  We prove the existence of small amplitude periodic solutions, with strongly
irrational frequency $ \om $ close to one, for completely resonant nonlinear
wave equations. We provide multiplicity results for both monotone and
nonmonotone nonlinearities. For $ \om $ close to one we prove the existence of
a large number $ N_\om $ of $ 2 \pi \slash \om $-periodic in time solutions $
u_1, ..., u_n, ..., u_N $: $ N_\om \to + \infty $ as $ \om \to 1 $. The minimal
period of the $n$-th solution $u_n $ is proved to be $2 \pi \slash n \om $. The
proofs are based on a Lyapunov-Schmidt reduction and variational arguments.
