Stabilizing the Benjamin-Feir Instability
Diane Henderson
Penn
State University
Here we
report experiments on permanent form gravity waves on deep
water propagating in both one and two horizontal dimensions. We find
that moderate amplitude, bi-periodic patterns are ``stable'' within the
length of our wave basin. This result is surprising in light of classic
instability results (the Benjamin-Feir instability) for deep-water
waves. And large amplitude experiments do show evidence of what appears
to be the Benjamin-Feir instability. However, recent numerical results
by Fuhrman & Madsen (2006) provide a different explanation. Our
further experiments show that their explanation is correct and the
patterns are indeed ``stable''. To explain the unexpected persistence
of these patterns mathematically, we reconsider the stability of a
uniform wavetrain using the nonlinear Schroedinger (NLS) equation
modified to include linear damping. We prove that the presence of
damping, no matter how small, stabilizes (with linear and nonlinear
stability) the uniform wavetrain solution. The predicted evolutions are
in excellent agreement with our experiments. These stability results
are then extended to the case of a permanent form solution of coupled
NLS equations that model wave patterns.Time: 3:30 pm
(Penn State) in Skiles 255
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