Solve the following PDE using Fourier transform Solve the following 3-D wave equation using Fourier transform $$PDE: u_{tt}=C^2[u_{xx}+u_{yy}+u_{zz}],\qquad-\infty<x,y,z<\infty,\qquad t>0$$ $$BC: u(x,y,z,t)\rightarrow 0\qquad as \qquad r^2=x^2+y^2+z^2\rightarrow \infty \qquad $$ $$IC: u(x,y,z,0)=f(r),\qquad  r=\sqrt{x^2+y^2+z^2} , \qquad u_t(x,y,z,0)=0 $$

Just take 3 dimensional Fourier transform on the equation, and then it will be solved. Let $\hat{u}$ be the 3-D Fourier transform of $u$ and the variables $x,y,z$ will transform to $s_1,s_2,s_3$. Take Fourier on the original equation, we have
$$
\hat{u}_{tt}=-C^2(s_1^2+s_2^2+s_3^2)\hat{u}\\
\hat{u}(s_1,s_2,s_3,0)=\hat{f}\\
\hat{u}_t(s_1,s_2,s_3,0)=0
$$
This is a 2nd order ODE. It can be solved by classical methods (without lose of generality we assume $C\ge0$):
$$
\hat{u}=\hat{f}\cos(Ct\sqrt{s_1^2+s_2^2+s_3^2})
$$
Then, take inverse Fourier transform on it, you can obtain the solution of original PDE. I omit it here because it is trivial.
PS: Note that when you take inverse Fourier transform, the zero boundary condition will be used.
