Forward and Inverse Flexural Wave Scattering with Applications to Arctic Ice Shelves

Flexural waves are involved in a number of important phenomena, including the bending and flexing of floating ice sheets driven by the ocean. These waves have been linked to large-scale iceberg calving events on several ice shelves in Antarctica. However, the interaction of these waves with complex ice shelf features, such as the surface rolls found in Arctic ice shelves, remains poorly understood, despite being a potential precursor to calving. Using a hierarchy of numerical models, we find that ice shelves with surface rolls are able to reflect incident wave energy from the ocean, which may help to explain their long-term survival.

These findings motivate the development of new tools for other forms of flexural wave scattering, including scattering off of solid obstacles. In particular, we explore boundary integral formulations for scattering in thin plates with clamped, supported, and free plate boundary conditions. We present an original integral representation for the free plate that uses the Hilbert transform to eliminate singularities that result from higher-order boundary conditions. These methods give fast and accurate solutions for flexural wave scattering problems with a wide range of applications in geophysics and engineering.

Finally, we use these forward solvers to address the inverse problem of reconstructing a supported cavity in a thin plate using measurements of the solution in the far field. We derive two different sampling methods for these problems: the linear sampling method (LSM) and direct sampling method (DSM). Both methods are based on the construction of an indicator function which has certain asymptotic properties away from the obstacle. We present a variety of numerical experiments to assess the performance of these methods and show that they are sufficient to recover the locations and shapes of various supported obstacles.