Showing Kirchhoff's Formula Indeed Solves the 3D Wave Equation

Question

I am currently trying to show that the Kirchhoff's Formula indeed solves the 3d Wave Equation, but am having trouble proving this. The formula is given by: $$u(\textbf x_0,t) = \frac{1}{4\pi c^2t^2}\iint_{\partial B(\textbf x_0,ct)}\phi(\textbf x)+\nabla\phi(\textbf x)(\textbf x-\textbf x_0)+t\psi(\textbf x)dS_{\textbf x}$$ Where the bolden variables denote vectors. Now, I tried taking the Laplacian in terms of $\textbf x_0$: $$\Delta_{\textbf x_0}u(\textbf x_0,t) = \frac{1}{4\pi c^2t^2}\iint_{\partial B(\textbf x_0,ct)}\nabla\phi(\textbf x)\Delta_{\textbf x_0}(\textbf x-\textbf x_0)dS_{\textbf x}$$ But taking two time derivatives with respect to $t$ is confusing me as it s also the radius of the sphere of which we are integrating over. How do I find $u_{tt}$, and how does that actually equal the Laplacian of $u$? Right now the Laplacian makes no sense to me, and I don't even recognize how taking 2 $t$-derivatives would yield the same result.