### 5.8 Spacetimes with waves

[170] presents a method to produce initial data configurations containing waves. Given a fixed
background metric, it shows how to superimpose a given gravitational wave content. The equations are
solved numerically using a multidomain spectral code based on [171, 167]. Space is covered by various
sphere-like shells and is described up to infinity. When no black hole is present, the origin is covered by a
square domain because regularity conditions at the origin, in spherical coordinates, are not handled
by [171, 167]. Such a setting is used to generate spacetimes containing i) pure quadrupolar waves ii) flat
space with an ingoing pulse and iii) a single black hole superimposed with an ingoing quadrupolar
wave.