I am given the wave equation in spherical coordinates for a wave function only depending on $r$: $\frac{1}{v^2} \frac{\partial^2 \xi(r,t)}{\partial t^2} = \left( \frac{\partial^2}{\partial r^2} + \frac{2}{r} \frac{\partial}{\partial r} \right) \xi(r,t).$ Also, I know that this equation can be solved using the ansatz $\xi(r,t) = A(r) f(kr \pm \omega t)$. Now I have to show that for any (twice differentiable) function $f$,
$A(r) = \frac{C}{r}$
with some constant $C$ must hold.
To prove this, I used the given ansatz and computed the partial derivatives of the wave function and plugged them into the wave equation. Doing this, I end up with the condition
A''(r) f(kr\pm\omega t) + A'(r)[2(f'(kr\pm \omega t)k + \frac{1}{r} f(kr\pm \omega t))] + A(r) \frac{2k}{r} f'(kr \pm \omega t) = 0,
where A', A'' and f' denote the derivatives of those functions.
My question is: How do I proceed?
Obviously, this equation holds for $f = 0$. If we have $f = m$ for some $m \in \mathbb{R} \setminus \{0\}$, the condition will simplify to the second order differential equation A''(r) + \frac{2 A'(r)}{r} = 0. Using substitution ($B(r) := A'(r)$), we find $A(r) = - \frac{c_1}{r} + c_2$ and since $A(r) \to 0$ as $r \to \infty$, we can conclude $c_2 = 0$.
But how can I similarly (?) show $A(r) = \frac{C}{r}$ for any function $f$?
Thank you very much in advance for any help.