Stationary Waves in Water. 109 



This will give a wave very closely resembling that of 

 Lord Rayleigh (Lamb, ; Hydrodynamics/ art. 248). 



The finite singularities of t- are given by 



cosh /uaw = 0, i. e. paw — i (2n -f l)7r/2, 



cosh 2 fiaw= 2a~ (9) 



X*=p«(X-l) (X-LVi, (10) 



where X = cosh 2 yua^', 



, 4a¥a 6 



Two main cases arise according as 2a 8 <1, i.'g. as c 2 ^2gh. 



Case (a). 2a 2 < 1, i. e. c* < 2#/i. 



The roots of (9) are 



fiaw—i(mr i«i) where cosa 1 = av/2, 0<a 1 <7r/2. 



Tlie roots of (10) are 



one negative root X= — S 5 2 , 



one positive root between and 2a 2 , say 8 2 2 > 



and two imaginary roots ^±irj. 



The condition 2a 2 < 1 prevents two other real roots > 1 

 occurring. 



Hence the singularities are 



fiaiv = i (2?i -f 1) 7r/2 + a- y where sinh a 5 = S 5 , 



lxaiv = i(mr + o(. 2 ) where cos a 2 = S 2 , 0;<a 2 <7r/2, 



fiaw=u + ij3. 



Xo motion of this type is therefore possible in an un- 

 obstructed liquid of infinite depth. 



There are no singularities on the real axis in the zoplane 

 and the smallest zeros on the </>-axis are 



±ia 2 , dhwtj, and on the stream-line ^ = ^ at </>=+«. 



Different cases arise according as /3$«i, and to the range 

 of values taken for yjr. 



The most interesting case is where (3<a 1 , and 0^->/r</3. 

 Evidently ^ = gives a free surface over which 



y = — [1 — a 2 ta nh 2 /*«<£]. 



