[Math] Fourier transform of the wave equation

Question

On the LHS side of the highlighted expression should it not read:

$\displaystyle \frac {d^2 \hat{u}}{dt^2}$

as the Leibniz Integral rule requires you to transform the partial derivative to a straight derivative when you move it outside the integral?

Answer 1

As you observed in your comment, $\omega$ and $t$ are independent.

The Fourier transform $u \mapsto \hat{u}$ is performed at every fixed $t$ as a transformation on the variable $x$, and since $x$ and $t$ are independent, so is $\omega$ and $t$.