  Impulsive gravitational waves are theoretical models of short but violent
bursts of gravitational radiation. They are commonly described by two distinct
spacetime metrics, one of local Lipschitz regularity, the other one even
distributional. These two metrics are thought to be `physically equivalent'
since they can be formally related by a `discontinuous coordinate
transformation'. In this paper we provide a mathematical analysis of this issue
for the entire class of nonexpanding impulsive gravitational waves propagating
in a background spacetime of constant curvature. We devise a natural geometric
regularisation procedure to show that the notorious change of variables arises
as the distributional limit of a family of smooth coordinate transformations.
In other words, we establish that both spacetimes arise as distributional
limits of a smooth sandwich wave taken in different coordinate systems which
are diffeomorphically related.
