(n) oj > n: 



Radiation and Dispersion of Internal Waves 



G(x,z, t ) 



^00 



1 /Sz-iat ikx- I z I //3^ +B^k^ f^k 



2t7B ^ *^o \B^ 



(80) 



where b^ = l - (N/c-j)^ . The argument of the functional dependence of G in this 

 case also has been correctly predicted earlier (see (62)). 



This problem has also been investigated by Gortler (14) and Wong (15). 



FUNDAMENTAL SOLUTION OF STEADY 

 INTERNAL WAVES 



The fundamental solution of steady two-dimensional internal waves due to 

 a point source moving underneath a free surface was discussed earlier by Mei 

 and Wu (16) for the cases (a) ;3 = const., (b) p arbitrary but g (/Su^) large. An 

 error contained in the solution of one case due to the indeterminancy of making 

 the necessary branch cuts has been corrected subsequently (Wu and Mei (17)). 

 The three-dimensional case has been discussed by Wu (18) for /: = const. We 

 propose to review here briefly the solution of three-dimensional internal waves 

 in order to make a comparison with the approach based on the propagation of 

 wave groups. The two-dimensional solution then appears to be deducible by a 

 descent in dimension. 



For the steady three-dimensional flow of a stratified fluid (with /3 = const, 

 and the fluid unbounded in extent) having a uniform velocity u past a point 

 source Q = S(x) S(y) S(z), the vertical velocity component w satisfies the equa- 

 tion (see (34)) 



Bx^ Bx^Bz Bx2 By2 



G = -^ S(x) S(y) 5(z) , 

 Bx 



w(x,y,z) = ^— - 2/3) G(x,y,z) , 



where 



a2 = n2/u2 = U/V^)(-p'o/Po) . /3 = - Y'°o/'°o 



By applying to (81a) the double Fourier transform 



1 , 



(81a) 

 (81b) 



(82) 



G(k 



.■S-) = J/^"""" 



i< ,y 

 ^ G(x, y , z) dx dy 



(83) 



569 



