  In this paper, we consider finite difference approximations of the second
order wave equation. We use finite difference operators satisfying the
summation-by-parts property to discretize the equation in space. Boundary
conditions and grid interface conditions are imposed by the
simultaneous-approximation-term technique. Typically, the truncation error is
larger at the grid points near a boundary or grid interface than that in the
interior. Normal mode analysis can be used to analyze how the large truncation
error affects the convergence rate of the underlying stable numerical scheme.
If the semi-discretized equation satisfies a determinant condition, two orders
are gained from the large truncation error. However, many interesting second
order equations do not satisfy the determinant condition. We then carefully
analyze the solution of the boundary system to derive a sharp estimate for the
error in the solution and acquire the gain in convergence rate. The result
shows that stability does not automatically yield a gain of two orders in
convergence rate. The accuracy analysis is verified by numerical experiments.
