Does a finite wave necessarily have to be non-monochromatic in reality, or is that implication just a result of the mathematical analysis? I always wonder at these sort of things that come out of a strictly mathematical analysis.
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Yes, it happens in reality too, nicely demonstrating that the Fourier analysis predictions are confirmed. An easy way to see this is to take an electrical sine wave signal, which is nice and monochromatic, and pulse it on and off. If you examine the spectrum of the pulsed wave on a spectrum analyser, you will see the spread of frequencies about the centre sine wave frequency.
Of course if you don't trust a spectrum analyzer because you suspect it's actually secretly doing an internal Fourier transform, you can directly measure the new frequencies created by the pulsing by applying narrow bandpass filters to the pulsed wave and measuring the output power of the filters with and without the pulsing.
twistor59 is right, and here's another way to demonstrate it. You need a tone generator that can generate multiple sine waves of close-together frequencies and add them together. Or, you can do it with a simple program, or even just pencil and graph paper.
A single sine wave is just that. It goes on "forever" with constant power.
Two sine waves close together produce a beat frequency (the "wow-wow" effect) where the power is highest when the two waves are in sync, and least when they are out of sync.
Three sine waves close together produce a pattern where the point of lowest power is spread out between the peaks, because the points where all three waves are in sync come farther apart.
The more sine waves close together in frequency that you add together, the farther apart come the points where they are all in sync.
If you add together an infinite number of sine waves, each differing from the next by a small increment of frequency, there is only one peak where they are all in sync, and everywhere else they cancel out. This is because the distance between peaks grows to infinity.
That illustrates how any waveform that is localized to a single segment of the X-axis, if it is expressed as a sum of sine waves, has to contain an infinite number of them.