The boundary condition then becomes 



d6 g 



— = e~ 3T sin 30, 



cty 3c 3 



t = 



which leads to 



/ 2iriw \ 



(34) 



(35) 



where w = ^ A-ty is the complex potential and c 2 = g\/2-, where A is the wave- 

 length. 



The condition for breaking at the crest is u — iv = when w = which 

 since it — iv — cer %m , leads to 3 A = 1. 



In the neighborhood of the crest the wave then forms a wedge of angle 120°. 



Moreover if we write A = l-xci/X where a is small we recover the ordinary 

 linearized theory. 



Thus the method discovered by Davies yields an approximation which applies 

 over the whole range from waves of small amplitude to those on the point of breaking 

 at the crest. 



t 



h 



i 



-► x 



\f/ = — ch 



FIG 9 



The method has also been applied by my assistant B. A. Packham [20] at the 

 Royal Naval College to obtain a unique solution in closed form of the problem of the 

 solitary wave, Fig. 9. The solution is so simple that I must give it 



e -3i U _ i _ s j n 2 ^c/j, sech 2 y 2 k(w — ich) 

 c 2 tanh kch 



gh 



kch 



< kch < %ir 



(36) 

 (37) 



In this result k — corresponds to rest and kch = tt/3 corresponds to breaking at 

 the crest. 



In connection with waves I should like to mention a striking phenomenon to 

 which Dr. Gawn called my attention. He also demonstrated it to me at Haslar. 



11 



