Stationary Waves in Water, 107 



Suppose this is the case. 



Put p 2 =-z— 7,: the roots of (4) are given by 



(X?-l)(2o\X-/3 2 )- P 2 = 0. 



There will always be one real root, say «.,, > ^r-r . I£ 

 there are three real roots the other two will lie between +1. 

 Now cos fj.w=a 2 gives fiw = 2n7r+i cosh -1 a 2 , 



cos/xiv= — — = aj gives /^ = 2>/7r + z cosh -1 a : , 



and a 1 <a 2 , /. e. cosh -1 a 1 <COsh~ 1 a 2 . 



Case (a). Two imaginary roots ff+M/. 



Let cos jjl (4> + tyr ) = £ + ?7 7- 



Hence the singularities all lie on the lines 



yjr= +cosh _1 a l5 + COsh -1 « 2 , + ^0? 



and different problems will be solved according to the range 

 of values taken for ty and the relative magnitudes of ^ 

 and a x . 



Let A/r ^cosh _1 aj and take 



O^^g^n- ^=0 is a free surface over which 



?/ = //o + Xcosyu,(/>. The stream-line -ty—^ is a rigid undu- 

 lating boundary without sharp bends. 



The extreme case arises when (4=) has two equal roots f 

 between +1. 



In such a case the range 0<\Jr<cosh _1 « t gives flow with 



a free surface ^ = 0, which is just on the point of breaking 

 at places >u(£ = + cos -1 £-\-2mr where the tangent is vertical. 



The stream-line -v/r=cosh -1 a 1 is rigid, with periodic undu- 

 lations, and sharp bends where 2 = 0, viz. at places //,</> = 2 ?i7r. 

 The maximum elevation occurs over such places. 



If cosh -1 a^-v/r^ cosh -1 « 2 be taken the flow is between 

 two rigid corrugated boundaries, 



and cosh _1 a 2 ^^^+ rJZ '• gives flow in a semi-infinite 

 liquid with a corrugated surface. 



