Consider a solution to the wave equation $ \psi\left(x,t\right) $, then using Fourier transform, we can represent:
$ \psi\left(x,t\right)=\left(\frac{1}{2\pi}\right)^{2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\widetilde{\psi}\left(k,\omega\right)e^{i\left(kx+wt\right)}dkdw $
Now if we'll apply this form into the wave equation $ \frac{\partial^{2}\psi}{\partial x^{2}}-\frac{1}{c^{2}}\frac{\partial^{2}\psi}{\partial t^{2}}=0 $
We'll get:
$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\widetilde{\psi}\left(k,\omega\right)e^{i\left(kx+wt\right)}\left(-k^{2}+\frac{\omega^{2}}{c^{2}}\right)dkdw=0 $
Now according to my book, this obligates the term $ \left(-k^{2}+\frac{\omega^{2}}{c^{2}}\right) $ to be $0$.
I really cant understand why. My knowledge in Fourier transform is very low, we've just learned maybe $ 2 $ hours just getting familier with the equations and applying it to some basic physics exercise. But I want to understand in a more profound way.
As I understand (with my basic knowledge of just year of math learnings), taking a fourier transform is equvivalent to representing a vector in a vector space using orthogonal basis. Which means $ e^{i\left(kx+wt\right)} $ those are forming the orthogonal vector basis and the inner product is probably the integrals $ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} d\omega dk$.
So why does $ \left(-k^{2}+\frac{\omega^{2}}{c^{2}}\right) $ has to be $ 0 $ in order for the equation to make sense? as I see it maybe the term $ \widetilde{\psi}\left(k,\omega\right) $ can also cause everything to be $ 0 $.
If someone can explain in a simple way which dosent require profound understanding of the math behind the scenes, I'll be greatful. Thanks
