Wave Propagation in Linear Media | re-examined

Interlude
Wave functions in graphical
representation
Wave functions are complicated things. They are most easily formulated as Fourier integrals
| provided, of course, we can nd the Fourier transform of some initial state of the wave
and have a closed expression for the dispersion relation. In many cases, these are only
minor hurdles, and the formulation of the wave integral is readily found. But this is already
where simplicity ends and troubles start, for the evaluation of the integral is under most
circumstances anything but straightforward. There are rare cases where the initial wave or
the dispersion relation are so well-behaved that the inverse Fourier transform can be calculated
right away or after application of some sophisticated mathematical tricks. Sometimes, special
values for x or t also make integration feasible. However, normally the wave functions balk
at analytical evaluation.
On the other hand, it is clearly desirable and valuable to be able to calculate the wave
integrals in order to depict at least snapshots of the wave and to underline analytical ndings
in a graphical way. The early investigators of wave propagation were hard-pressed to tackle
this problem and had no choice other than to resort to approximation techniques. For the
examination of the wave front and the rst precursor in a Lorentz medium, Sommerfeld [12]
used a high-frequency expansion of the dispersion relation, which then led to integratable
expressions. This approximation, however, was valid only about the wave front where highfrequency
components are dominating. Brillouin [19] and Baerwald [21, 22], who looked into
the evaluation of the signal behind the wave front, employed the method of saddle point
integration where they made use of two approximations: the rst one being the fact that
the relevant contributions to the wave integral come from the vicinities of the saddle points,
and the second one being the approximation of the integrand about these saddle points. By
using these techniques, the authors could reduce the integrals to combinations of well-known
functions and eventually obtain graphical representations.
With the advent of electronic computing machines, the problems with the evaluation of wave
integrals were alleviated. First, computers allow us to make more accurate plots of given
functions since they can calculate function values at a much larger number of points much
faster than it was possible with the tedious and error-prone look-up methods using mathematical
tables. Second and more importantly, numerical methods are available to evaluate
integrals that would be impossible be by hand calculations. Still, numerics are no general
remedy for all problems raised by the investigation of wave functions. Actually, the di culty
of nding a closed form for the integral at all is only transformed into a question of computing
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