Displacement within velocity effect in gravitational wave memory

Particles initially at rest hit by a passing sandwich gravitational wave exhibit, in general, the velocity memory effect (VM): they fly apart with constant velocity. For specific values of the wave parameters their motion can however become pure displacement (DM) as suggested by Zel’dovich and Polnarev. For such a “miraculous” value, the particle trajectory is composed of an integer number of (approximate) standing half-waves. Our statements are illustrated numerically by a Gaussian, and analytically by the Pöschl–Teller profiles.