We have just started learning small oscillations at grad level and our professor pulled out the technique of Fourier Analysis to illustrate its power by applying it to the case of a simple harmonic oscillator.
The differential equation is: $\ddot x+w_o^2x=0$
At this point he said, using the definition of fourier transform i.e. $$x(t)=\int_{-\infty}^{+\infty}\tilde x(w)\exp(iwt)\ dw$$
we get: $$(-w^2+w_0^2)\tilde x(w)=0$$ or $\tilde{x}(w)=A\delta(w-w_o)+B\delta(w+w_o)$
My problems:
- What does $\tilde x(w)$ mean physically?
- I have looked at other answers from the site and I can't understand the meaning of a negative $w$. It would be great if this doubt is cleared as well.
Any help is appreciated.