The numerical error associated with finite-difference simulation of wave propagation in discontinuous media consists of two components. The first component is a higher order error that leads to grid dispersion; it can be controlled by higher-order methods. The second component results from misalignment between numerical grids and material interfaces. We provide an explicit estimate of the interface misalignment error for the second order in time and space staggered finite-difference scheme applied to the acoustic wave equation. Our analysis, confirmed by numerical experiments, demonstrates that the interface error results in a first-order time shift proportional to the distance between the interface and computational grids. A two-dimensional experiment shows that the interface error cannot be suppressed by higher-order methods and indicates that our one-dimensional analysis gives a good prediction about the behavior of the numerical solution in higher dimensions.