Rubber strings, real strings, cameras and phones

When I first saw demonstrations of string vibration using low tension rubber/elastic strings my instiinct was that the behaviour must be very different from the vibration of strings in real musical instruments such as a guitar. Assuming the rubber is stretched enough, however, it turns out that rubber strings do satify the wave equation to a good approximation just like real musical instruments. Here’s an example of my slap bass technqiue and what it does to a rubber string:

Super Slow Motion Struck String (slap bass technique) at the University of St Andrews

This demonstration shows the string stationary or moving at constant speed when a portion of the string is a straight line and accelerating when the string is curved (as required for linear wave equation behaviour).

So what are the differences between this rubber string and strings in an instrument like a guitar from the point of view of physics? The main difference is that the rubber has a much lower Young’s Modulus. The Young’s Modulus of rubber is of the order of 1 MPa whereas Nylon has a Young’s Modulus around a thousand times higher and steel has a Young’s Modulus about 200 times larger than that of Nylon. This means that a rubber string picks up much less tension when it is stretched by a given fraction of its length. To get linear wave equation behaviour from any string we need the elongation (change in length) to get the string up to its rest tension to be much bigger than the additional elogation we add when we pluck, strike or bow the string. If this requirement is met by stretching the rubber enough then the tension in the string will be approximately the same through the entire motion and a linear wave equation behaviour will be a good approximation. For the rubber string the very low Young’s modulus means we can apply a very large elongation to the string without the tension going up much at all and therefore we can do demonstrations with large transverse displacements and the resulting motion will still have approximately constant tension and therefore be linear (and the string won’t snap). The low tension means the vibrations are low enough in frequency for a reasonably priced high speed camera to pick up and the large displacements are easier to see.

Real musical instruments typically require higher tensions to give audio frequencies for us to hear. The frequencies are too high and the displacements too low for reasonably priced cameras to pick up properly. For acoustic instruments the high tensions also mean high forces are applied to the bridge of the instrument when the angle of the string changes there (and this creates more volume).

As a side note, you may have seen footage such as that below of strings vibrating as taken on a smartphone and this is of course not picking up the actual shape of the string in each frame, rather it is based on the progressive scanning within the image sensor. Each row of pixels is the result of looking at a different phase of the string motion and the shape of the string in the resulting video is analgous to an oscilloscope plot of the motion. Note high light levels are requried for this to work as the shutter-speed has to be fast to prevent blurring. Someone should write an app to make a smartphone into a polyphonic (and probably gnarly) guitar pickup. Does anyone want to try this?

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