Can a prism separate two different waveforms with the same spectrum?

[stringbean]

Suppose you have an arbitrary waveform made up of different frequencies of light. In books the different frequencies that add together to make the arbitrary waveform start and stop at the same place as all the other frequencies, say at 0 and 2 pi. Well their wavelengths divide 2 pi evenly if you know what I mean. I just wanted to ask if it matters if they start and stop together. Will you get a different waveform if you use the same composition of frequencies with them offset from each other in some different arbitrary way. This kind of confuses me because if it depends on whether they are initially in phase with each other, you could expect two different specific waveforms of light to pass through a prism and separate into the same spectrum except that two bands of light from the two different waveforms could have the same frequency but would be out of phase by some amount. Is that right? So two waveforms can seem to have the same spectrum when passed through a prism and yet be completely different? What really confuses me is if an arbitrary waveform can be represented by two different compositions of frequencies they would separate differently by a prism, but that is wrong, correct? An arbitrary waveform is composed of only one unique set of frequencies. So if you shifted the arbitrary waveform over by some amount and did harmonic analysis on it, would it still be the same set of frequencies just shifted over themselves or could these shifted frequencies themselves be composed of another barrage of frequencies? I'm guessing they can't because then they would separate differently by a prism. Help please?

 

[mikeph]

A prism only separates different wavelengths, not wavelengths of different phases.

And if you shift the phase of a single mode within the field you will get a different field. Consider for example sin(x) + sin(2x) =/= sin(x) + sin(2x+1)! The phase of the second is shifted by 1 and it is certainly not producing the same amplitude.

 

[stringbean]

I'm really having trouble explaining what I mean, but I've got another question that might help you figure it out. How do you express the function sin(x + 1) by a Fourier series of the form A_n sin(nx) + B_n cos(nx). I mean expressing sin(x) this way would be easy, but what about sin(x + 1). Would there be many frequencies adding together. If so would a prism break the light up into the many frequencies or would it just bend the one frequency of light. My guess is that it only bends the light according to just the one frequency and that the different colors from a prism come from different frequencies of light that come through the prism at different times rather than the light being literally split up into different frequencies all at once. I would suppose that you can only see the colors from a prism all at once because light travels so fast and there is a variety of signals during that time. Maybe everyone else already thinks about it this way, but I've been thinking about it the other way for a long time. If true, it's good I finally got that through my head. Tell me if I'm wrong okay.