Relatively Undistorted Solutions to the Wave Equation: A Review

A review of simple solutions to the linear wave equation in three spatial variables, involving arbitrary functions, is given. Physicists’ interest in these solutions is primarily motivated by the need for a convenient description of spatiotemporally localized wave beams and packets. Four families of closed-form solutions with arbitrary functions—relatively undistorted waves—and some of their applications, particularly in optics, are examined within a unified framework. These are classical plane waves and their simple generalizations, classical and complexified spherical waves known as “complex source fields”, the Bateman wave and its various generalizations, and quasi-spherical waves, which have recently been introduced into consideration. In connection with the latter, we discuss the relatively new concept of the unidirectionality of solutions. We also briefly review families of solutions generated from spherical, quasi-spherical, and Bateman-type waves by Lorentz transformations. The review updates literature from the past two decades to assist in modeling spatiotemporally localized wave phenomena, such as laser pulses, using arbitrary functions instead of complex simulations. The review focus on few-cycle pulses and localized beams matches crucial ongoing developments in laser physics. It also bridges the gap between recent exact solutions and modern paraxial and non-paraxial optics and fills a significant literature void.