Calculation of Wave Resistance. 520 
We have, for example, 
0 fo) 
| ae | ey dy 
—h —o ha 
ue ds (ihe ae(" (f (8)}2 cosh? x (z + h) (coth «ch — rh cosech? xh) 
us Kg sinh? Kh (1 + sin? 0 — koh sech? kh) 
x sin? («a’ cos @) x2 cos? 6 d0 
coth? xh — «*h? cosech! ch 
i (ca os O) eon! Od, 
o(l + sin? 6 — x gh sech? xh) 
(34) 
= ret [" (Ff (O)}? - 
Evaluating the remaining terms in (25), we obtain after a little reduction the 
result 
$1 
ee np0*| {f (8)}2 (coth xh — xh cosech? xh) cos® 6 dé, (35) 
0 
with « given in terms of 0 by (31). 
This may be compared with (17) for the similar wave pattern in deep water. 
For a horizontal doublet M at depth f in water of depth h, an expression for 
the complete surface elevation can be derived from results given previously.* 
We have 
— M I, ( cosh x (h —f) gir (2’ cos 0-+ysin 8) 
re J _» 
Ee J ee id 36 
Jo cosh xh ( — x, sec? 0 tanh xh - ip sec 6) Sie 
where we take the limiting value of the real part for » > 0. 
From this we may easily deduce the free wave pattern to which the dis- 
turbance approximates at a great distance in the rear. It is given by 
4x 2M (#" coshe« (h —f) tanh? xh sect 0 ; : 
= — g i) 6)} dé. 
‘ c | _3, cosh Kh (1 — xgh sec? 8 sech? xh) Sn (e(e nce ery ne) 
(37) 
From (35) this gives 
chr 2 = 
ne 16M | peleas Weel re (0 =) ap, (38) 
9 cosh? kh (1 — Kh sec? 6 sech? xh) 
It will be found that this agrees with the result obtained by a different 
method in the paper just quoted, when the previous expression is corrected 
for an obvious slip ; in formula (37) of that paper 32 should be replaced by 16 
and tanh «ch (1 + tanh xh) by (1 + tanh «h)?. 
**Proc. Roy. Soc.,’ A., vol. 118, p. 33 (1928). [This paper is No. 22 of this 
collection and the error mentioned above has been corrected. —Editor. | 
396 
