EM wave through free space propagation

Questions
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1

Hello!

I am trying to model a microwave radiation passing through an inhomogeneous body (human head). To start with something, I try to obtain solution for a simple EM wave propagation through a free space 2D disk. However, there appears to be a trouble already at this stage, as the DSolve does not produce a solution, NDSolve and NDSolveValue complain about insufficient initial conditions (even though they should be set using the DirichletCondition at t==0), consume a lot of memory and do not end cell evaluation when run.

I have prepared a simplified example that captures said problems in the attached notebook. Admittedly, my experience with Mathematica is rather limited, so any help or advice would be appreciated.

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2

Regarding your project to model microwave radiation passing through an inhomogeneous body (human head), back in 1997 I corresponded with Le-Wei (Joshua) Li (unfortunately now deceased) of the Antennas & Scattering Lab & EM Research Group at the National University of Singapore regarding his paper A Spheroidal Vector Wave Function Analysis of Field and SAR Distributions in a Dielectric Prolate Spheroidal Human Head Model and, later, his book Spheroidal Wave Functions in Electromagnetic Theory.

Le-Wei used Mathematica and described his project as follows:

Primarily it is utilized in the analysis of airborne antenna radomes
and antennas of the oblate and prolate shapes, as well as the
interaction between human head (modeled as an inhomogenous spheroidal
media) and a mobile phone where closed form solution of the electrical
fields and specific absorption ratio (SAR) can be found.

The reason for my interest was that my MSc student, Peter Falloon, wrote his computational thesis on the Theory and Computation of the Spheroidal Harmonics with General Complex Arguments which, eventually, resulted in the built-in spheroidal functionality. Interestingly, Peter’s thesis was submitted and examined as a series of Mathematica Notebooks.