  We prove existence of infinitely many classical periodic solutions with
periodic boundary conditions for a class of monotone semilinear wave equations.
Our argument relies on some new estimates for the linear problem with periodic
boundary conditions by combining Littewood-Paley techniques, the
Hausdorff-Young theorem and a variational formulation due to Rabinowitz. We
also develop a new approach to the regularity of distributional solutions by
differentiating the equations and employing Gagliardo-Nirenberg estimates.
