Displacement within velocity effect in gravitational wave memory

Sandwich gravitational waves exhibit the velocity memory effect (VM) which however can become, for specific values of the wave parameters, pure displacement (DM) as suggested by Zel'dovich and Polnarev. Fixing such a "miraculous" value, the particle trajectory is an (approximate) standing wave characterized by a unique integer m, for which the particle does not absorb any energy from the passing wave. Our statements are illustrated by a simple Gaussian and by the Pöschl-Teller potential as profiles.