I am starting to learn about the complex exponential form of waves and SHM and was wondering what the significance of the imaginary part is. I know this question has been asked many times in different ways, but I think my question here will be slightly different.
From what I have seen, we use the complex representation because of its convenience as many calculations are simplified (like integration, differentiation, wave superposition and enveloping etc) but we have to extract the real part to get the physical reporesentation. The complex part is useful for this also because it consists of the real and imaginary part which are orthogogonal and so when we apply linear operators such as derivatives we can take the real part before or after applying these. This is not so for non[linear operators, such as getting the power/energy/intensity of a wvae which all depend on the square of the amplitude; the real part must be taken before squaring.
I also know that there is some significance of the imaginary part in quantum mechanics as probabilities are found by multiplying the wavefunction with its complex conjugate, which does not give the same result as just squaring the real part.
So it seems to me that the complex representation can just encode extra information in a convenient way. From what I have read I have not seen there being any extra phsyical interpretation of the imaginary part. I was wondering if someone could clarify or add to this? Parhaps I am also missing something- some physical phenomena where the complex nature of waves becomes apparent and physical results cannot be explained simply from a real representation of waves (whether in quantum mechanics or not)?