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Combining multiple arbitrarily chosen frequencies of sound makes a complicated wave, not a new sine wave. Doesn't light also do this? If I add a 650nm (reddish) wave and a 550nm (greenish) wave, I get a complicated wave. If I shine this light on a prism, presumably it would resolve into red and green, I guess because the two wavelengths have different refractive indexes. But neither of those wavelengths is going into the prism. Or are they? Does nature do some kind of Fourier transform to white (or multiple-mixed-frequency) light? Or does white/mixed light somehow retain the component frequencies?

There are a lot of questions out here that are closely related to mine, but none of them seem to ask this particular question, and none of the answers shed any...light...on the problem, perhaps only because I lack the needed physics and mathematics background.

A very closely related question, maybe even a duplicate, but asked very differently.

SaganRitual
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If I add a 650nm (reddish) wave and a 550nm (greenish) wave, I get a complicated wave. If I shine this light on a prism, presumably it would resolve into red and green, I guess because the two wavelengths have different refractive indexes. But neither of those wavelengths is going into the prism. Or are they?

They are. Your "complicated wave" is still made up of red components and green components. No indigo or yellow or blue components are created when you spatially overlap a beam of red light with a beam of green light.

Or does white/mixed light somehow retain the component frequencies?

Yes. White light is just a mix of light of different colors that happen to fall on the same spot. That might be because they came from the same source, but it might just be because three or more monochromatic beams were shone on the same spot (this situation wouldn't actually be "white" as physicists define it, but it could be arranged to appear white to your eye).

The Photon
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Your assumed difference between behaviour of EM waves and sound waves is not real. Sound waves experience refraction at the interface of two media and if at least one of the two media is dispersive for the acoustic waves you will have a separation of frequencies. However, whereas common media like glass and water are dispersive enough in the visible range to make the effect visible, most comon media for sound propagation, including air, water, solid metals, show very small dispersion for audible acoustic waves. But the dispersion (so, sepration of the frequencies) can be found and measured in some media, including bones.

Also, to actually see the waveform of a complex sound is enough to use your computer or phone, with a simple microphone and a free app. To see the same for visible light of mixed frqencies you need a very rapid and wide band detector and an oscilloscope working in the hundreds of terahertz range.

nasu
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Sine waves are eigenfunctions of the basic wave equation. When we do "Fourier analysis" of a wavefunction, what we're really doing is diagonalizing the wave equation operator. The set of sine and cosine waves is just a basis for wavefunctions. The math of applying an operator is simpler if the wavefunction is given in terms of a basis that consists of the eigenfunctions of the operator, but the final result is the same regardless of how the calculation is performed. Fourier analysis is just a method for doing the calculation; nature just finds the right answer, it doesn't have a specific computational algorithm. The wavefunction is the sum of the two frequencies, so in that sense it does "retain" those components.

Acccumulation
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Photons don't ever mix. They maintain their own frequencies. The mixing is when your brain interprets a simultaneous combination of frequencies. This video https://www.youtube.com/watch?v=Y3HXR2OYEqo shows how separate colors appear to mix and make white but the photons of different frequencies travel independently and only become a perceived mixture when they hit your eyes.

Bill Alsept
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