  In this paper we give a survey on various multiscale methods for the
numerical solution of second order hyperbolic equations in highly heterogeneous
media. We concentrate on the wave equation and distinguish between two classes
of applications. First we discuss numerical methods for the wave equation in
heterogeneous media without scale separation. Such a setting is for instance
encountered in the geosciences, where natural structures often exhibit a
continuum of different scales, that all need to be resolved numerically to get
meaningful approximations. Approaches tailored for these settings typically
involve the construction of generalized finite element spaces, where the basis
functions incorporate information about the data variations. In the second part
of the paper, we discuss numerical methods for the case of structured media
with scale separation. This setting is for instance encountered in engineering
sciences, where materials are often artificially designed. If this is the case,
the structure and the scale separation can be explicitly exploited to compute
appropriate homogenized/upscaled wave models that only exhibit a single coarse
scale and that can be hence solved at significantly reduced computational
costs.
