Phillips 



where 



72 = k2 sin20 = C-^- l^j (a2 + /32) (D4) 



and 



f = -^U(z). (D5) 



Note that in the absence of the steady current ( f = 0) Eq. (D3) shows y to be 

 simply the vertical wavenumber, and Eq. (D4) gives the dispersion relation 



\ a'' + /3'^ + 7^ / 



which was presented as Eq. (4) in the paper. 



ANALYSIS 



If f(z) is any periodic function, then Eq. (D3) becomes a form of Hill's 

 equation and resort may be made to the extensive theory available for equations 

 of this general type. However, let us keep to the simple example treated in the 

 paper. Thus we take 



U = u cos kj z , (D7) 



and further assume that is small in comparison with the phase velocity n/a in 

 the X direction, so that f = au/n « l. Hence Eq. (D3) becomes approximately 



— + (7^ + kj2 a cos kjZ) w = , (D8) 



dz 2 



with 



^ ^' - 1^ f . (D9) 



sin 25 V^ 



This is Mathieu's equation, and so the problem is now cast in a very familiar 

 form. 



According to the well-known theory of Mathieu's equation,* the solutions of 

 Eq. (D8) are periodic for most values of 7^ > when a is small. In this case, 

 free transmission upward or downward is possible. The exceptional case oc- 

 curs when the parameters ^'y'^/\<.^ and a lie in certain ranges indicated by the 

 shaded areas in the following diagram: 



*See, for example, Chapter IV of "Theory and Application of Mathieu Functions," 

 by N. W. McLachlan (Oxford University Press, 1947). 



546 



