Numerical simulations of wave propagation produce different errors and the most well known is numerical dispersion, which is only valid for homogeneous media. However, there is a lack of error studies for heterogeneous media or even for the canonical case of media that have two constant velocity layers. The error associated with media that have two layers is called an interface error, and it typically converges to zero with a lower order of convergence compared to the theoretical convergence rate of the finite-difference schemes (FDS) for homogeneous media. We evaluated a detailed numerical study of the interface error for three staggered-grid FDS that are commonly used in the simulation of seismic-wave propagation. We determined that a standard staggered-grid scheme (SSGS) (also known as the Virieux scheme), a rotated staggered-grid scheme (RSGS), and a Lebedev scheme (LS) preserve the second order of convergence at horizontal/vertical solid-solid interfaces when the medium parameters have been properly modified, such as by harmonic averaging of finely layered media for the stiffness tensor and arithmetic mean for the density. However, for a fluid-solid interface aligned with the grid line, a second-order convergence can only be achieved by an SSGS. In addition, the presence of a fluid-solid interface reduces the order of convergence for the LS and the RSGS to a first order of convergence. The presence of inclined interfaces makes high-order (second and more) convergence impossible.