Two-dimensional free surface flows may be formulated through boundary integrals which allows analytic continuation in the unphysical complex plane. For two dimensional water waves, Tanveer showed that the only form of singularity in the unphysical plane is of the square-root type. One may wonder what influence these singularities may have on the behavior of water waves. This thesis resolves some of these questions by tracking singularities in the unphysical domain and relating their close approach to the real axis with wave breaking.
The main result is the direct verification of Tanveer’s singularity result. A boundary
integral technique is used to simulate deep water wave motion. A spectral procedure is used to form-fit the Fourier spectrum of the curvature of the wave profile to a prescribed asymptotic expression. The form-fit provides information on the power and location of the closest singularity to the real axis. The power of the curvature singularities is found to be -3/2 when the curvature is expressed as a function of the Lagrangian variable. This singularity is associated with a pole singularity in the complex arclength plane, and is not an artifact of the parametrization. The singularity approaches the real axis when a plunging breaker occurs. For nonbreaking waves, the singularity wanders above some level in the unphysical plane. It is then established that this curvature singularity is theoretically equivalent to Tanveer’s one-half power singularity. When the surface elevation is viewed as a function of horizontal distance, a different type of singularity arises. It is a square root type singularity that takes the form of a breaking wave when it reaches the real axis of the horizontal coordinate.
Nonlinear interactions among various wavelengths are considered important in random ocean waves. A particularly important nonlinear interaction is the Benjamin-Feir instability. For moderate initial amplitudes, the end-state of this instability is either wave breaking or the Fermi-Pasta-Ulam recurrence. Clearly, singularities in the unphysical domain will play a role. Starting with initial conditions that contain several singularities away from the real axis, their trajectories are studied. When breaking occurs, results show that it is one of the crests in the wave train that breaks like a plunging breaker while others remain moderately flat. One of the singularities moves close to the real axis while the other singularities stay far away. When recurrence occurs, evidence indicates that the closest singularity remains far away from the real axis for all time. The hope and the possibility is that a study of singularities in Benjamin-Feir instability may lead to insight into nonlinear wave interactions in general.