The wave equation in $\mathbb{R}^3$ , ie $u_{tt}-\Delta u=0 $ for $x\in \mathbb{R}^3, t>0 $
$u=g, u_t=h$ for $x\in \mathbb{R}^3, t=0$
Define an average: $U(x,r,t)= \frac {1}{|\partial B|} \int_{\partial B(x,r) }u(y,t)dS_y $ and similarly $G(x,r,t) =\frac {1}{|\partial B|} \int_{\partial B(x,r) }g(y)dS_y$
and
$H(x,r,t) =\frac {1}{|\partial B|} \int_{\partial B(x,r) }h(y)dS_y$
We fix $x \in R^n , n\ge2$ and suppose $u \in C^m(R^m\times \mathbb{R}_+$ for $m\ge2$
Claim : $U$ solves the following initial value problem . $U_{tt}-U_{rr}-\frac {(n-1)}{r} U_r=0, r>0, t>0$
$U=G, U_t=H , r>0, t=0$
Proof : $U_r =\frac {\partial}{\partial r}\frac {1}{|\partial B|} \int_{\partial B(x,r) }u(y,t)dS_y $. First derivative was easy for me to find but now My aim is to find $U_{rr}$ . There is another way of proving the claim without computation of second derivative . But i am looking forward to know how to differentiate it twice and many more times .
Any help will be appreciated.