Wave Propagation in Linear Media | re-examined

Part II
Numerical aspects of wave
equations
In the rst part, we focused on the physical meanings of energy velocity and wave propagation.
To this end, we took the numerical solutions of the wave equations for granted and left aside
the particular problems associated with these computations. It is now time to make up for this
de ciency and demonstrate how we obtained the numerical answers to our physical questions.
The second part of this work is therefore dedicated to the computation of in nite integrals
by means of numerical algorithms.
In the following chapter we shall rst give anoverview of the basics of this branch ofnumerical
mathematics and various approaches that have been published over the years. Unfortunately,
we shall discover that none of the commonly available software algorithms is suited to help
us solve our problems. Consequently, the next chapters deal with general thoughts on the
particular di culties and nally the implementation of a suitable quadrature strategy to
compute the type of wave integrals we are interested in. Although the implementation of the
quadrature routine itself is application-oriented, it is still independent of the actual examples
we examined in the rst part. Therefore, we shall also present how the general functions are
applied to the problem of the tunnelling particle.
The programming language used for the actual implementation of the functions was Mathematica
| despite FORTRAN's still dominating role in the world of numerical mathematics.
There were several reasons for this choice: rstly, Mathematica o ers a high-level functional
programming environment which makes it ideally suited to the prototyping of algorithms.
Secondly, it can perform symbolic as well as arbitrary-precision numerical calculations. Although
quadrature functions naturally place the main emphasis on numerical computation,
the symbolic capabilities are also needed for auxiliary functions and the testing of the implementation.
An additional advantage is that | in contrast to FORTRAN | Mathematica
can cope with complex-valued integrands in arbitrary precision, which allows a very convenient
programming of certain wave integrals. Finally, the visualisation of the results requires
no extra e orts and is readily accomplished from within Mathematica. The cumbersome
and time-consuming computations could of course be implemented in the signi cantly faster
FORTRAN at a later date, but this was beyond the scope of this work.
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