waves - Slinky reverb: the origin of the iconic Star Wars blaster sound

This is a fun problem that I came across recently, which I'm posting here for your delectation. We all love a good slinky: they can be used for all sorts of fun demos in physics. One example is the "spring reverb" effect, which is well known in the film sound effects industry. If you couple a microphone to a slinky, and then tap the slinky, it produces a wonderful chirping noise. This is actually how they made the sound of the blasters in the Star Wars film franchise.

Watch and enjoy this video for a scaled-up example. The slinky in this case is a coil of 3.5mm galvanised steel, 100m in length. A piezo transducer is used to pick up the sound. What property of the coil allows it to produce such an interesting noise?

Answer

I think I've got this one, inspired by this recording of sound transmitted through ice. What you hear sounds much like a Star Wars laser but is also quite clearly the downwards chirp caused by nonzero dispersion in the ice: shorter waves travel faster, so that sounds propagated through enough ice will start with higher pitches and end with the lowest ones.

From a very short excursion on the internet, I gather that slinkies have a dispersion relation of the type $$\omega=ck\cdot kr,$$ where the dimensional information comes probably from the slinky radius $r$. (My one reference: Slinky‐whistler dispersion relation from ‘‘scaling’’ (Frank S. Crawford. Am. J. Phys. 58 no. 10, pp. 916-917 (1990).)

This means the phase velocity is $c\cdot kr$ and increases with frequency. If you disturb the slinky in one place and listen in another, the higher tones will get there first and you will hear a downwards chirp.

I've done some numerical playing in Mathematica and it does look like it's the case. For a nice example, if you have MM, try

Sound[{Play[
   Re[E^(-((10000 I)/(4 10^-6 I + 60 t)))/Sqrt[10^-6 - 15 I t]], {t, 0, 15}]}]

though I don't have a solid enough justification for it yet. (This is the result of a disturbance of the form $\exp\left(-\frac14(\frac{x}{1\,\text{mm}})^2\right)$ at $t=0$ heard from $x=100\,\text m$ away on a slinky of radius $r=5\,\text{cm}$ and speed of sound $c=300\,\text m/\text s$, with the dispersion relation as above. Unfortunately if you play it from $t<0$ you also get the sound of the left-bound wavepacket, which I can't yet eliminate. But the physics seems right.)