Some Boundary-Value Problems for Water Waves

Abstract

Euler's equations describe the evolution of waves on the surface of an ideal incompressible fluid. In this dissertation, I discuss some boundary-value problems associated with Euler's equations. My approach is motivated by the ideas generated by Fokas and collaborators, particularly the notion of a global relation for boundary-value problems for partial differential equations. I introduce a new method to compute the evolution of the free surface of a water wave based on a reinterpretation of the relevant global relation. Next I consider the bathymetry reconstruction problem <italic>i.e.<\italic>, the reconstruction of the bottom boundary of a fluid from measurements of the free-surface elevation alone. By analyzing the global relation for the water-wave problem, I derive an exact, fully nonlinear equation which is solved for the bottom boundary. Finally, I present a method of reconstructing the free surface of a water wave using measurements of the pressure at the bottom boundary. Using this reconstruction, I obtain several new asymptotic approximations of the surface elevation in terms of the pressure at the bottom. Comparisons with numerical and experimental data show excellent agreement with my predicted reconstructions.