General plane or spherical electromagnetic waves with electric and magnetic fields parallel to each other

•Maxwell’s equations with electric and magnetic fields mutually parallel are studied.•General plane- and spherical-wave solutions are found.•Other wave solutions cannot be as general as the plane- or spherical-wave solution.

General plane-wave and spherical-wave solutions are found for Maxwell’s equations with electric and magnetic fields parallel to each other everywhere in a uniform isotropic sourceless medium. For deriving them, Beltrami’s and Chang–Carovillano–Low’s results about a Beltrami flow (or a force-free field) with a non-constant proportionality factor are applied to Uehara–Kawai–Shimoda’s equations. It is proved that other wave solutions, such as a cylindrical-wave solution, cannot be as general as the plane-wave or spherical-wave solution.