In special relativity, the transformation between inertial frames for an electromagnetic plane wave is usually derived for the time-harmonic case (the field is a sinusoid of infinite duration), even though all practical waves are of finite duration and may not even contain a dominant sinusoid. This paper presents an alternative derivation in which the time dependence of the field is general; for example, it could be a Gaussian pulse in time. The changes that occur on transformation in the quantities that describe the wave are obtained and discussed. These changes include the temporal behaviour, direction of propagation, electromagnetic field, energy, and linear momentum. The derivation uses only elementary principles from special relativity, so it is suitable for an introductory course on the subject or a course on electrodynamics. (Contains 1 table and 3 figures.)