Author
Xie, Chao

Year
2009

Advisor
Baker, Gregory

Abstract
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.

Thesis
Xie Chao.pdf