Radiation and Dispersion of Internal Waves 

 steepest descent. The final result for a >> /? may be cited here as 



G - ^ e^^ H(x) A(x,y,z) sin (a^) 



where H(x) is the step function, 



^PCx.y.z) 



1 + — 



ar 



(90) 



(91) 



A(x,y, z) 



(x2+y2+ z2) 



1/ 2 



[(y2+ z2)^ + x2y2] 



(92) 



and there are no internal waves on the upstream side (x < o). A characteristic 

 wavelength of G is 



27T U / ^0 

 = 277 - = 277U - — -T 



a N \ i/^o 



1/2 



(93) 



which agrees with (65b). The equiphase surfaces v^ = const, are given by 



(n=0, 1, 2, ...). . (94) 



'^n - 4 n + T 



they being the wave crests. The surface w^ = const, is plotted in Fig. 8, which 

 is typical of all v^ = const. They all descend downward in the streamwise di- 

 rection to form a long-trenched valley, starting from x^ = (2n + i)\/4 to oo. The 

 steepest path of descent is the circular arc in the symmetry plane y = . The 

 level lines on w^ = const, are hyperbolas, along each of which the amplitude A 

 decreases monotonically with increasing x. These salient features are in quali- 

 tative agreement with the previous discussion of the wave field based on the 

 propagation of wave energy, and it should be noted that some of the fine structure 



Fig. 8 - The surface of * = constant 

 571 



