672 Prof. T. H. Havelock on the Damping of the 
8. We may now attempt to apply these results to the pitching motion 
of a ship. For a long narrow ship the appropriate source distribution 
could be taken over the longitudinal vertical section of the ship as in 
the theory of wave resistance for such forms. On the other hand, the 
keel of the ship may play a large part in wave production in pitching. 
As a suitable example for calculation, we choose a rectangular form with 
vertical sides, of length 27 and beam 26, and floating immersed to a 
depth f; such a form will clearly exaggerate the wave-making effects 
of bow and stern. We suppose the form to have an angular pitching 
oscillation given by 0=6, sin pt. We neglect the effect of the vertical 
ends and consider only the flat base. With the present procedure, we 
take the source strength at each point such that 47m cos pt is equal to 
the normal velocity, that is, equal to x’ in the notation of (22). Hence. 
from (22), we have in this case 
290 (! ® hae eek 
P+iq—ho ae | wletoftiva(e’cost+y' sin) Jy? | |, (27) 
J = =i 
__ tpg sin (Ky D sin) , . a an “ep 
= ae Ga {sin (9! cos 0)—K pl Cos @ cox (Kol cos A) }e 7, 
(28) 
From (23) this gives 
Aa 
15 
| 
sin? («gb sin @) 
Fa atncaeHe {sin (xl cos) —k 9! cos @ cos (il cos A) }* dd. 
0 hs JIE 
(29) 
Ko? 
= se ea] 
For the pitching of a ship, as for heaving, «,b !s a moderately small 
quantity ; (29) then reduces to a simpler form, which might have been 
derived directly by assuming a line distribution of sources and sinks. 
We have then 
r 
bo 
Spp°b?6,° dé 
Oo © . 
= alee eof | {sin (cl Cos 6)—x gl cos @ cos (Kgl cos #)}2 —_. 
TK ¢: Ja cos! 
(30) 
EK 
For pitching oscillations of a ship, the usual equation for natural 
pitching in still water is 
I6+Né+gpVmd=0, . ..... . (31) 
where I is the total effective moment of inertia of the ship, V the displaced 
volume, and m the longitudinal metacentric height. As before, we 
estimate N by equating the mean value of N6? to the value of E given by 
(30), with p equal to the natural frequency o. There do not seem to be 
any direct determinations or calculations for the added moment of 
inertia. We shall therefore derive the effective value of T from the 
relation 
o*l=gpVm, WA) MOT on Wy Anta a eon am (D2) 
498 
