Let us now see how the Maxwell equations (17.2)–(17.5) predict the existence of electromagnetic waves. For simplicity we will consider a region of space and time in which there are no sources (i.e., we consider the propagation of electromagnetic waves in vacuum). Thus we set p = 0 = j in our space-time region of interest. Now all the Maxwell equations are linear, homogeneous.
maxwell equation, space, time, electromagnetic, chapter 18
Torre, Charles G., "18 The Electromagnetic Wave Equation" (2012). Foundations of Wave Phenomena. Book 5.