Oe = con 0.0| 
TT 
33 Wave Resistance. 
This satisfies the condition d¢/dz = 0 at z = — h; we note that the first term 
represents the original doublet and its image in the bed im an analytical form 
valid for z+ f > 0, and therefore suitable for applying the boundary con- 
dition (2) at the free surface. This yields 
2e—*" cosh « (h — f) Kk + kK, sec? 0 + ip sec 8 
F (0 = EE Ne, ee ee, (CB 
(8, ) cosh Kh kK — Ky sec? § tanh xh + 2p sec 8 (Se) 
consequently the required surface values are given by the real parts of 
aes a a a0 e—*" cosh {«(—f)} (+tanh + £)) ice. g 
Ky 
Te 0 K — Ky Sec” 6 tanh ch +i sec 0 
Su 
hee) 
e— "cosh {« (h — f)} (1 + tanh ch) (+ tp sec 8) ies ede, (34) 
0 K — Ky sec? @ tanh kh + iu sec 0 
where o = x cos 0+ ysin 0. 
Comparing these with the corresponding values for deep water given in (16) 
and (17), we can write down the expression for the wave resistance as 
: pr ”8e-2 xh cosh2{x(h—f)} (1+tanh Kh)? 
R = Lim 160x,M2 10) Siete LCOSMENS ze sa sta aC) 
n> 0 ao “|, Jo © (k—ko sec? 6 tanh xh)?2+ 2 sec? 0 
(35) 
There are two points to notice in evaluating this limit. The result is only 
different from zero when 
Kk — kK, sec? 0 tanh kh = 0 (36) 
has a real positive root ; and this occurs only for «gh sec? 06> 1. Further we 
must introduce in the denominator d.(« — ky sec? 0 tanh xh) /dx. We may sum 
up the result in this form 
7/2 «3e—-2«Kh cosh2{x(h—f)} (1+tanh xh)? 
& 1—xoh sec? 6 sech?2 kh cosh 6 dé, (37) 
R= 6rpyM? | 
where « is the positive root of (386); further, the lower limit 0p, is given by 
OQ, =O, ror p>, or & <a, 
6, = are cos +/ (kh), for c?> gh. 
We may note that the change in the lower limit occurs at the so-called critical 
velocity ~/ (gh) for the given depth. From (37), R may be graphed as a function 
of the velocity for various ratios of f to h; the calculations may be carried 
out by numerical and graphical methods. A similar expression in the case of 
a certain distribution of surface pressure was examined in detail in a previous 
paper,* and it may be anticipated that (37) would give somewhat similar 
curves. 
* “ Roy. Soc. Proc.,’ A, vol. 100, p. 503 (1922). 
HaRRIson AnD Sons, Ltd., Printers in Ordinary to His Majesty, St. Martin’s Lane. 
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