  In this paper a novel method to solve the constant coefficient wave equation,
subject to interface jump conditions, is presented. In general, such problems
pose issues for standard finite difference solvers, as the inherent
discontinuity in the solution results in erroneous derivative information
wherever the stencils straddle the given interface. Here, however, the recently
proposed Correction Function Method (CFM) is used, in which correction terms
are computed from the interface conditions, and added to affected nodes to
compensate for the discontinuity. In contrast to existing methods, these
corrections are not simply defined at affected nodes, but rather generalized to
a continuous function within a small region surrounding the interface. As a
result, the correction function may be defined in terms of its own governing
partial differential equation (PDE) which may be solved, in principle, to
arbitrary order of accuracy. The resulting scheme is not only arbitrarily high
order, but also robust, having already seen application to Poisson problems and
the heat equation. By extending the CFM to this new class of PDEs, the
treatment of wave interface discontinuities in homogeneous media becomes
possible. This allows, for example, for the straightforward treatment of
infinitesimal source terms and sharp boundaries, free of staircasing errors.
Additionally, new modifications to the CFM are derived, allowing compatibility
with explicit multi-step methods, such as Runge-Kutta (RK4), without a
reduction in accuracy. These results are then verified through numerous
numerical experiments in one and two spatial dimensions.
