(n+1) + (\rho_i/\rho)/n)(R/h)$  and $N=(n-1)(n+2)$ for integer $n$ corresponding to the different overtones. Here $R$ is the shell radius, $h$ the thickness, Young's modulus $E$, Poisson's ratio $\nu$, $\rho$ the shell density, and $\rho_i,\rho_e$ the air density inside and outs, and, finally, $T$ is the tension along the surface.
While I doubt this formula is at all useful for the marimba, it does show how the tension makes the frequency go up.

The problem with this solution is that it is for a spherical shell and the coke bottle has a more complex shape. In particular, it likely has a bunch of "bending" modes, and might be more similar to a closed tubular bell. This likely explains why the pressure has a bigger effect than on the sphere (where the $n=1$ mode is pressure independent!) but is likely very hard to solve mathematically even for a cylinder, let alone a coke bottle.
In the end, I suspect a dimensional analysis argument might work best to estimate the pressure dependency of fundamental frequency.
A: Most important:
First, I would suggest, WEAR SAFETY GLASSES.  If a small piece of plastic embeds itself anywhere in your body, the damage will almost always be reparable, unless it's in your eye.  It really just can't stressed enough -- if there's ever any chance of flying material, as there is here, wear safety glasses; and don't let a few experiences that don't result in injury lull you into complacency on this issue.
Physics:
The vibrational modes of an object depend dramatically on it's shape, and, say a sphere and a cylinder from the same material and thickness will sound completely different.  To estimate anything here, we need to either guess or measure what's moving and making the sound.  Consider, for example, a drum head, which doesn't have a nice resonance vs the same material if it were made into a linear band.
Based on the bottles and the sound, I would guess that the primary resonant modes are being created by the smooth uniform cylinder that composes the top 1/4 of the bottle, just below the half dome for the cap.
The reason that I think this is the important region is that it's notable that the tone of the bottle is relatively pure, which implies a region of the bottle where I nice resonant standing wave could be set up, and where the overtones would be harmonic.  It's also interesting that these bottles were carefully selected and that they happen to have this broad, simple, flexible, uniform band of material that looks perfect for setting up a nice vibrational mode.  Also, it's interesting that the bottle needs to be under pressure to make a sound, which implies that the restoring force is not purely due to a rigid part of the bottle (like for a bell) or an acoustic resonance of the air within the bottle (like for a woodwind), but, instead, that it's a flexible region of the bottle, where the restoring force is created by a tension setup by the pressure.
The specific resonance mode I suggest here is a standing wave that encircles the bottle. This then will have a nice resonance and with overtones that are approximately harmonic, and therefore sound nice.  It's similar to the standing waves on a guitar string, but if the two ends of the string were attached for form a ring.
Like a guitar string, then, the frequency will be proportional to the square root of the tension, and the tension should be proportional to the pressure, so,
$$f \propto \sqrt{P}$$
There are lots of interesting things to measure:

*

*Try using the mallet to tap in various locations and various ways to see what types of tapping excites the best resonant modes. My guess is that tapping will work best when on the smooth uniform region I've described.

*Once you've found a good way to tap, you can lightly touch other places to force a node (like playing harmonics on a guitar, so it takes the right "touch"), but then see where the nodes are.

*You can set the bottle by a speaker that's playing pure tones and then examine what modes are excited by looking carefully for vibrations, putting sand on or near the bottle (admittedly difficult for a bottle), bouncing a laser pointer off the bottle, etc.

*Measure the pressure and see how the frequency varies with the pressure.

*If you find that the vibrational modes of the cylindrical band are approximately right for this, it's convenient because these will be relatively mathematically tractable.

