Journal of Fluid Mechanics

Resonant interactions between waves. The case of discrete oscillations

F. P.  Bretherton a1
a1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge

Article author query
bretherton fp   [Google Scholar] 


The mathematical basis for resonance is investigated using a model equation describing one-dimensional dispersive waves interacting weakly through a quadratic term. If suitable time-invariant boundary conditions are imposed, possible oscillations of infinitesimal amplitude are restricted to a discrete set of wave-numbers. An asymptotic expansion valid for small amplitude shows that oscillations of different wave-number interact primarily in independent resonant trios. Energy is redistributed between members of a trio over a characteristic time inversely proportional to the amplitude of the oscillations in a periodic manner. The period depends on the initial conditions but is in general finite. Cubic interactions through resonant quartets are also discussed. The methods used are valid for a fairly wide class of equations describing weakly non-linear dispersive waves, but the expansion procedure used here fails for a continuous spectrum.

(Published Online March 28 2006)
(Received October 14 1963)
(Revised May 2 1964)