## Publication Date

Fall 12-20-2019

## Abstract

The Maxwell equations may be viewed as evolution equations which develop an initial state of the electromagnetic field forward in time. Such evolution can be simulated numerically, that is modeled on a computer, in which case the domain of simulation is typically finite in extent. Nonetheless, one is often interested in the electromagnetic waves which reach infinity (of course which is outside of the simulation domain). Thus we are interested in near-to-far field signal propagation, that is a mathematical process where a signal or solution recorded at a finite radius r = r1 can be converted to a signal at r = r2 > r1. We achieve such a conversion via application of convolution kernels in the time-domain, although the derivation of the appropriate kernels relies on Laplace transform arguments. Decomposing the wave and Maxwell equations using scalar and vector spherical harmonics respectively, we have solved the equations on the assumption that the source and initial data are compactly supported. We further assume that we work at a large distance outside of the supports. We develop from a theoretical standpoint signal-conversion formulas for the 3d wave and Maxwell equations and these generalize the simple time delay associated with the propagation between two radii of a solution to the 1d wave equation.

## Degree Name

Mathematics

## Level of Degree

Masters

## Department Name

Mathematics & Statistics

## First Committee Member (Chair)

Stephen Lau

## Second Committee Member

Jens Lorenz

## Third Committee Member

Rouzbeh Allaverdi

## Language

English

## Keywords

wave propagation, Maxwell equations, far field

## Document Type

Thesis

## Recommended Citation

Ahmed, Alhassan. "Near-to-Far Field Signal Propagation for the Wave and Maxwell Equations." (2019). https://digitalrepository.unm.edu/math_etds/167