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When I strike a string on my guitar and look at the audio spectrum, I can see the fundamental frequency as a large peak. I can also see the harmonic frequencies as a train of little peaks at increasingly higher frequencies. When I tune an oscillator to produce a pure tone and drive a speaker, predictably I don't see any harmonics.

Do harmonics occur in the radio spectrum reserved for communications?

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A guitar string produces harmonics because it vibrates in a non-linear fashion. An electronic oscillator can be made to generate a much purer form of vibration (near sinusoidal) than a mechanical device such as the guitar string. Hence its harmonic level, while not zero, is much lower. For example, the harmonic distortion of a guitar string is probably on the order of 10% or so. A good electronic oscillator might be 0.01% or even lower. It has harmonics but they are hard to see on a spectrum analyzer because they are so low. For 0.01% the harmonic levels are -80 dB or less below the fundamental and may not be visible unless the spectrum analyzer has a dynamic range of at least 80 dB. That being said, the existence of harmonics in the radio spectrum is a function of the quality of the oscillator producing the communication signal as well as any power amplifiers between the oscillator and the antenna. There is nothing inherent about having harmonics in the radio spectrum. Most communications devices use high quality oscillators and power amplifiers, and often have a bandpass filter just before the antenna, to minimize harmonics. FCC requirements specify the permissible harmonic levels depending on the frequency band, power level, type of modulation, and other characteristics. Typically harmonics are kept at least 40 dB below the fundamental which is equivalent to a power level ratio of 1 to 10,000.

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Guitar strings have harmonics because they are tied at both ends, leading to a countable (one for every positive integer) set of sinusoidal solutions. Because the highest-frequency vibrations attenuate the most quickly, no matter how you push or pull on the guitar string to get it started it will quickly end up in a vibration state that can be described almost entirely in terms of the first few normal modes.

When radio waves are tied at both ends (like in an antenna that is not infinitely long), you can find similar resonance phenomena. This is why the size of the antenna is often chosen based on the intended wavelength of reception. If a broad-band pulse were emitted near a radio antenna that wasn't driving any load, the resistance in the wire would quickly filter out the higher modes, leaving an analogue of the guitar string situation. However broadband pulses are avoided as much as possible and disconnected antennas are rare, so this situation is not often considered.

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