When people talk about harmonic oscillators it seems to me that they always assume either that the relationship of force and displacement is linear, or that it behaves in some sinusoidal fashion. Do you always have to assume one to be able to arrive at the other? Or is there something I'm missing? I feel like the starting point is always assuming something based on experiments; is that correct? Or is it just that the actual derivation of harmonic oscillators formulas are beyond the scope of the lectures I've been taking?