Radiation and Dispersion of Internal Waves 



is more important. Strictly speaking, this is true only in a region beyond a 

 certain distance from the disturbance, for the inertial effect cannot be negligible 

 at small distances. For large values of Fr, D^Bwj/Bz is dominant near the dis- 

 turbance. This relative importance of the inertial effect for large Fr , however, 

 is not uniformly valid at infinity. The third term in the square bracket of (34) 

 represents the effect of shear, which depends linearly on the shear gradient u" , 

 V" and is independent of the shear scale u' , V . The latter, however, may play 

 a role in the interaction with the effect of density stratification. 



DISPERSION OF INTERNAL GRAVITY WAVES 



The simplest case of density stratification is the uniform density gradient 

 such that for a positive constant /?, 



Po = const e-^^" or -p'^/p^ = (/3>0). (36) 



For periodic motions in this medium (which is otherwise at rest), 



U=V=0, wj = w(x) e"'"^* , Q = Qo(x) e^^'"* , (37) 



Eq. (34) further reduces to 

 where 



N' = - ;^ Po = 2/3g . (38b) 



Her e the Vaisala frequency N is constant, corresponding to a period r = 2n/N = 

 277 x/g/H (H= 1/2/S), the same as that of a pendulum of length H. The partial dif- 

 ferential Eq. (37) of w(x,y,z) has constant coefficients; its type is 



elliptic > 



parabolic according as w^ = n^ . (39) 



hjrperbolic 



For 0)2 < N^, Eq. (37) has two real mathematical characteristics (straight lines): 



for plane flows 



, \ 1/2 



for axisymmetric flows 



(x2+y2)^/^ t(N! - l)'''z = const. (40b) 



557 



