  We analyze two types of summation-by-parts finite difference operators for
approximating the second derivative with variable coefficient. The first type
uses ghost points, while the second type does not use any ghost points. A
previously unexplored relation between the two types of summation-by-parts
operators is investigated. By combining them we develop a new fourth order
accurate finite difference discretization with hanging nodes on the mesh
refinement interface. We take the model problem as the two-dimensional acoustic
wave equation in second order form in terms of acoustic pressure, and prove
energy stability for the proposed method. Compared to previous approaches using
ghost points, the proposed method leads to a smaller system of linear equations
that needs to be solved for the ghost point values. Another attractive feature
of the proposed method is that the explicit time step does not need to be
reduced relative to the corresponding periodic problem. Numerical experiments,
both for smoothly varying and discontinuous material properties, demonstrate
that the proposed method converges to fourth order accuracy. A detailed
comparison of the accuracy and the time-step restriction with the
simultaneous-approximation-term penalty method is also presented.
