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 numerical treatments of non-conforming grid interfaces and non-conforming mesh blocks. Interface conditions are imposed weakly by the simultaneous approximation term technique in combination with interface operators, which move discrete solutions between grids at an interface. In particular, we consider an interpolation approach and a projection approach with corresponding operators. A norm-compatible condition of the interface operators leads to energy stability for first order hyperbolic systems. By imposing an additional constraint on the interface operators, we derive an energy estimate of the numerical scheme for the second order wave equation. We carry out eigenvalue analyses to investigate the additional constraint and its relation to stability. In addition, a truncation error analysis is performed, and discussed in relation to convergence properties of the numerical schemes. In the numerical experiments, stability and accuracy properties of the numerical scheme are further explored, and the practical usefulness of non-conforming grid interfaces and mesh blocks is discussed in two practical examples. © 2016, Springer Science+Business Media New York.