  We use high order finite difference methods to solve the wave equation in the
second order form. The spatial discretization is performed by finite difference
operators satisfying a summation-by-parts property. The focus of this work is
on the numerical treatment of non-conforming grid interfaces. The interface
conditions are imposed weakly by the simultaneous approximation term technique
in combination with interface operators, which move the discrete solutions
between the grids on the interface. In particular, we consider interpolation
operators and projection operators. A norm-compatibility condition, which leads
to stability for first order hyperbolic systems, does not suffice for second
order wave equations. An extra constraint on the interface operators must be
satisfied to derive an energy estimate for stability. We carry out eigenvalue
analyses to investigate the additional constraint and how it is related to
stability, and find that the projection operators have better stability
properties than the interpolation operators. In addition, a truncation error
analysis is performed to study the convergence property of the numerical
schemes. In the numerical experiments, the stability and accuracy properties of
the numerical schemes are further explored, and the practical usefulness of
non-conforming grid interfaces is presented and discussed in two efficiency
studies.
