A string is stretched out between two points under a constant tension of 23 Newtons. If both ends are fixed to rigid supports 13 meters apart and the string is simultaneously plucked at several points, what are the first three frequencies that you would expect the system to exhibit? The string between the supports has a mass of .5 kg.
A Fourier analysis is used to determine the frequencies present in a set of data. For example if a certain harmonic 'jerks' the end of the object 30 times per second, a Fourier analysis will clearly reveal this frequency, even if many other frequencies and a lot of random 'noise' are present.
Any disturbance introduced by the pluck which does not reinforce itself will end up interfering with its own reflections, and will therefore disappear relatively quickly. Only those disturbances which are very close to the natural harmonics of the object will maintain any significant presence much beyond a couple of reflection cycles.
The harmonics for this object are determined by the fact that there must be nodes at the ends, separated by the 13- meter length of the object. The first three harmonics therefore have 2, 3, and 4 nodes in the 13 meters, so they have 1, 2, and 3 half wavelengths in the same distance. So we have:
Knowing the wavelengths, we can obtain the frequencies if we know the velocity of wave propagation. This velocity is given by
The first wave has wavelength 26 m.
Reasoning similarly, we find that the second and third harmonics (these are the first and second overtones) will have frequencies of 1.88 Hz and 2.821 Hz.
Note that we could have used our knowledge of the ratios 2/1, 3/2, 4/3, ... of frequencies for this node-at-each-end configuration to obtain the same results from the .9403 Hz fundamental frequency.
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