Periodic Irrotational Waves of Finite Herght. 46 
5. Returning to the general equations (15), we consider a wave short 
of the highest and we select the case e~**= 3. We shall find that this 
corresponds to a value of about } for Stokes’ parameter 6. 
The coefficients B,,2.+1 have to be calculated from the relations (14); 
the hypergeometric series are, of course, convergent, and the values can 
be obtained to any required degree of accuracy. Substituting the numerical 
values in (15) we obtain the equations 
1—1:258,;—0:582+3'75B—3:2561B2—1:2583—0°7583 
= g (1:081 — 25848; —1'5383—1-:2818s), 
581 —45B2— 22581? + 9:258)82+ 58:3—0°75B3 
= 9 (00166 + 2:1338) — 2'32282—1:42983), 
8B2+4B.2—7°75B8i182—7°'75B3 = g (—0°0157 + 027678; + 2°237 B,— 2:27483) 
1183+11f:82 = g (—0°0125 + 0:08658; + 0:3243 834 2:25483). 
(24) 
We carry out now the successive approximations described in the previous 
section. At the third stage, we find 
= 09246, 6, =0:00273, B2= —0:0034, 63 = —00013. (25) 
Comparing these values with those for the highest wave given in (21), 
we see that the §’s are much smaller; on the other hand, there may be 
greater difficulty in obtaining their values accurately, because of the later 
stage at which the §’s begin to diminish steadily in absolute value. We 
shall find this impression confirmed later when we try smaller values 
olvem=s3 
To find the ratio 4/1 for this wave, we have 
(velocity at crest)? = 2773 (1 —e~**)73(1+4+ 61+ 82+ B3+...)?, 
(velocity at trough)? = 2-98 (1 +e) (1—B, + B2—B3+...)”. 
Taking the difference, and dividing by 2g, we find ; and since L = 21, 
we have h/L = 00898. Stokes’ parameter 6 is, to a first approximation, 
ah]; hence this wave corresponds to b equal to 7 nearly. 
6. We have now two methods for a wave of finite height, namely, that 
described above and Stokes’ method. The two can be shown to be in 
agreement in any particular case. 
From (8), we have, on the wave surface yr = «, 
gon a = (107 28e2i8)-18 (14 Biot + Boetio-t 2, (26) 
For any wave below the highest possible, that is provided « is not zero, 
the first factor on the right of (26) can be expanded in a series valid for 
140 
