Ship Waves 419 
The reduction factor f(A) is to mean a diminution of effective velocity 
from the value c at the bow to a smaller value @c at the stern. In order 
to allow the calculations to be made in terms of known functions, we 
shall suppose the diminution to take place uniformly over the front half 
of the model; thus we assume 
f(A) "Ba (ab) is SO GhiG 
Using (5) and (4) in (3) and carrying out the integration with respect to 
h, we obtain 
Db (e , @ Adk 
Sy EN gece tO rg Gu | iets ihe es ee ee SED 6 
: sal ee cemene anes nate 0) (6) 
where 
A = {2(1 — 8) sec? 6 + (2 — 8) ix] sec 0 — 12/2} ete LD cose+y sin] 
—= 9» (1 ee B) sec2 8 ex (cos 6-+-y sin @) 
= (iB«l sec 0 -++ Bx?/?) ex {(@+l) cos 6+y sin 6] (7) 
This expression gives finite and continuous values for the surface elevation. 
It is convenient, for purposes of calculation, to separate it into finite 
and continuous expressions associated respectively with the bow (x = /), 
amidships (x = 0), and the stern (x = —/). Further, for points on the 
central line y = 0, we can express these in terms of known functions. 
Writing 
Cr oO aK cos é 
G@ =i 0 d0 eee AGES 8 
@ "| a ( Kk — ky sec? 8 + ip sec 0 (8) 
=n 
Gy (q) = Ks Qa GA@g= | ‘Go (@) a4, (9) 
and so on, it can readily be shown that (6) gives, for the wave profile 
along y = 0, 
t= — 7. (PGy(e— 1) + 2— BIG, @—) +201 — 8) Ge —D) 
ae AU CnC) = Caesar) Gn esis (tO) 
In the limit, when we take p zero, we have* 
G(q)=7{Hy(k.g)—Y o(kog)}, g>0 
= —1}H,(ko@)—Y¥ q(kq9)}-40?Y,(k 9g), g<0. - (il) 
** Proc. Roy. Soc.,’ A, vol. 135, p. 5 (1932). 
400 
