NONLINEAR PROCESSES IN 

 LONG-CRESTED WAVE TRAINS 



Brooke Benjamin and J. E. Feir 



University of Cambridge 

 Cambridge, England 



ABSTRACT 



This paper is a commentary on some recent findings in water-wave 

 theory which have been reported in previous papers by the present au- 

 thors, and which relate closely to ideas developed recently by Whitham 

 and by Lighthill, The principal aim here is to clarify the practical 

 significance of the new results, with regard both to ocean waves and to 

 waves fornned in model basins. 



The original object of the work to be reviewed was to explain an im- 

 pressive, clearly nonlinear phenomenon that was discovered during ex- 

 periments with a wave tank. To demonstrate this phenomenon, a long- 

 crested progressive wave train of fairly large amplitude needs to be 

 generated on deep water by awavemaker whose reciprocating motion is 

 slightly miodulated either in amplitude or frequency. It may then be ob- 

 served that the modulations imposed on the basic wave train undergo 

 enormous amplifications with distance from the origin, the final result 

 being a complete disintegration of the wave train and redistribution of 

 its energy over a broad spectrum, 



A theory agreeing very satisfactorily with the experimental observa- 

 tions has been worked out for the initial phase of these events, during 

 which the modulations can be represented adequately by two infinitesi- 

 mal wave components at side-band frequencies separated by a fraction 

 S from the fundamental frequency. The theory shows in effect that, on 

 sufficiently deep water, the waves of permanent form described by the 

 well-known Stokes approximation are unstable to perturbations of this 

 type over a certain range of S. Nonlinear interaction between the side- 

 band components and the second harmonic of the basic wave train is 

 crucial to the mechanism of instability. If S is not too large, a balance 

 can gradually be attained between the nonlinear effect of the phase ve- 

 locities of the side -band components and the effect of frequency dis- 

 persion on them, so that thereafter the interaction is resonant; each 

 side -band component is then subject to a synchronous forcing action 

 proportional to the amplitude of the other, and thus the two grow mutu- 

 ally at an exponential rate. 



In an application of the theory to wave trains on water of arbitrary 

 depth h, it has been found that instability is possible only if kh > 1.363, 

 where k is the fundamental wavenumber (= Ztt/ wavelength). This crit- 

 ical value of kh has also been discovered by Whitham. By means of a 

 variational principle using an averaged fornn of Lagrangian function, he 



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