Wave Manipulation by Topology Optimization - Solid Mechanics

Chapter 2
Time-harmonic acoustic and electromagnetic
wave propagation
In this chapter acoustic and electromagnetic waves are introduced. It is shown how
the physics of time-harmonic acoustic waves and time-harmonic electromagnetic
waves are governed by second order differential equations. The governing differential
equations with appropriate boundary conditions form boundary value problems,
which for the majority of the problems covered by this thesis are solved by the finite
element method.
2.1 Acoustic and electromagnetic waves
Acoustic waves are a special form of elastic waves propagating in inviscid fluids with
zero shear modulus[1]. The inviscid fluid particles oscillate back and forth about
their equilibrium positions creating regions of high density (compression) and low
density (decompression). This motion effectively creates a traveling longitudinal
pressure wave with oscillations parallel to the direction of propagation. Just as for
mechanical waves, it is the disturbance that travels not the individual particles in
the medium. Because acoustic waves work by means of compressing matter it can
only propagate in a medium and not in vacuum.
In contrast to acoustic waves, electromagnetic waves are transverse in character.
The electric and magnetic fields oscillate perpendicular to each other and to the
propagation direction[2]. Electromagnetic waves also differ from acoustic waves in
the sense that they can propagate in vacuum as well as in a medium. Electromagnetic
waves are caused by the interaction of a time-varying electric field and a time
varying magnetic field.
In the following detailed description of wave propagation we will focus our attention
on electromagnetic waves because manipulation of acoustic waves constitutes
only a minor part of this thesis. However we will in our derivations end up with
the well-known duality between acoustic and in-plane electromagnetic wave propagation
in two dimensions (e.g. [3]). That is, the two different wave phenomenons
are governed by the same scalar wave equation.
2.2 Electromagnetic vector wave equations
Electromagnetic wave propagation is governed by Maxwells equations[4]. The magnetic
and electric field are coupled through Maxwell-Ampere’s and Faraday’s laws,
which in source-free regions with isotropic, linear and inhomogeneous material can
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