Trapped modes in non-uniform elastic waveguides: asymptotic and numerical methods

Abstract

Trapped modes within elastic waveguides are investigated employing asymptotic and numerical methods. The problems considered in this thesis concentrate on linear elastic waves in thickened/thinned and curved waveguides. The localised modes are propagating within some region that is characterized by a small parameter but are cut-off for geometric reasons exterior to that region, and thereafter exponentially decay with distance along the waveguide. Given this physical interpretation long wave theories become appropriate. The general approach is as follows: an asymptotic scheme is developed to analyse whether trapped modes 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 that encapsulates the essential physics. Then, numerical simulations based on spectral methods are performed for this reduced equation and for the full elasticity equations to validate the asymptotic scheme and demonstrate its accuracy. The thesis begins with an investigation of trapping due to thickness variations. The long-wave model for trapped modes is derived and it is shown that this model is functionally the same as that for a bent plate. Careful computations of the exact governing equations are compared with the asymptotic theory to demonstrate that the theories tie together. Different boundary conditions upon the guide walls and the importance of the sign of the group velocity are discussed in detail. Then, it is shown that boundary conditions also play a crucial role in the possible existence of trapped modes. The possibility of trapped modes is considered in nonuniform elastic/ ocean/ quantum waveguides where the guide has one wall with Dirichlet (clamped) boundary conditions and the other Neumann (stress-free) boundary conditions. For bent waveguides, 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 guided along, and localised to the vicinity of, welds. These waves propagate unattenuated along the weld and exponentially decay with distance transverse to it. Three-dimensional geometries introduce additional complications but, again, asymptotic analysis is possible. The long-wave model provides numerical values of the trapped mode frequencies and gives conditions at which trapping can occur; these depend on the components of the wave number in different directions and variations of the plate thickness. To mimic the guide stretching out to infinity a perfectly matched layer (PML) technique originally developed by Berenger for electromagnetic wave propagation is employed. The method is illustrated on the example of topographically varying and bent acoustic guides, and numerically implemented in the spectral scheme to construct dispersion curves for a two-dimensional circular elastic annulus immersed in infinite fluid. This numerical scheme is new and more efficient than direct root-finding methods for the exact dispersion relation involving the Bessel functions. In the final chapter, the influence of external fluid on trapping within elastic waveguides is considered. A long-wave scheme for a curved and thickening plates in infinite fluid is derived, conditions of existence of trapping are analysed and compared with those for plates in vacuum.