Fourier transform of homogeneous wave equation

Question

I have the following PDE $u_{tt} - \Delta_xu = 0, x \in \mathbb{R}_n, t>0$. I suppose $\Delta_{(x_1,...,x_n)}u \equiv u_{x_1x_1} + ... + u_{x_nx_n}$, because it wasn't defined in the textbook. The purpose is to study wave equation. Now, both sides are multiplied by $e^{-i(x,\xi)}, \xi = (\xi_1, \xi_2, \xi_3) \in \mathbb{R}_3$ and the equation is integrated by $x$ over $\mathbb{R}_3$. The textbook then, states that from here we obtain $\tilde{u}_{tt} + |\xi|^2 \tilde{u}(\xi, t) = 0$, where $\tilde{u}(\xi, t) = \int_{\mathbb{R}_3} {e^{-i(x,\xi)}u(x,t)dx}$ is the Fourier transformation of $u(x,t)$. Here is what I don't understand. How was $|\xi|^2 \tilde{u}(\xi, t)$ obtained? where did it come from? I know I am missing something basic, but I can't understand.

Answer 1

Hint: When you integrate $e^{-i(x,\xi)} u_{x_j,x_j}$ over $x_j$ ($j=1,2,3$), you can integrate by part twice, and each time you get a factor of $i\xi_j$ for each coordinate, which gives you $-\xi_j^2$. Adding them up gives you what you want.