Applying perturbation methods, = 0^' +0*^' + • • ■, the first-order equation 

 reads 



2^(1) 



2^(1) 



d''4> 



dt 



+ oj^<f>'" = (6) 



with the solution 



n 

 These are mutually independent waves. The second-order equation is 



21 



(7) 



r5 2^(2) 



-— — +a;V^*='^*^*^- ^ ^ r, . J(co^ + ojjt (8) 



dt 



m n 



The secondary waves ^S'^) are therefore forced waves with frequencies co^ + <^n- 

 As long as these combination frequencies «„ + con are not the same as a natural 

 frequency of the system, the amplitudes of the secondary waves will remain 

 small. However, when one of the combination frequencies w^ + «„ is the same 

 as one of the natural frequencies co of the system, this wave never gets out of 

 phase with the forcing wave, and a continuous energy transfer, only limited by the 

 available energy in the primary waves, is possible. If this occurs, the solution 

 of (8) is of the type <;;!)* 2>oe/, that is, (?!)* 2) increases linearly with time. This is the 

 resonance case. Its physical meaning is that energy from frequencies co„ and ojnj 

 is continuously fed into the frequency range co^ + con- ^^ ^^^^ study the problem 

 of whether energy from surface waves in the frequency bands com and wn can be 

 transferred into internal waves in the frequency band cD^tcon- Obviously, if com 

 and (On are swell frequencies, a transfer to the frequency band com — (^n is pos- 

 sible only because internal waves have a cutoff at the Vaisala frequency, which 

 is considerably lower than the swell frequencies. Additionally, the frequencies 

 ojjy^ and aj„ must be very close together in order to feed energy into a frequency 

 band corresponding to periods 2 to 20 minutes. 



EQUATIONS AND BOUNDARY CONDITIONS 



The hydrodynamic equations of a nonviscous, incompressible, stably 

 stratified fluid in a Cartesian frame of reference with z pointing downward may 

 be written as 



du ^ _^ 

 p— + pu-Vu -= -VP - pW^ (9) 



dt 



10 



