Mathematics of Waves, The Wave Equation, Electromagnetic Waves

The goal of the paper Waves is to show how electromagnetic waves (EM) follow as a consequence of the experimentally established Maxwell equations which were discussed in the previous blog entry. We first review the physical characteristics and properties of waves including the mathematical description given in the linear wave equation in one (spatial) dimension. We discuss how this naturally extends to higher dimensions. Using Maxwell's equations, we show how the properties of EM waves follow as a consequence of those equations where the electric and magnetic fields associated with the EM waves are solutions of the three (spatial) dimensional wave equation. Included in the paper are many illustrative worked examples and a solution of the one dimensional linear wave equation is included as an appendix. In particular, we show that an EM wave in a vacuum propagates at the speed of light and carries a magnetic field and an electric field where the two field vectors are always (that is, at all points in space and time) orthogonal to each other and the (propagation) direction of the wave.