I have a question about my approach to another question asked here:
$$u_{tt}=C^2[u_{xx}+u_{yy}+u_{zz}],\qquad-\infty<x,y,z<\infty,\qquad t>0$$ $$BC: u(x,y,z,t)\rightarrow 0\qquad as \qquad r^2=x^2+y^2+z^2\rightarrow \infty \qquad $$ $$IC: u(x,y,z,0)=f(r),\qquad r=\sqrt{x^2+y^2+z^2} , \qquad u_t(x,y,z,0)=0 $$
Since there is radial symmetry I use the radial component of the laplacian:
$$\frac{\partial^2u}{\partial t^2}=c^2\frac{1}{r^2}\frac{\partial}{\partial r}\bigg(r^2\frac{\partial u}{\partial r}\bigg)$$
I laplace transform the time. Also $U'=\frac{\partial U}{\partial r}$
$$s^2U-s f(r)=c^2U''+\frac{2c^2}{r}U'$$
$$U''+\frac{2}{r}U'-\frac{s^2}{c^2}U=-sf(r)$$
The homogeneous solution (because I can't solve for a function I do not know) to this equation is
$${exp\bigg({\pm\frac{s}{c}r}}\bigg)\rightarrow \frac{\delta(t\pm \frac rc)}{r}$$
Which has some resemblance of a correct solution (time convolution with any function takes the correct form) because the general solution should look like
$$\frac{f(t\pm \frac rc)}{r}$$
What am I missing, which step am I skipping, doing wrong?