What do physics and music have in common? Both have much to do with motion: Physics could be defined as the study of motion (in a broad sense of the word). The beautiful sounds from which music is built up, on the other hand, result from the subtle motion of musical instruments. We can bring physics and music together by visualizing this very special kind of motion.
We start with the motion of strings which are the central ingredient of many musical instruments. The low E guitar string, for example, vibrates mostly at around 80Hz. Since humans can distinguish only up to around 12 distinct pictures per second, the vibration of the low E guitar string will appear blurred and stationary--a kind of average picture of the string's position in time. To get a clear picture of the motion, we turn off the lights and illuminate the string with a stroboscope tuned close to the frequency of the string. The eye (or a camera with low frame rate) will then see a drastically slowed down vibrating string, moving at the difference frequency of stroboscope and string:
(note: the moving dark and bright vertical stripes are an artefact of the camera and depend on the relation between camera frame rate and stroboscope frequency).
The low E guitar string vibrates most strongly at around 80Hz in the so-called fundamental mode. But we know that the specific guitar sound depends essentially on more complex motion of the string which occurs at so-called higher harmonic modes of the fundamental mode. To make these higher modes visible we can tune a function generator with a loud speaker connected to it to twice (or threee times etc) the frequency of the fundamental mode (a skilled guitar player will actually be able to selectively excite higher modes directly). Again we use a stroboscope to slow down the apparent motion. It is quite hard to excite the higher modes directly so the resulting video shows an almost imperceptibly subtle motion:
We can use a freely available algorithm to exaggerate the subtle second harmonic motion of the string:
In the amplified version of the video one can see clearly that the second harmonic has a node exactly on top of fret number twelve; not surprisingly that's the fret you press to get the octave ie double the frequency. The algorithm and more details on how to use it are available at http://people.csail.mit.edu/mrub/vidmag/
Of course, many instruments don't involve strings. The sound of timpani, for example, results from the movement of a large circular sheet stretched over a large copper bowl. Instead of the one-dimensional string we are now looking at the movement of a two-dimensional surface. And as usual, things get more interesting with increasing dimensionality.
Since we did not have timpani at our disposal we used a broad thin rubber sheet stretched across an empty (and cleaned) tomato can. We again excite different modes of motion using a function generator and a loud speaker. We get a first idea of the possible motions by scattering some sand or salt on the rubber sheet. When we hit one of the resonance frequencies of the rubber sheet the salt will gather in the anti-nodes (where the sheet does not move):
Then we can use stroboscope and camera as before to record a video of the corresponding mode of motion:
As before we can amplify the motion to make it clearer:
You see that visualizing the motion of musical instruments is not too hard. No doubt even more interesting videos can be recorded with some patience and creativity.
Jason Hölscher-Obermaier und Jonas Schmöle
(Gruppe Markus Aspelmeyer, Universität Wien)
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