GROWING WAVES DUE TO TRANSVERSE VELOCITIES 123 



a deep minimum for k = 0, so that there are still four real roots unless z 

 is very large. For z fixed, as 5^ increases, the depth of the minimum de- 

 creases and there will finally occur a 5" for which the minimum is so shal- 

 low that two of the real roots disappear. Call z(0) the value of ziork = 0, 

 write the sum as 2(5^ k^) and suppose that 2(5o^ 0) = —irziO), then for 

 small k we have 



S(5^ e) = -«(0) + (6^ - 8o') §, + k'§,= -«(0) -"^ k 



do^ dk^ Ua 



as 



dB dk^ 



^ = ^± / ".^(^-^0^) + 



'^ a/ 

 dk' y 



The roots become complex when 



aA-2 



S.2 J 2 (Ul/Uo) 



= do — 



52 as 



d8^ dB 



Since Ui/uq may be considered small (say 10 per cent) it is sufficient to 



look for the values of 5o^. 

 When k = we have 



-TZ = 2X) 



2z 



z (n + y,y 



z^ + 52 



irz" 



z'-\-in-\- y^r (n -1- y^y- - s' 



' H ^ + i ^ 



\in + 3^)2 - 52 ^ (n + 1^)2 + zy 

 (5 tan -Kb -\- z tanh irz) 



z" + 52 



Fig. 4 shows the solution of this equation for various 2(0) or oiyo/iruo . 

 Clearly the threshold 5 is rather insensitive to variations in uyo/ir^io . 



Equation (2.28) may be examined by a similar method, but here some 

 complications arise. Fig. 5 shows the infinity curves for n = 0, 1, 2, 3; 

 the n = term being of the form k^/k^ — 8^. The lowest critical region 

 in 5^ is the neighborhood of the point fc^ = 6^ = ]^i, which is the intersec- 

 tion of the n = and n = 1 lines. To obtain an idea of the behavior of 



