We consider the periodic solutions   of a semi-linear variable coefficient wave equation arising from the forced   vibrations of a nonhomogeneous string and the propagation of seismic waves in   non-isotropic media. The variable coefficient characterizes the inhomogeneity   of media and its presence usually leads to the destruction of the compactness   of the inverse of the linear wave operator with periodic-Dirichlet boundary   conditions on its range. In the pioneering work of Barbu and Pavel (1997),   they gave the existence and regularity of the periodic solution for   Lipschitz, non-resonant and monotone nonlinearity under the strong convexity   assumption on the coefficient. In this work, by developing the invariant   subspace method and using the complete reduction technique and Leray-Schauder   theory, we obtain the existence of periodic solutions for such a problem when   the nonlinear term satisfies the asymptotic non-resonance conditions. Our   result needs neither requirements on the coefficient except the natural   positivity assumption nor the monotonicity assumption on the nonlinearity.