S  C  H  O  O  L    OF    P  H  Y  S  I  C  S

Spring 2007

January  29, 2007 (Monday)
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 UNLESS OTHERWISE NOTED.

reserve time with the speaker

Phone: (404) 385-2500 Fax: (404) 385-2506
Address: 837 State Street, Atlanta, GA 30332-0430