Phase Space Analysis of the Telegraph Equation

We present a new approach to the study of the telegraph equation, which is based on analyzing the equation in phase space. We consider the phase space of time and frequency, as well as of position and spatial frequency (or wavenumber). We show that considerable simplification, both conceptual and mathematical, results from this approach. The basic reason why a phase space approach is fruitful is that when there is dispersion, the group velocity is not constant but changes with frequency, and therefore the wave changes as it propagates over space and time. Simple but effective and insightful phase space approximations to the solution of the telegraph equation are presented, with examples.