Alien Math · Toys
Three sound toys
A pipe that keeps one note forever while every other sound runs out of the ends. A wooden bar that only sounds sweet once you carve an arch under it. Sand that draws the shape of a sound. Strike, carve and shake them here.
All three are known physics. The simulations, the pictures and the checks are ours; the people who worked these out are named at the bottom.
Toy 1 · A pipe, a plate, a note
The note that stays
Below is a long pipe seen from the side, with a thin flat plate fixed down its middle. A little hummer in the centre sings one pitch for a while, pushing up above the plate while it pushes down below. Then it stops. Any sound that can travel runs off both ends into the soft dark zones and never comes back. Watch what stays.
Pick a plate and a pitch to start.
Sound can only travel down a pipe this wide if it pushes the same way on both sides of the middle, or if its pitch is above a cut-off. The trapped note does neither: it sways up on one side of the plate and down on the other, at a pitch just below the cut-off. So once it’s there, it can’t leave. Hum it and nearly all of it stays. Hum 5% off and the pipe never really catches it. Take the plate out and nothing stays at all.
Heard and solved
Hum 5% low and stop. The little that stays rings at 1.038, the plate’s own pitch, not the hum’s. A separate eigenvalue solve on the same grid (trapped.py) gives 1.038 too, against a cut-off of 1.571.
Finer grids, higher note
Halving the grid twice moves the medium plate’s note to 1.0464; the trend points to about 1.049 on a perfect grid (trapped.json).
No plate, no note
Same hummer, same pitch, empty pipe: when the hum stops, the sound near the middle drops to almost nothing. The plate makes the trap.
What the model leaves out: the pipe is flat (two dimensions), the air has no friction, and the plate has no thickness. In a real duct the trapped note still loses a little energy to friction, so it rings for a long time instead of forever.
Toy 2 · Wood, an arch, two octaves
Carve a marimba bar
A plain wooden bar rings with overtones that don’t fit any musical scale: about 2.76 and 5.40 times its main note. That clangs. Marimba makers carve an arch under the middle of each bar until the first overtone lands exactly two octaves up, four times the main note. Carve it yourself.
Carving the middle makes it thinner and floppier, which lowers the main note a lot, because that’s where the main note bends most. The first overtone bends near the ends too, so it drops less. Carve deep enough and the gap between them opens to exactly two octaves. Strike the plain bar, then the tuned one, and listen to the clang turn into a tone.
What the model leaves out: the bar is a thin beam with no twisting and no air around it, and wood is treated as the same in every direction. Real makers also tune the second overtone, usually near ten times the main note, with a second cut; one arch depth can’t fix both, which is why this one lands near 9.3.
Toy 3 · Sand, a plate, a violin bow
Sand on a plate
Sprinkle sand on a metal plate and make it ring with a violin bow. The sand bounces off the parts that move and collects on the lines that stay still. Each note draws its own picture. Pick a note.
What the model leaves out: the plate’s shape of motion here is the simple textbook formula (two crossed waves added or subtracted), which gets the look of the patterns right. A real square plate held at its centre bends in a more complicated way, and its exact figures differ in the details. The sand moves by random kicks scaled to the plate’s motion, with no gravity or bouncing.
Prior art
Who got there first
The note that stays
- D. V. Evans, M. Levitin and D. Vassiliev, Existence theorems for trapped modes, Journal of Fluid Mechanics (1994): any body symmetric about the middle of a channel traps at least one such note.
The marimba bar
- Marimba bars are tuned with the first overtone two octaves up, xylophone bars an octave and a fifth up (HyperPhysics: xylophone and marimba).
- The plain-bar ratios 2.76 and 5.40 are the textbook free-beam values; the page computes them as its control.
Sand on a plate
- Ernst Chladni, Entdeckungen über die Theorie des Klanges (1787), where the sand figures were first published.
- Walther Ritz worked out the square plate’s figures properly in 1909, with the method now named after him (Ernst Chladni).
The code behind every number: pipe.js (the pipe simulation, the hummer and the ringing pitch), trapped.py (the separate eigenvalue solve, which writes trapped.json), bars.js (the bar’s vibrations), chladni.js (the sand).
Next click down
The trapped note is the same physics as the long-lived ringing in jet-engine ducts and heat exchangers, where plates sit in moving air. Adding a slow flow down the pipe, and seeing whether it feeds the note, is the obvious next build.
More toys: three toys that shouldn’t work and three shape puzzles. Back to the Alien Math wing or the Lab.