A Theoretical and Numerical Analysis of the Faraday Wave Experiment

This thesis focuses on the well-posedness and stability of the Water Wave Equation in the context of the Faraday wave experiment. We prove that the solutions to the viscous water wave equation formulated by Dias, Dyachenko and Zakharov (DDZ) and supplemented with viscosity, surface tension and small vertical forcing is well-posed. We then derive some numerical stability results in the case of larger forcing and solve these equations numerically by using High Order Perturbation of Surfaces (HOPS) methods. We validate our code by first testing it in the case of traveling waves where exact solutions can be derived and by testing our approach against methods present in the literature.