Radiation and Dispersion of Internal Waves 



_ N M^ CSC e _ -N cos e csc'^6 



^g. ~ \r(f)\ ,3/2 ' '^gj ~ \ ,3/2 ■ (53) 



^y^> (1 + m2) (1 + m2) 



The wavelet for each specific e is propagated along the straight line 



f=^=-4cot^. (54) 



and all these waves therefore exist only in the sector outside the vertical cone 

 bounded by r = ± Mz . Since the waves are all outgoing, with wave front normal 

 to k, the slope of the constant phase surfaces (crests) is 



■57 = - tan 5 . (55) 



Upon elimination of 6 from (54) and (55), and integrating the resulting equation, 

 we obtain the constant-phase surfaces as 



M^z^ = r2- r^^, (56) 



which are hyperbolic surfaces of revolution. Furthermore, c has the mag- 

 nitude 



c^^ — (M^ sin^^ - cos^C)^^^ (M'* sin^^ + cos^^)^^^ csc^^ . (57) 



For large t (» N" '), the extent penetrated by these waves will be (Cgt), which 

 depends on i9, as illustrated in Fig. 4. The solution is singular as -► ^^ (or 

 r = ± Mz), which are the mathematical characteristics. 



Furthermore, there are also waves with w < N that propagate in horizontal 

 directions, with their amplitudes varying exponentially in z. The wave number 

 k^ and exponential factor a satisfy the relation (see (45b)) 



M^k^^ ^ ^2 ^ ^2 (0 < k^ < /3/M, < |a| < /3) . (58) 



Hence all real values of k^ lying in (0, /3/m) are admissible to this class of 

 waves, and the solution will be of the form 



I = e 



ySz-iwt 



./3/M 



•o" I z I + ik , r 



f(k J dk. , (59) 



in which a = (/J^- M^k^^^ 1/2 assumes the positive branch, and f(k^) depends on 

 the description of the singularity. For large values of r and | z | , the above in- 

 tegral may be evaluated by usual asymptotic methods, such as the method of 

 steepest descent, by which it can be shown that 



563 



