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I'm sure this is a trivial question for someone who knows something about electromagnetic radiation, but: how do experimenters measure the wavelength/frequency of light? For example, how do we know that red light has $650-700~\text{nm}$ wavelength?

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The earliest accurate determination of wavelength was, I think, by Michelson. Using his invention, the Michelson Interferometer, he could turn a micrometer dial and actually count how many wavelengths he moved a mirror. Reasonable monochromatic light could be had at the time from mercury vapor (or other elemental) discharge tubes or from a monochromator (a spectroscope with a slit on the output to select a color). This was around 1880. I confess I don't know for sure. He was determined to measure the speed of light. Exactly when he worked on wavelength I don't know. I'm sure someone here can add that info.

http://physical-optics.blogspot.com/2011/06/michelsons-interferometer.html

Michelson was able to count a lot of wavelengths so that the mirror moved enough to get a good average from the mechanical measurement. He was able to measure the wavelength of precisely known colors so that the results were easily reproduced by others. At the time there was a lot of interest in the spectra of excited atoms of elements and of the sun and stars through the new medium of photography. Photographic spectra of a star was done first in 1863.

Once you have a wavelength and the speed, which Michelson also determined to a high degree of accuracy by refining the the rotating mirror method, the frequency is just f=velocity/wavelength. The frequencies are crazy big numbers like the red in a helium-neon laser is 4.7376 x 10^14 per second or 473.76 THz. That's tera-Hertz and it is nice that tera- is also trillion. This is why people use wavelength in nanometers, so that the red from the laser is described as 632.8nm, which is a lot easier. If you read older material you will see that we used a slightly more convenient measure, the Angstrom, which is 1/10 a nanometer. The same light is 6328 $\overset{\circ}{A} $. The Angstrom is abbreviated as a capital 'A' with a little dot or circle over it. (It is in the UTF8 character set but I'm not sure will render for everyone, so I faked it in LaTeX.)

I think I got that frequency calculation right. By the way, it is accepted to use a Greek lambda $\lambda $ for wavelength and nu $\nu $ for frequency. Then $velocity\; =\; \lambda \nu $.

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For a rough measurement, you could set up essentially any experiment whose results depend on the wavelength. For example, reflect a beam off of a diffraction grating, and measure the angle of reflection. This essentially means building a type of spectrometer.

An instrument that measures the wavelength of a near-monochromatic beam very precisely is called a wavemeter. Wavemeters can be built on several different principles, but common ones include the Michelson inteferometer and the Fizeau interferometer.

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