Journal of Fluid Mechanics

A non-linear equation incorporating damping and dispersion

R. S.  Johnson a1p1
a1 Department of Aeronautics, Imperial College

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The steady-state solution of the non-linear equation \[ h_t + hh_x + h_{xxx} = \delta h_{xx} \] with both damping and dispersion is examined in the phase plane. For small damping an averaging technique is used to obtain an oscillatory asymptotic solution. This solution becomes invalid as the period of the oscillation approaches infinity, and is matched to a straightforward expansion solution. The results obtained are compared with a numerical integration of the equation.

(Published Online March 29 2006)
(Received May 13 1969)

p1 Present address: School of Mathematics, University of Newcastle Upon Tyne.