Expanding impulsive gravitational waves

We demonstrate explicitly that the known solutions for expanding impulsive spherical gravitational waves that have been obtained by a 'cut and paste' method may be considered to be impulsive limits of the Robinson-Trautman vacuum type-N solutions. We extend these results to all the generically distinct subclasses of these solutions in Minkowski, de Sitter and anti-de Sitter backgrounds. For these we express the solutions in terms of a continuous metric. Finally, we also extend the class of spherical shock gravitational waves to include a non-zero cosmological constant.