  In this paper, we present and analyze an energy-conserving and linearly
implicit scheme for solving the nonlinear wave equations. Optimal error
estimates in time and superconvergent error estimates in space are established
without time-step dependent on the spatial mesh size. The key is to estimate
directly the solution bounds in the $H^2$-norm for both the nonlinear wave
equation and the corresponding fully discrete scheme, while the previous
investigations rely on the temporal-spatial error splitting approach. Numerical
examples are presented to confirm energy-conserving properties, unconditional
convergence, and optimal error estimates, respectively, of the proposed fully
discrete schemes.
