108 Prof. A. R. Richardson on 



Case (b). Two real roots between ±1. 



cos /jlw = % 2 , cos//,w = £ 3 , say, 



i. e. fiw = 2nir ± cos -1 f 2 



fjLiv = 2nir ±cos' 



(5) 



are the singularities additional to those in Case (a). 



These lie on yfr = 0. 



Hence, since sm/jiiv changes sign on passage of ulw 

 through nir\ over a range containing four consecutive 

 values of (5), the stream-line must cut itself. 



Two-dimensional irrotational motion is impossible, the 

 explanation being that the wave-surface curls over when 

 the tangent to it is vertical. 



It is interesting to note that this occurs before the limiting 

 height ft 2 = 2g\ is reached. 



Hence the motion breaks up as soon as equation (4) has 

 all its roots real. 



To complete the discussion, suppose /3 2 <2g\. 



There are now zeros of (3 2 — 2g\ cos /jlw on ^ = 0, viz. at 



S 2 

 fiw = 2n7r + cos l a Y where ^i—^-z-. 



In Case (a) these are the only singularities on ^ = 0, and 



passage through consecutive zeros will change the sign of 



dx 



-j— and the surface will break as before. 



aw 



A similar remark applies to Case (b) since a 3 <a 4 <aj, 

 so that the above argument holds good. 



It should be noticed that the limiting form when the bed 

 of the stream is fixed in shape has not been discussed ; the 

 method is evidently unsuitable. 



III. Standing waves of Elevation. 



Jn view of the prominence given to the consideration of 

 standing waves take in (1) 



F(^)=-(l-a 2 tanhV^) where a 2 =.l-^, (6) 



F(w)= _2^ si h 



' g cosh 6 fiaw v ' 



and c 2 + 2gh - 2g¥ {w)=c 2 (1 — 2a 2 sech 2 fiaw). ... (8) 



