Portions of this entry contributed by Waldyr A. Rodrigues, Jr.

Among the infinite family of exact superluminal solutions of the Maxwell equations are waves known as X-waves (Rodrigues and Lu 1997). Scalar X-waves have been measured experimentally by Lu and Greenleaf (1992), and subsequently by Lu, who showed that the peak of a finite aperture approximation to an acoustical X-wave can travel with speed greater than the sound speed parameter appearing in the homogeneous wave equation (Rodrigues and Lu 1997). Rodrigues and Lu (1997) also performed several simulations for the propagation of X-waves, showing that their peaks can move with superluminal speed, an effect subsequently verified by Saari and Reivelt (1997).

These results do not violate special relativity because all the produced superluminal X-waves have wavefronts that travel with the speed parameter c (the speed of light) that appears in the corresponding wave equation. The superluminal motion of the peak is therefore a transitory phenomenon similar to the reshaping phenomenon that occurs (under very special conditions) for waves in dispersive media with absorption or gain and which is in this case responsible for superluminal (or even negative) group velocities (Maiorino and Rodrigues 1999).

Superluminal