assn4 - 1.685J/2.0345/18.377J Nonlinear Dynamics and Waves...

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1.685J/2.0345/18.377J Nonlinear Dynamics and Waves Spring 2007 Problem Set No. 4 Out: 12, 2007 Due: 26, 2007 in class Problem 1 Consider the system and t) = = 1 for - oo < x < oo, t 2 0 = ii - u = ii+ul with normal modes of the form where 4 = + l)rx}, s = a) = a (1 - a2) - + + a2)/~ for n = 0, I,. .. and any real wavenumber a. Deduce that the mode is stable if and only if R 5 (a) = + + (1 - a2). Hence show that the flow is stable if and only if R 5 R, = Ro(a,) = 41.5 where = + rP2)'l2 - 1) = 0.488.
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Show that the Landau equation for the amplitude A(t) of the most unstable mode for R 3 Rc7 where Problem 2 A particle of mass ml is attached to a light rigid rod of length
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assn4 - 1.685J/2.0345/18.377J Nonlinear Dynamics and Waves...

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