In this paper, we provide the optimal convergence rate of a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the second-order wave equation in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [W. Cao, D. Li and Z. Zhang, Commun. Comput. Phys. 21 (1) (2017) 211-236]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the $L^2$-norm to $(p+1)$-degree right and left Radau interpolating polynomials under mesh refinement. The order of convergence is proved to be $p+2$, when piecewise polynomials of degree at most $p$ are used. We use these results to show that the leading error terms on each element for the solution and its derivative are proportional to $(p+1)$-degree right and left Radau polynomials. These new results enable us to construct residual-based a posteriori error estimates of the spatial errors. We further prove that, for smooth solutions, these a posteriori LDG error estimates converge, at a fixed time, to the true spatial errors in the $L^2$-norm at $\mathcal{O}(h^{p+2})$ rate. Finally, we show that the global effectivity indices in the $L^2$-norm converge to unity at $\mathcal{O}(h)$ rate. The current results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be $p+3/2$ and $1/2$, respectively. Our proofs are valid for arbitrary regular meshes using $P^p$ polynomials with $p\geq1$. Several numerical experiments are performed to validate the theoretical results.