The propagation of sound waves produced by a moving sphere of radius $a$ in an uniform, infinite fluid, is given by the following equation: $$ r^2 \frac{\partial^2{u}}{\partial t^2} = c^2 \frac{\partial}{\partial r} \left (r^2 \frac{\partial u}{\partial r} \right) \qquad \qquad (1) $$
Where $t$ represents time, $r$ the distance from the center of the sphere and $c$ the wave propagation speed. The following conditions are known:
$$ u(r,0) = \frac{\partial u(r,0)}{\partial t} = 0 \qquad \qquad (2)$$ and $$\frac{\partial u(a,t)}{\partial r} = f(t) \qquad \qquad (3)$$ and its also known that $u\to 0$ when $r\to \infty$.
The first step of the solution is to consider a variable $v = ru$ and, from that, rewrite the first equation as
$$ \frac{\partial^2 v}{\partial t^2} = c^2 \frac{\partial^2 v}{\partial r^2} \qquad \qquad (4) $$
then using Laplace's Transform...
I could not see how this rewriting of the first equation is valid because if $v=ru$ and assuming $$\frac{\partial r}{\partial t} = \frac{\partial t}{\partial r} = 0 \qquad \qquad (5)$$
then
$$\frac{\partial v^2}{\partial t^2} = r \frac{\partial u^2}{\partial t^2} \qquad \qquad (6)$$
and
$$c^2 \frac{\partial v^2}{\partial r^2} = c^2 \frac{\partial}{\partial r} \left( r \frac{\partial u}{\partial r} + u \right) = c^2 \left( r \frac{\partial^2 u}{\partial r^2} + 2 \frac{\partial u}{\partial r} \right ) \qquad \qquad (7)$$
I tried to consider $v = r^2u$ and got:
$$\frac{\partial v^2}{\partial t^2} = r^2 \frac{\partial u^2}{\partial t^2} \qquad \qquad (8)$$
and
$$c^2 \frac{\partial v^2}{\partial r^2} = c^2 \frac{\partial}{\partial r} \left( r^2 \frac{\partial u}{\partial r} + 2ru \right) = c^2 \left( r^2 \frac{\partial^2 u}{\partial r^2} + 2r + 2u + 2r \frac{\partial u}{\partial r} \right ) \qquad \qquad (9)$$
The equality of the first and fourth equations does not hold yet. Am I missing something?