WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Chapter 1
Introduction
This thesis is devoted to waves and vibrations in inhomogeneous structures and
specifically to the investigation of bandgap phenomena and optimization of bandgap
structures. The effect of a bandgap on the propagation of waves is illustrated in
Fig. 1.1 for a simple one-dimensional structure. A single inclusion in an otherwise
homogeneousstructurereflectsapartofthewave pulse(showninthethreefiguresto
theleft). Asequence ofequidistantly spacedinclusions (4inclusions inthisexample)
causes in-phase reflections which lead to a larger part of the wave being reflected
(shown in the three plots to the right). In order for in-phase reflections to occur and
a bandgap condition to be created, the spacing between the inclusions must match
certainconditions. Theseconditionsaregovernedbythemainfrequency/wavelength
of the wave pulse and also by the contrast between the material properties of the
inclusions and the background material. This usually results in a few inclusions per
wavelength. If the bandgap condition is fulfilled the amplitude of the transmitted
wave decays exponentially with the number of inclusions and if an infinite number
of inclusions were present (a periodic material) the incoming wave will, in theory,
be totally reflected.
In the following the background and motivation for the work leading to this
thesis will be outlined.
Phononic and photonic bandgaps
The research on bandgap materials and structures has had a rather peculiar history
during the last century. To the author’s knowledge the first report on the bandgap
phenomenon was made in 1887 by Lord Rayleigh (Rayleigh, 1887) who found that
in-phase reflections from certain periodic arrangements of different elastic materials
may lead to frequency ranges for which waves will be totally reflected. However,
Lord Rayleigh did not use the term bandgap to describe this effect. Neither did
Brillouin in his book on wave propagation in lattice structures (Brillouin, 1953).
Instead the terms passband and stopband were used to describe the frequency ranges
for which waves either could or could not propagate. With his pioneering work,
Brillouin paved the way for much of today’s research in the field of bandgaps with
his treatment of symmetry conditions and the concept of Brillouin zones. These
concepts are nowadays an integrated part of the description of wave propagation
through periodic materials through the use of the band diagrams.
A band diagram for a certain periodic material is illustrated in Fig. 1.2. The
solid lines in the plot indicate propagating modes with the frequency of the wave
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