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The phenomena of reflection and refraction of electromagnetic fields at the boundary between two electrically different media are well known from optics {Ibn Sahl, 984 and W. Snell, 1618}. The easiest analytical solutions of Maxwell’s differential equations, satisfying boundary conditions, were found in the years following 1900 { history see e.g. Sommerfeld [1]}. These were plane waves. They still are a common far field approximation, valid at least in the frequency range from some 10^5 Hz to about 10^18 Hz. However, the first classical solutions failed, regarding at least these two aspects :
- At angles of incidence larger than a limiting ‘cutoff’ angle, no refracted wave exists. Sommerfeld knew already that he must introduce ‘surface waves’ [1]. He proposed complex wave vectors – but only for the ideal case without attenuation/damping.
- No simple refracted plane wave can exist in a medium, if its electric conductivity κ does not vanish! Solutions cannot be TEM. In this paper, the author presents an analytical solution. Classical surface waves are possible, as well as generalized waves refracted into absorbing media, propagating as TM or TE modes.
For the investigation of high frequency electromagnetic waves scattered at metallic bodies it is necessary to use algorithms with high accuracy. Moreover, ordinary differential equation systems play an important role in this context. Thus, the authors present an ODE-Solver based on the method of Lie Series.
The algorithm is computing in arbitrary precision. An adaptive step width control enables improvement in computation time and precision. Furthermore, the authors show tests of one classical ODE-Problem plus geodesics on an ellipsoid.