Abstract:
This thesis concerns the development of a general theory for the wave behaviour in
one-dimensional, periodic, mono-coupled waveguides. The theory is also extended
to waveguides where the effects of near-field waves incident on the periodic discontinuity
are negligible (quasi-mono-coupled). Systems of coupled waveguides are then
considered with locking and veering in general one-dimensional periodic waveguides
demonstrated, and the general solution for two periodically coupled beam systems
derived.
Solutions for the propagation constants in quasi-mono-coupled waveguides are derived
in terms of the phase of the transmission coefficient and another phase variable.
Expressions for the bounds of band-gaps, attenuation and bandwidth are presented
in terms of these variables. The mechanism of band-gap formation is proposed in
terms of transmission at the discontinuity, with a discussion on the mechanisms of
Bragg-scattering and local-resonance. Results show that Bragg-scattering and localresonance
are both described by the transmission coefficient phase, and are special
cases of the more general wave interaction at a periodic discontinuity.
The behaviour of periodically coupled one-dimensional waveguides is investigated.
Locking and veering is demonstrated for the case of periodic structures where, for
cases where there are two or more branches of the dispersion curves, as frequency
increases, two propagation constants lock, forming complex conjugate pairs when
the uncoupled propagation constants intersect with slopes of opposite sign. Veering,
where the coupled propagation constants are repelled from each other, occurring
when the slopes intersect with the same sign.
The case of two periodically coupled Euler-Bernoulli beams is discussed. The dispersion
relation is derived and analysed with expressions for the bounds, bandwidth
and nature of band-gaps presented. For a given frequency, if the intersections of the
uncoupled propagation constants with the Brillouin Zone occur on the same side of
the Brillouin Zone, then all solutions are propagating and no band-gap forms. If they
intersect at opposite ends, all solutions are evanescent and a band-gap is guaranteed
to form. A parametric study demonstrates that to maximise the bandwidth of this
band-gap, one beam must be heavier and less stiff than the other.
Experimental measurements of propagation constants for a beam with periodic masses
and two beams periodically coupled with springs were taken, demonstrating good
agreement between experiment and theory.