The d’Alembert solution in hyperboloidal foliations

Abstract

We explicitly construct the analogue of the d’Alembert solution to the 1+1 wave equation in an hyperboloidal setting. This hyperboloidal d’Alembert solution is used, in turn, to gain intuition into the behaviour of solutions to the wave equation in a hyperboloidal foliation and to explain an apparently anomalous permanent displacement of the solution in numerical simulations discussed in the literature.