Periodic Irrotational Waves of Finite Herght. 40 
downwards, with O at a crest. Let the successive crests be given by 
¢ = +n, n integral; and let the upper surface bey =0. If @ be the 
inclination of the wave line to the horizontal at a point ¢, assume 
GF = fob c08 2p + 1 008 4 +... (2) 
This is equivalent to assuming that under the given conditions for the surface 
pressure, the ratio of the curvature of the wave line to the velocity is finite 
and continuous throughout a wave-length; in that case 00/dd can be expanded 
in a uniformly convergent Fourier series. In the numerical calculations 
which are needed later, the practical success of the method of approximation 
depends upon a, a, ..., being small compared with a, and, in fact, upon the 
series converging rapidly. 
With the notation, w = $+ wy, z = x+7y, U = log (dz/dw), Michell showed 
that 
x —1 (a +467” + age” +...), (3) 
is a function which is real over the surface y= 0, and possesses only 
simple poles, which are at the wave crests. 
Suppose that near a crest, say w=0, we have dz/dw= Aw”, then 
q? = const. x r-*"/"*), where g is fluid velocity and 7 is distance from the 
summit. But, since the pressure is constant in the neighbourhood of the 
crests, we have g? = 2gy, and hence n= —}. It follows that the function 
(3) differs by only a constant from the quantity —}2(w—mm). Hence, 
after adjusting the constants and integrating, we find for dz/dw the form 
— = (—Ztsin w) Ve (1 4 ce? 4 coe + ...), (4) 
the real root of (—isin w)¥ being taken along ¢ = 0. The units are such 
that the wave velocity V, or the velocity at y= oo, and the wave-length L 
are given by 
W = 2s L=a/V = 2'*q, 
It is convenient to invert (4) and write 
— = (—Zsin w)3el3 (1 + bye” + boct” + ze ...). (5) 
3. The coefficients 0), b2, ..., are now to be determined by the pressure 
condition at the free surface. So far, we have stipulated only that the 
pressure at y = 0 shall be finite, continuous, periodic, and stationary at the 
points ¢ =m. For our present purpose we shall leave this pressure 
distribution undetermined, except for these conditions. We shall assume 
134 
