We investigate the smoothness properties of the solutions of one-dimensional wave equations with nonlinear random forcing. We define singularities as anomalies in the local modulus of continuity of the solutions. We prove the existence of such singularities and their propagation along the characteristic curves. When the space variable is restricted to a bounded interval, we impose the Dirichlet boundary condition at the endpoints and we show how the singularities are reflected at the boundary.
Rene Carmona. David Nualart. "Random Nonlinear Wave Equations: Propagation of Singularities." Ann. Probab. 16 (2) 730 - 751, April, 1988. https://doi.org/10.1214/aop/1176991784