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?

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    $\begingroup$ Great demo. Spring lines, with an acoustic transducer at each end, were always used for the reverb effect in electronic music before the advent of digital delay lines. They made a great noise when you hit them (Deep Purple, ELP). Pretty sure the answer will be related to the acoustic delay and multiple reflections due to the impedance mismatches at the ends. $\endgroup$
    – twistor59
    May 8 '13 at 18:35

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

   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.)

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    $\begingroup$ So, are we going to see "Computational Foley Artist - Emilio Pisanty" in the credits at the cinema any time soon? That sound was rather special! $\endgroup$ Aug 11 '13 at 0:51
  • $\begingroup$ Naw - next time I need to foley up some lasers I'll go record some ice. $\endgroup$ Aug 11 '13 at 0:55
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    $\begingroup$ Very good! You could also add that the chirping sound comes from transverse modes of the slinky: the longitudinal modes are dispersionless. Transverse vibrations bend the wire away from its equilibrium configuration and are described by wave solutions of the dynamic Euler-Bernouilli equation, for wavevectors much larger than the slinky curvature. Unlike the traditional wave equation, this one is second order in time but fourth order in space, and so naturally results in a quadratic dispersion relation $\endgroup$ Aug 12 '13 at 12:49
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    $\begingroup$ I also heartily recommend the original and marvelously irreverent paper on ``slinky whistlers" by Frank S. Crawford (Am. J. Phys. 55, 130-134 (1987)), who apparently wrote prolifically about the physics of everyday phenomena. $\endgroup$ Aug 12 '13 at 12:52

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