We consider the problem of multiple existence of 2 π -periodic weak solutions to wave equations □ u ( x , t ) = h ( x , t , u ( x , t ) ) + f ( x , t ) of space dimension 1, where h ( x , t , ξ ) is asymptotically linear in ξ both as ξ → 0 and as | ξ | → ∞ . It is shown by variational methods that there exist at least three solutions under several conditions on h ( x , t , ξ ) if f is sufficiently small. One of the results reads as follows. Let b ≔ lim | ξ | → ∞ ∂ h / ∂ ξ ( x , t , ξ ) and assume that the convergence is uniform with respect to ( x , t ) and that b ∉ σ ( □ ) (non-resonant case). Then the following conditions guarantee the existence of at least three solutions for sufficiently small f: (a) h ( x , t , ξ ) - ξ ∂ h / ∂ ξ ( x , t , 0 ) is non-decreasing (resp. non-increasing) in ξ , and sup ( x , t , ξ ) ∂ h ∂ ξ ( x , t , ξ ) < min { λ ∈ σ ( □ ) ; b < λ } resp . inf ( x , t , ξ ) ∂ h ∂ ξ ( x , t , ξ ) > max { λ ∈ σ ( □ ) ; b > λ } . To obtain these results, we prove that a C 1 -class functional, which is bounded from above and satisfies a condition similar to the linking condition, has at least three critical points under ( WPS ) * condition.