7. Spherical waves Prove that, among all possible dimensions, only in three dimensions can one have distortionless spherical wave propagation with attenuation. This means the following. A spherical wave in $n$-dimensional space satisfies the PDE $$ u_{t t}=c^{2}\left(u_{r r}+\frac{n-1}{r} u_{r} ight), $$ where $r$ is the distance from the origin. Consider solutions of the special form $$ ucr, t)=\alpha(r) f(t-\beta(r)), $$ where $\alpha(o)$ is called the attenuation and $\beta(r)$ the delay. The question is whether such solutions exist for general functions $f$. i. Plug the special form into the PDE to get an ODE for $f$. ii. Set the coefficients of $f, f^{\prime]$ and $f^{\prime \prime} $ equal to zero. iii. Solve the ODEs to see that $n=1$ or $n=3$ (unless $u \equiv 0$). iv. If $n=1$, show that $\alpha(r$ is a constant (so there is no attenuation).cs.SD. 111||