(Reprinted from the PROCEEDINGS OF THE Royat Socrety, A. Vol. 95] 
Periodic Irrotational Waves of Finite Height. 
By T. H. Havetock, F.R.S. 
(Received May 21, 1918.) 
1. The method of Stokes for waves of finite height on deep water consists in 
working upwards by successive steps from the infinitesimal wave towards the 
highest possible wave. Although lacking formal proof of convergence, it is 
generally accepted that the method is valid, but that it does not include the 
highest possible wave when the crests form wedges of 120°. 
For the highest wave itself we have Michell’s analysis by a distinct method, 
also involving an infinite series whose convergence has to be assumed.* 
The theoretical position of Stokes’ method has been stated concisely by 
Prof. Burnside in a recent criticism} :— 
“The complete result would be to express the co-ordinates x and y in terms 
of @ and yw in the form 
2 = —btbeV sin 6+ S0"P, (6) e™ sin nd, 
2 
y = —p+ber cos $ + S0"Q, (De cos np, (1) 
where P,, (0), Q, (6) are power series in 0. 
“These results have a meaning and can be used for actual approximate 
calculation only, if P,, Q, are convergent power series when 6 does not exceed 
some value, say 0, while for suitable values of 6 and for real negative values 
of y, the series for z and y are convergent. 
“ Until the form of the power series P, and Q, have been determined, it is 
impossible to deal with their convergence. Assuming that they are 
convergent, it is clear from physical considerations that there must be an 
* J. H. Michell, ‘ Phil. Mag.,’ Ser. 5, vol. 36, p. 480 (1898). 
+ W. Burnside, ‘Lond. Math. Soc. Proc.,’ Ser, 2, vol. 15, p. 26 (1916). 
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