In this paper we consider the nonlinear wave equation problem $$displaylines{ u_{tt}-Big(|u|_0^2,|u_{r}|_0^2ig)(u_{rr}+frac{1}{r}u_{r}) =f(r,t,u,u_{r}),quad 0less than r less than 1,; 0 less than t less than T, ig|lim_{ro 0^+}sqrt{r}u_{r}(r,t)ig| less than infty, u_{r}(1,t)+hu(1,t)=0, u(r,0)=widetilde{u}_0(r), u_{t}(r,0)=widetilde{u}_1(r). }$$ To this problem, we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved, in weighted Sobolev using standard compactness arguments. In the latter part, we give sufficient conditions for quadratic convergence to the solution of the original problem, for an autonomous right-hand side independent on $u_{r}$ and a coefficient function $B$ of the form $B=B(|u|_0^2)=b_0+|u|_0^2$ with $b_0$ greater than 0.