Showing $(\Delta-\frac{1}{c^2(x)}\partial^2_t)u(x,t)=0\leadsto(k^2V(x)-\Delta)\hat{u}(x,k)=k^2\hat{u}(x,k)$

Question

Let's start with the velocity wave equation

$$\left(\Delta-\frac{1}{c^2(x)}\frac{\partial^2}{\partial t^2}\right)u(x,t)=0.\qquad(\star)$$

Then Roger G. Newton claims that we can apply the Fourier transform with respect to the $t$ variable, $\mathscr{F}_t$, which yields

$$(k^2V(x)-\Delta)\hat{u}(x,k)=k^2\hat{u}(x,k),\qquad (\bigstar)$$ where $V(x)=1-1/c^2(x)$.

I cannot seem to show this, however, since applying $\mathscr{F}_t$ to $(\star)$ yields

$$\left(\Delta+\frac{1}{c^2(x)}k^2\right)\hat{u}(x,k)=0.$$

Then how would I introduce $V$ here to yield $(\bigstar)$?

Answer 1

$$ \begin{aligned} (\Delta + \frac{1}{c^2 (x)} k^2 ) )\hat{u} (x,k) &= 0\\ (\Delta + \frac{1}{c^2 (x)} k^2 + k^2 - k^2) )\hat{u} (x,k) &= 0\\ (\Delta + \frac{1}{c^2 (x)} k^2 - k^2 ) )\hat{u} (x,k) &= - k^2 \hat{u}(x,k)\\ (\Delta + k^2 ( \frac{1}{c^2 (x)} - 1) ) )\hat{u} (x,k) &= - k^2 \hat{u}(x,k)\\ (\Delta + k^2 (-V(x)) )\hat{u} (x,k) &= - k^2 \hat{u}(x,k)\\ (k^2 V(x) - \Delta )\hat{u} (x,k) &= k^2 \hat{u}(x,k) \end{aligned} $$