420 T. H. Havelock 
In the notation used in previous work, we have 
Qo (u) = 5 | {How — Yo (w} ds 
AH=(°WHd, QW = [Que du, 
Po) = — 5 | Yo (u) du, (12) 
Pot (u) = | Po (w) du= 1+ Pr 
P, (u) = | Po (u) du = u + Pu). 
Summing up these results, we obtain finally for the wave profile 
c= —# ‘S Fo (K041) fe +h (Koga) + 2c F, (ko) 
— 2058 Fe (xoq2) + oF (koqs3) — = ae (ods) (13) 
Tiana fon = NOH fa) == Real, Oh == 28 — IE Oy = 255 Wace Also we have 
FLKW=Q@), u>d 
= Q, (—u) — 4P, (—), u< 0, 
FiwWM=Qa@, u>0 
=—Q,(u+4Pot(-u), u<9, 
F, (vy) = Q. (), u >O 
= Q, (—u) — 4P) 7 (—»), u< 0. 
Using tables and graphs of the various P and Q functions, the wave 
profile can now be found, for any speed, for any assigned value of . 
We have chosen the value 8 = 0-6, and calculations have been made for 
a sufficient number of values of x to give the wave profile for two different 
speeds; the speeds are those for which «l= 6 and «,/ = 3, or for 
c/,/(gl) equal to 0-408 and 0-577 respectively. The wave profile has 
also been calculated at these speeds for the value 6 = 1, that is for the 
usual theory without any allowance for frictional effects. The four 
curves are shown in fig. 1, the full curves being for 8 = 1 and the dotted 
curves for 8 = 0-6. 
These curves may be compared with some given recently by Wigley* 
in a comparison of experimental and calculated wave profiles. 
** Proc. Roy. Soc.,’ A, vol. 144, p. 144 1934). 
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401 
