Fourier expansion and transform - what about the phase of the waves that i am adding?

Question

Say we have a wave on the surface of the water and we want to describe it as a sum of other waves. So we use Fourier expansion to add waves of different wavelengths. For simplicity, say we have to just add two waves in order to get the mathematical description of the wave we have. If we add the two waves, the resulting wave will largely depend on the phase difference of the waves we added. For example, if the two waves are sinusoidal , if they have zero phase difference they will add up to a different wave than if they had a phase difference of π
So, how do we consider the phase of each wave that we add up in the Fourier expansion and Fourier transform?

Answer 1

In Fourier analysis, we mostly do not use $\cos$ and $\sin$ functions, but instead use $e^{i\xi x}$.

The two approaches are equivalent, but with the complex exponentials, we have a huge advantage: If we change the phase (I.e. if we translate), we get $e^{i\xi (x+x_0)}=e^{i \xi x_0} e^{i \xi x}$, i.e. translation/change of phase corresponds to multiplying with a complex number of modulus 1 (a phase factor). Thus, the different phases can be encoded in the different coefficients of the individual "waves".

This behaviour is one of the main points of Fourier analysis, since it implies that translation invariant operators (convolution, differentiation) are diagonalised by the Fourier transform.