﻿354 Dr. J. McCowan on the Highest 



and therefore if bp denote the excess of pressure at any point 

 on the surface over that at the mean level, we have 



8p=ip(W-g*)-gpv ; .... (11) 



whence, on substituting for rj and q 2 from (7) and (9), we get 



8p/pTJ 2 = 2ctP€- m *{cos mh-g/mJJ 2 . sin mh) + &c. . (12) 



Now for a free surface hp ought to vanish; therefore for a first 

 approximation we must take 



cos mh —g/mU 2 . sin mh = ; 

 that is, 



U 2 =g/mtanmh; (13) 



and (12) becomes, writing it out to the next term, 



$p/ P XJ 2 = 2k 2 e- 2m *{{2u- /3k 2 ) sin 2 mh-U l U 1 }. . . (14) 

 The coefficient of e~ 2mx in (14) ought of course to be made 

 to vanish; but it will be preferable for our present purpose to 

 retain it as a small pressure-error, and so leave another of our 

 constants available to satisfy the conditions in the neighbour- 

 hood of the crest to which we proceed. 



} 



3. The Surface-Pressure near the Crest. 



Put z — c — £, so that £ vanishes at the crest ; then, writing 

 for brevity 



p = tan \ mc, 



A=l-/, 



B={(l + ll/)/-(l + 3/)}/8p, 



we get, on expanding (1) in powers of f— la, 



-(u+iw')/TJ= /pm(5-M?){A + Bm(?-^) + &c.} ; (16) 



whence, integrating with respect to f— ix, we get 



m^ + L^/lJ -mf /JJ=2^p {m^-Lx)\H^A+ ^Bm^-ix)}. 



where i/r is the value of yjr at the crest. 

 Put 



% = r cos 3, #=rsin$, (18) 



then (18) gives for the surface, determined by -\Jr=^ , 



J-Acosf$ + JBwircosf& + &c.:=0. . . . (19) 



rrr 



Thus when r = 0, 3= + — showing that the crest is formed 



by two branches equally inclined to the bottom cutting at an 

 angle of 120°. 







7T 



Put then ^==- +0- ; 



o 



