# Fourier optics question

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?