Phillips 



instability can be interpreted in terms of the theory of resonant interactions and 

 that it is not restricted to purely two-dimensional motion. 



Other manifestations of resonant interactions among surface waves have 

 been studied theoretically by Benney, Hasselmann, Saffman and others; there 

 have been unambiguous expei'imental verification of the existence of the inter- 

 action and the confirmation of its salient properties by both Longuet-Higgins 

 and Smith (3) and by McGoldrick, Phillips, Huang, and Hodgson (4). The same 

 type of interaction occurs in other dispersive wave motions and some have been 

 considered in detail. In various contexts, they can be responsible for the gen- 

 eration of internal waves at considerable depth by the interaction between sur- 

 face waves of nearly equal wavenumber (5,6), for the transfer of energy from 

 capillary to longer but still short gravity waves (7), and for the transfer of en- 

 ergy to longer waves in a wind-gene^-ated sea (8). 



In this paper, I propose to describe the resonant interactions among inter- 

 nal gravity waves in a fluid of constant stratification (i.e., constant Brunt- 

 Vaisala frequency). One simple but important case will be considered in detail, 

 namely, the case when one of the wave components reduces to a zero frequency 

 wave, or a steady horizontal shearing motion with a periodic variation of veloc- 

 ity along the vertical. Motions of this kind appear to be widespread in stratified 

 fluids, and it is found that their effect can be to prevent the vertical propagation 

 of other internal wave modes —to restrict a disturbance to a zone surrounding 

 the horizontal plane to the height of the source. This effect is clearly relevant 

 to questions involving the vertical spread of internal wave energy in both oceans 

 and atmosphere. It is described later in this paper, but the first step is to set 

 up the general interaction equation from which these results are specialized. 



THE INTERACTION EQUATION 



Suppose that, in the ambient state, the fluid is stratified with a uniform 

 Brunt -Vaisala frequency 



the z axis being vertically upward and p^ being the reference density. If the 

 velocity field is represented as u ::: (q.w), where q represents the horizontal 

 vectorial component of the velocity and w the vertical component, then the dis- 

 turbances to the fluid are governed (9) by the equation 





^' V=w*N'V,= '' ' '""^ "-r 3b 3 / ^ 



3t2 ^ 3x„Bz3t \ J Bxj / *" L ^ 9x. 3t \ J 9x. 



(2) 



where v^^^ = b^/Bx^ + B^/By^, and where the summation convention is used with j 

 taking values 1, 2, 3 and a taking values 1, 2. The coordinate specifications 

 x.y.z and Xj.Xj.Xj are used interchangeably. The fluctuation b in buoyancy is 

 related to that p ' in density by 



536 



