Internal Wave Interactions 



the application of general interaction theory made in the paper. Use is made of 

 a method that has previously been applied to various other problems of wave 

 propagation in periodically structured media, as discussed, for example, in 

 Brillouin's well-known treatise.* It will be shown that, for a given current dis- 

 tribution and given wave frequency, channeling of waves occurs not only at the 

 particular wavenumber determined in the paper but over a set of wavenumber 

 bands whose widths increase with the amplitude of the current. 



PERTURBATION EQUATIONS 



The basic state is taken to be a stratified fluid with constant Vaisala fre- 

 quency N, as defined by Eq. (1) in the paper, and with a steady velocity U(z) in 

 the horizontal x direction. The wave motion is considered as a small perturba- 

 tion from this basic state: thus the velocity vector is expressed in the form 



u = hU(z) + u'(x,y, z, t) 



(Dl) 



where h is a unit vector in the 

 linearized in u ' . 



X direction, and then the equations of motion are 



Let us now assume u' to be periodic in time and in the horizontal coordi- 

 nates X and y, hence seek to determine its possible dependence on the vertical 

 coordinate z . When u ' is found to be a periodic function of z , free propagation 

 in a direction with a vertical component is in effect demonstrated. But the ap- 

 pearance of solutions attenuating with + z indicates that horizontal channeling of 

 the wave motion must occur. 



Accordingly we write 



= u'(z) 



,i(ax + /3y-nt) 



(D2) 



and from the linearized equations of motion we may derive a set of ordinary 

 differential equations for the components (u,v,w) of u'(z). Some simplification 

 of the resulting equations is justified on the basis of the assumption (made im- 

 plicitly in the paper) that -p^^ dp/dz = N^ g « k, where k is a typical wavenum- 

 ber. (Alternatively, we may consider the linearized form of Eq. (1) in the paper, 

 also using the incompressibility condition v • u' = and the equation 



w = , 



*L,. Brillouin, "Wave Propagation in Periodic Structures," Dover, 1953. 



545 



(D3) 



