For Dirichlet-periodic and double periodic boundary conditions, we prove the existence of solutions to aforced semilinear wave equation with large forcing terms not flat oncharacteristics. The nonlinearity is assumed to be non-monotone, asymptotically linear, and not resonanant.We prove that the solutions are in $L^{p}$, $(p\geq 2)$, when the forcing term is in $L^{p}$. This is optimal;even in the linear case there are $L^p$ forcing terms for which the solutions are only in $L^p$. Our resultsextend those in [9] where the forcing term is assumed to be in $L_{\infty}$, andare in contrast withthose in [6] where the non-existence of continuous solutions is established for $C^{\infty}$ forcing terms flat on characteristics. 200 words.