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I have a string which is vibrating at a fixed frequency, say 1 kHz. I know it is vibrating in a purely first harmonic mode because of the (sinusoidal) drive function I am using. With a microphone I measure the sound wave and do a frequency analysis of this. I measure several harmonics in this spectrum. Is this possible? Does the environment of the string generate higher harmonics?

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  • $\begingroup$ How could it not be possible? You have a source for a Time, during such time the Frequency modulates to reflect the amplitude of its position with respect to the observation, during such time the oscillations from the Frequency interfere with itself and some cancel and others cause some spontaneous new waves to be created in a coherent vector to the initial ones. You can see your Frequency, Its compliments and derivates over the course of time of their radiation or emission. I would think more importantly depending on your microphone that your also picking up a lot of other noise. $\endgroup$
    – Jay
    Jan 17, 2021 at 2:07
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    $\begingroup$ You can only verify that the string is vibrating at its fundamental frequency if the standing wave on the string is only 1/2 of a wavelength. A sinusoidal driving function can produce many harmonics on the string (fundamental and higher harmonics), so the fact that you are using a sinusoidal driving function does NOT by itself mean that you are seeing the first harmonic. $\endgroup$
    – David White
    Jan 17, 2021 at 2:56
  • $\begingroup$ The "string" is a conducting wire in a homogeneous magnetic field. An AC current is run through the wire that is tuned to the fundamental frequency of the wire. I would think this excited only the fundamental mode. I can measure the induced current in the wire and Fourier analyze it, confirms fundamental mode. Microphone picks up harmonics that must be generated by the cavity, walls, enclosure? Can David White please comment? Thx. $\endgroup$
    – JoseA
    Jan 17, 2021 at 5:22
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    $\begingroup$ How do you produce your driving AC? Have you cheched the spectrum of the driving voltage or even of the current through the wire? $\endgroup$
    – nasu
    Jan 17, 2021 at 6:21
  • $\begingroup$ Let us assume that the wire is vibrating PURELY in the fundamental harmonic. For the moment let's not focus on the driving mechanism. I do measure the induced current on the wire, and establish by Fourier analysis that it is vibrating in the fundamental mode only. My question relates to the measurement of the acoustic wave, where a Fourier analysis of the sound wave picked up by the microphone shows first, second, and some third harmonic components. Can the harmonics be generated by the sound wave interacting with the environment surrounding the wire? $\endgroup$
    – JoseA
    Jan 17, 2021 at 16:38

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A linear system driven by pure sine wave will produce a response only at the fundamental frequency. There will be no harmonics or overtones. Harmonics are multiples of the fundamental frequency. Overtones are higher order resonant modes of the system. For a vibrating string, the overtones are very close in frequency to multiples of the fundamental frequency.
In order for harmonics to be present, either the source is not a perfect sine wave or there is some nonlinearity in the system. An even-order nonlinearity (e.g. quadratic) will create only even harmonics. An odd-order nonlinearity (e.g. cubic) will create only odd harmonics. The nonlinearity could be, for example, that Hooke's Law is not obeyed perfectly.
In order for overtones to be excited, there must be some mode-coupling present so that energy can flow from the fundamental mode into higher-order modes. This can occur in any real system with damping and especially in situations where energy is strongly focused in a small region (e.g. bending at the ends of the string).

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