2 THE DAMPING OF 
might, for practical purposes, be taken as unity if o? L/g exceeds 
about 8. 
In addition to this approximation to unity above these 
respective values of o” L/g, a specially interesting feature of the 
curves is the rapid fall in both ratios for smaller values of the 
parameter. 
It has already been remarked that these results can only be 
taken as suggestive when applied to surface ships; however, it is 
of interest to see what are the relevant ranges of the parameter 
in such cases, referring in particular to work in which the damping 
coefficients have been calculated by the strip method over a 
range of frequencies. 
For free oscillations at the natural frequencies, there are data 
in the paper by St. Denis) for a ship of length 600 ft. and beam 
81 ft. The values of o for free oscillations are given as 0-706 
and 0:82! for heaving and pitching respectively. The corre- 
sponding values of o? L/g are 10 and 12-6 and these both lie 
within the ranges given above where no correction factor is 
needed. A similar remark would, no doubt, apply to the experi- 
ments with 5 ft. models used by Korvin-Kroukovsky, although 
the natural frequencies do not seem to be given in the paper. 
However, the present work may be taken to confirm his experi- 
mental result that for natural oscillations the two-dimensional 
calculation does not require any appreciable correcting factor 
either for heaving or for pitching. 
For forced oscillations we have a wider range of frequencies. 
In work for which data are available, the heaving and pitching 
are produced by driving the model at given speed through 
regular waves of given wavelength. For instance, from St. 
Denis, 2) for a 600-ft. ship moving against waves with A/L =1-25, 
o ranges from 0-518 at zero speed to 0-77 at 30 knots; thus 
o? L/g ranges from 5 to 11. From the curves in Fig. 1, heaving 
may be said to require no correcting factor; but for pitching, the 
lower values are well within the critical range where a large 
correction is needed and where it changes rapidly. 
There are similar data from Korvin-Kroukovsky™ for 5-ft. 
models. With one model and A/L = 1 the parameter ranges 
from 6-3 to 20, and with another model and A/L = 1-5, it ranges 
from 4-3 to 12-5. Here again the pitching calculation seems to 
require considerable correction at the lower speeds. 
It should be noted again that one can only expect general 
indications in applying the present results to surface ships. For 
one thing, a spheroid is not a normal ship form. A more 
important point is that the flow round a completely submerged 
solid may differ considerably from that round a floating body. 
However, it is possible that the strip method and the three- 
dimensional calculation might be affected in much the same way; 
if so, the ratios for the two methods may not be so far astray. 
Finally, in all calculations for forced oscillations due to 
advancing through waves, it is assumed that the only effect of 
the speed is to alter the frequency of encounter. But a satis- 
factory theory of heaving and pitching including the effect of 
speed of advance, for anything like a normal ship form, is one 
of the main outstanding problems. The corresponding theory 
for a wholly submerged body might prove more tractable, and 
it may be possible later to extend the present work to a sub- 
merged spheroid which is moving forward while making heaving 
and pitching oscillations. 
APPENDIX 
1. The underlying theory was given in a previous paper© for 
a source distribution; it is convenient to give now explicit 
expressions for a distribution of vertical dipoles. 
Take the origin O in the free surface of the water, with O x 
and Oy horizontal and O z vertically downwards. If there is 
a vertical dipole of moment M cos of at the point (A, k, f) in 
the water, the velocity potential of the fluid motion is given by 
HEAVE AND PITCH 
¢=Meocosct z 
oo 
= J @ 
—_ a ant 2 Ky 706) nee die 
vai We} 0 
0 
—27KiM Jo (ky @’) e~ E+ sin o t 
(1) 
where 
R= (x—h?+0-b?+E- 
B=D2+E4N% aK 
The motion as w’ —> © is given by 
b> — 27M Q/rky @)'2e—K0E+/ sin (o t — ky @ + 7/4) 
(2) 
representing circular waves travelling outwards. For a given 
distribution of vertical dipoles all at the same depth f, we obtain 
the velocity potential from (1) or from (2) by integrating with 
respect to A and k over the given distribution. The rate of flow 
of energy outwards through a vertical cylindrical surface of 
radius @ is given by the rate of work of the fluid pressure over 
this surface, namely 
o) 27 
og dg_ 
-| az| p3! sane 
tt) t) 
Taking the radius of the cylinder large, we only need ¢ to the 
order w—!/2 as @ — 00. 
If in (2) we put 
2-32 +@—fP 
(3) 
x=a@cos@: y=a@sin8@; 
w’2 = @2—2ha@cos6—2kosinO + h?2 + k? 
then, to the required order, (2) gives 
b> — 273 M( z 
TI Ky @ 
1/2 
) e—kole+/) 
sin (or— Ky @ +7 + ko h cos 8 + kok sin) (4) 
Hence, for a given distribution, we shall have 
$ > @—1/2 e-eaciP] A sin (¢ t—kK)@ + ) 
+ Beos (01 —xyo +) | (5) 
with A, B known functions of 0. 
Putting this form into (3) and taking the mean value with 
respect to the time, we get for the mean rate of flow of energy 
outwards 
(6) 
Finally, inserting the forms for A and B obtained by integrating 
(4) over the given distribution, 
E =tpo(A2 + B?) 
Qn 
B= 2apaxgente | (P2 + Q2)d@ (7) 
0 
(8) 
In the present work, we shall not need any more general expres- 
sions, but an obvious extension would give similar results for any 
distribution of dipoles not necessarily in a horizontal plane. 
2. Consider now a spheroid, of length 2a and equatorial 
diameter 2 6, immersed with its axis horizontal and at a depth f 
below the free surface. Suppose the spheroid made to describe 
small vertical oscillations, the velocity at any instant being 
Vcosof. It would be possible, theoretically, to proceed step- 
with P+iQ =| M(h, k) ei Kolhcos0 + ksin) dh dk 
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