The Lorentz transformation applies directly to the kinematics of moving particles viewed as geometric points. Wave propagation, on the other hand, involves moving planes which are extended objects defined by simultaneity. By treating a plane wave as a geometric object moving at the phase velocity, novel results are obtained that illustrate the difference between wave and point kinematics. In particular, the so-called Carroll kinematics replaces Galilean relativity at low frame velocities when events are nonlocal.