It is shown that, for a relativistic Doppler effect to depend on one‐dimensional source and observer motions only through their relative velocity, then the wave speed must equal the speed of light in vacuum or satisfy the dispersion relation for a collisionless electron gas. The three‐dimensional case is briefly examined, showing that the same results apply. The appropriate Doppler formula is derived, showing the explicit dependence upon the relative velocity; the one‐dimensional and vacuum optical cases are readily deduced as corollaries.