Since a wave disturbance propagates without dissipation, there are no spatially bounded solutions to the wave equation in source and absorber free space. Pulse packets in localized in space-time are potential solutions, but such solutions diffractively dissipate on propagation. There are no simple exact solutions of finite energy. This section reviews the basic nature of solutions and builds intuition for wave propagation. Lecture 3 considers solutions in more detail using Fourier methods.
Plane waves are the simplest solutions to the wave equation.
Plane waves are unbounded and constant in two dimensions and propagate along a third dimension. Note that for the plane wave to satisfy Gauss's law the z component of the field must be zero. Eqn. has many possible solutions, the simplest being where and is any twice differentiable function. We consider some simple animations to provide some feeling for solutions in the optical domain. The pulse oscillates at a center frequency of at approximately Hz.