Wave Propagation in Linear Media | re-examined

Part I
Wave propagation phenomena
Soon after the formulation of Einstein's theory of relativity, a discussion arose whether or not
certain types of waves can propagate faster than light. These are evanescent electromagnetic
waves and tunnelling quantum particles that are formally equivalent to some electromagnetic
cases. In recent years, the fairly old debate grew in intensity when experimental results
seemed to support `superluminal' theories. The rst part of this work is therefore dedicated
to the many facets of the discussion.
To provide an insightinto the subject, the rst chapters give a brief overview on the de nitions
of wave propagation velocities in general and on the faster-than-light issue in particular. The
variety of opinions and theories is rather unsatisfactory from a practical point of view, and so
the subsequent chapters present anumber of one-dimensional case studies to explore the problems
further. The rst among these are concerned with the propagation of electromagnetic
waves with a special focus on evanescent modes. We shall begin with simple monochromatic
examples where closed expressions for a propagation velocity can still be found. We then
extend the monochromatic investigation to dispersive media and nally solve the wave equation
for broad-band signals in a lossless plasma as well as in a rectangular wave guide. These
last examples will demonstrate that sudden signal changes travel at a wave front velocity no
faster than the speed of light.
The second portion of these case studies is dedicated to the classical quantum mechanical
tunnel e ect. Like in the sections before, we start with formulating the problem for a plane
wave, but immediately extend the results to the general case of signals with arbitrary bandwidth.
We shall then explore the scattering of a particle o a potential step. Subsequently,
after having looked into quantum mechanical tunnelling time de nitions for plane waves, we
deal with a particle tunnelling through a rectangular barrier. In all these cases, we must
employ numerical techniques to solve the Fourier integrals that constitute the solutions of the
wave equations. The examples will show that for very small-band pulses, the results could
be interpreted as superluminal propagation of a wave packet. However, as the bandwidth is
increased, the tunnelling time becomes even negative, which is quite an absurd result. So we
eventually arrive at the still open question whether a signal pulse is adequately described by
its peak or centre of mass, or what actually forms the information being transmitted.
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