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Title: Trapped modes in non-uniform elastic waveguides : asymptotic and numerical methods
Author: Postnova, Julia A.
ISNI:       0000 0004 2681 5435
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2009
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Trapped modes within elastic waveguides are investigated employing asymptotic and numericalmethods. The problems considered in this thesis concentrate on linear elastic wavesin thickened/thinned and curved waveguides. The localised modes are propagating withinsome region that is characterized by a small parameter but are cut-off for geometric reasonsexterior to that region, and thereafter exponentially decay with distance along the waveguide. Given this physical interpretation long wave theories become appropriate. The generalapproach is as follows: an asymptotic scheme is developed to analyse whether trappedmodes should be expected and to obtain the frequencies at which trapped modes are excited. The asymptotic approach leads to an ordinary differential equation eigenvalue problem thatencapsulates the essential physics. Then, numerical simulations based on spectral methodsare performed for this reduced equation and for the full elasticity equations to validate theasymptotic scheme and demonstrate its accuracy. The thesis begins with an investigation of trapping due to thickness variations. Thelong-wave model for trapped modes is derived and it is shown that this model is functionallythe same as that for a bent plate. Careful computations of the exact governing equations arecompared with the asymptotic theory to demonstrate that the theories tie together. Differentboundary conditions upon the guide walls and the importance of the sign of the groupvelocity are discussed in detail. Then, it is shown that boundary conditions also play a crucial role in the possible existenceof trapped modes. The possibility of trapped modes is considered in nonuniformelastic/ ocean/ quantum waveguides where the guide has one wall with Dirichlet (clamped)boundary conditions and the other Neumann (stress-free) boundary conditions. For bentwaveguides, with such boundary conditions, the sign of the curvature function is shown to play an important role in the possibility of trapping. Trapped modes in 3D elastic plates are considered as a model of waves that are guidedalong, and localised to the vicinity of, welds. These waves propagate unattenuated alongthe weld and exponentially decay with distance transverse to it. Three-dimensional geometriesintroduce additional complications but, again, asymptotic analysis is possible. Thelong-wave model provides numerical values of the trapped mode frequencies and givesconditions at which trapping can occur; these depend on the components of the wave numberin different directions and variations of the plate thickness. To mimic the guide stretching out to infinity a perfectly matched layer (PML) techniqueoriginally developed by Berenger for electromagnetic wave propagation is employed. Themethod is illustrated on the example of topographically varying and bent acoustic guides,and numerically implemented in the spectral scheme to construct dispersion curves for atwo-dimensional circular elastic annulus immersed in infinite fluid. This numerical schemeis new and more efficient than direct root-finding methods for the exact dispersion relationinvolving the Bessel functions. In the final chapter, the influence of external fluid on trapping within elastic waveguidesis considered. A long-wave scheme for a curved and thickening plates in infinite fluid isderived, conditions of existence of trapping are analysed and compared with those for platesin vacuum.
Supervisor: Craster, Richard Sponsor: ORSAS
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral