Consider the second order wave equation $u_{tt}=u_{rr}+1/ru_{r}+1/r^2u_{\theta\theta}$ with the boundary data
- $u(r, \theta, 0)=f(r)$
- $u_t(r, \theta, 0)=0$
- $u(1, \theta, t)=0.$ Assuming that $u(r, \theta, t)=R(r)T(t)$ (by radial symmetry), we can conclude that
$$T''/T=(R''+1/rR')/R=c.$$ My question is about determining the sign of $c$. Wikipedia says that since we expect $T$ to be periodic rather than decaying or growing exponentially, $c$ has to be a negative number. However I am trying to understand the following alternative argument given in Knapp's advanced analysis.
"The indicial equation is $s^2 = 0$. One solution is given by a power series in $r$, while another involves $\log r$. We discard the solution with the logarithm because it would represent a singularity at the middle of the drum. To get at the sign of $c$, we use the condition $R(1) = 0$ and argue as follows: Without loss of generality, $R(0)$ is positive. Suppose $c > 0$, and let $r_1 \le 1$ be the first value of $r > 0$ where $R(r_1 ) = 0$. From the equation $r^{−1} (r R' )' = c R$ and the inequality $R(r) > 0$ for $0 < r < r_1$ , we see that $r R'$ is strictly increasing for $0 < r < r_1$. Examining the power series expansion for $R(r)$, we see that $R' (0) = 0$. Thus $R' (r ) > 0$ for $0 < r < r_1$. But $R(0) > 0$ and $R(r_1 ) = 0$ imply, by the Mean Value Theorem, that $R' (r)< 0$ somewhere in between, and we have a contradiction."
I have not been able to understand the part where he says "Examining the power series expansion for $R(r)$, we see that $R' (0) = 0$." Can someone please explain why $R'(0)$ has to be zero?
