(^(T ,t) = A^ COS (k^-'x - oj^t) , l^Jx ,t} = A^cos (k^'x - cj^t) (1) 

 with A, » Ar,, the resulting wave can be described by 



C(T.t) = C^(T, t) + 42 Ix. t) = A OT, tj cos (k'^'T- a>^ t - cP(7, t) (2) 



where the amplitude AlT, t) and the phase (h(x. t) are given according to 



A(T,tj =\a^^ + A^^ + 2A^A^cos\(T^-~k^).'x-(co^-co2)t\ (3) 



and 



_^ A.-, sin\(k,-k.-j)-x ~ (clj, - ojJ t \ 

 cf>(x, t) = arctan -^ ! — —r^ — =«^ — =^ — r ^^' 



thus representing a swell which travels in a direction given by the wave vector 

 k ^ with an amplitude changing between Ai ± A2 and a variable phase. Wave 

 measurements at the NEL Tower indicate the reality of such a swell. 



ENERGY TRANSFER FROM SURFACE WAVES 

 TO INTERNAL WAVES BY RESONANCE 



The process of resonant interaction is easy to comprehend, but the detailed 

 analysis involves a great deal of algebra. A short outline of the main ideas may 

 help in understanding the following computations. 



The hydrodynamic equations are nonlinear. They include terms of the form 

 ir-Vi^ir being the velocity vector. In internal wave theory, these equations ai-e 

 linearized by perturbation methods. The result is a sum of mutually independent 

 waves, each one conserving its own energy if friction is neglected. Many fea- 

 tures are adequately described by this linear theory, but there exist circumstances 

 under which the nonlinear terms can give rise to significant energy transfer be- 

 tween these primary wave solutions. 



Surface waves are zero mode solutions of the internal wave equation. In- 

 ternal waves may be of any mode lai'ger than zero. 



The computations in the following sections are based on this idea. Consi- 

 der the simple nonlinear wave equation 



^+^2^_<^2^0 (5) 



