In this note we consider semidiscrete and multistep fully discrete Galerkin approximations (in the space of smooth periodic splines $S^\mu ,\mu \geqq 2$, on a uniform mesh with mesh length h) to the solution of the initial-periodic boundary value problem for a second-order hyperbolic equation with space-varying coefficients. We show that a suitable choice of the initial conditions for the Galerkin equations leads to increased accuracy (superconvergence) of the approximate solution which is $O(h^{2\mu - 2} )$-accurate when compared with a certain quasiinterpolant of the exact solution. We also analyze the effect of numerical integration on the superconvergence of the error estimates. We present the results of numerical experiments with cubic splines in one dimension and bicubic splines on squares coupled with the Stormer-Numerov time-differencing method.