Wu 



^ ^ e/32-i-t fn^^VM^z^ - r2 \ (r<M|z|), (60) 



provided f(k^) satisfies certain conditions of regularity. 



For a; > N, all admissible waves behave exponentially in z since 



.{^J > o] . (61) 



r2 _ 1,2 1, 2 



B^k/ + /S^ 



B^ = 1 



Hence all positive real values of k^ may be used to construct the solution as for- 

 mally expressed in (59), except the upper limit of integration is cd. From this it 

 follows that, as can be verified by using the method of steepest descent, 



^ -^ e/3z-i-t fn(-^yr2+ B^z^ \ (fi^/rU z^ » 1^ . (62) 



The dependence of ^ on r, z as given by (60) and (62) will be evaluated again 

 analytically later. 



INTERNAL WAVES IN STEADY FLOWS 



Suppose a localized disturbance, such as one produced by an obstacle of 

 fixed shape, moves horizontally with a constant velocity u in the negative x- 

 direction, while the fluid (with /S = const.) was originally at rest; and suppose 

 the point disturbance reaches the origin at time t = . At an instant t earlier, 

 the moving disturbance passed by the point Q, located at x = ut , and emitted at 

 that instant all components of internal gravity waves. These waves have been 

 propagating away from Q in all directions after the disturbance left Q. For 

 each specific wave number k, the waves with this k, if they had been carried 

 along with phase velocity c (see (46c)), would be situated at t = on a surface 

 of revolution which is obtained by revolving the circle having a horizontal diam- 

 eter PQ about the vertical axis through Q, where PQ = c^t = Nt/(k^ + /3^)'^. The 

 waves with different k will be located at the present instant on a different such 

 surface of revolution. The farthermost penetration of these waves is reached 

 by the limiting component with k = (infinite wavelength), corresponding to 

 PqQ = Nt//5 . Of all the waves generated at Q, those that may remain stationary 

 relative to the moving disturbance must have their wave number vector k sat- 

 isfy the relation 



kU = w = ck , (63) 



where U is the velocity vector of the moving disturbance, so that the frequency 

 of these stationary waves will appear to be zero with respect to the disturbance. 

 But U-k = -Uk sin 6 cos qp (see Fig. 5), hence from (63) 



U cos cp = -N/(k2 + /32)i/2 . (64) 



The solution k of (64) is real if u |cos(p| < N//3 and 77/2 < cp < i-n/i; then 



564 



