# WKB approximation for a particle on a ring $(E>V)$

You don't use $$\psi(s) = \psi(s + n L)$$ for $$n$$ integer, because the ring always has length $$L$$, its length has nothing to do with $$n$$. Instead you impose the condition $$\psi(s) = \psi(s + L)$$, which implies that $$\phi(L) - \phi(0) = 2 \pi n$$ which $$n$$ is the energy level. Then you get $$n h = \int_0^L \sqrt{2 m (E_n - V(x))} \, dx$$ which is a typical WKB quantization integral, from which you compute the $$E_n$$ in the usual way.