Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions

We consider the Laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the curves tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the curvature radii of the Neumann boundary to the Dirichlet one is the biggest. We also show that the asymptotics can be obtained from a form of norm-resolvent convergence which takes into account the width-dependence of the domain of definition of the operators involved.Comment: 18 pages, LaTeX with 1 EPS figure; to be published in ESAIM: COCV at http://www.esaim-cocv.org

Isobath variation and trapping of continental shelf waves

Since Trösch (Proceedings of the 4th International Conference on Applied Numerical Modeling, Tainan, Taiwan, 1984 (ed. H. M. Hsia, Y. L. Chou, S. Y. Wang & S. J. Hsieh) Science and Technology Series, vol. 63, 1986, pp. 307–311. American Astronautical Society) found trapped sub-inertial oscillations in computations of low-frequency variability in the Lake of Lugano, models of trapping have generally considered evenly spaced isobaths parallel to shorelines with approximate boundary conditions at any shelf–ocean boundary. Here an asymptotic analysis for slowly varying topography and accurate spectral computations demonstrate trapping on non-rectilinear shelves. It is shown that changes in any of three factors, isobath curvature, distance from the coast and the shelf-break, and the slope at the shelf-break, are sufficient on their own to give trapping. Continental shelves that abut smoothly onto the open ocean are considered thus avoiding the shelf–ocean boundary condition approximation and allowing the accuracy of previous approximations to be assessed

Existence of eigenvalues of a linear operator pencil in a curved waveguide - localized shelf waves on a curved coast

The question of the existence of nonpropagating, trapped continental shelf waves (CSWs) along curved coasts reduces mathematically to a spectral problem for a self-adjoint operator pencil in a curved strip. Using methods developed for the waveguide trapped mode problem, we show that such CSWs exist for a wide class of coast curvature and depth profiles

Computation of scattering matrices and resonances for waveguides

Waveguides in Euclidian space are piecewise path connected subsets of R^n that can be written as the union of a compact domain with boundary and their cylindrical ends. The compact and non-compact parts share a common boundary. This boundary is assumed to be Lipschitz, piecewise smooth and piecewise path connected. The ends can be thought of as the cartesian product of the boundary with the positive real half-line. A notable feature of Euclidian waveguides is that the scattering matrix admits a meromorphic continuation to a certain Riemann surface with a countably infinite number of leaves [2], which we will describe in detail and deal with. In order to construct this meromorphic continuation, one usually first constructs a meromorphic continuation of the resolvent for the Laplace operator. In order to do this, we will use a well known glueing construction (see for example [5]), which we adapt to waveguides. The construction makes use of the meromorphic Fredholm theorem and the fact that the resolvent for the Neumann Laplace operator on the ends of the waveguide can be easily computed as an integral kernel. The resolvent can then be used to construct generalised eigenfunctions and, from them, the scattering matrix.Being in possession of the scattering matrix allows us to calculate resonances; poles of the scattering matrix. We are able to do this using a combination of numerical contour integration and Newton s method

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

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