Maths of chords

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I have a few naive questions on music theory. Let us assume that I have two pitches A and C with certain frequencies. Then the corresponding sound waves are pure sinusoidal waves. But what happens if I vibrate the chord A-C? To get the resulting wave, I should simply sum up mathematically the corresponding sin waves? Conversely, if someone gives me a wave, how can I say that is, for instance, an A-C chord?

Any remarks welcomed...