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NOTES ON THE THEORY OF HEAVING AND PITCHING 
By Professor T. H. HAveLock, M.A., D.Sc., F.R.S., Honorary Member.* 
Summary 
The main points in the paper are (i) a calculation of the 
damping of heaving and pitching due to the waves produced 
by the motion of the ship, (ii) an examination of the extra 
resistance caused by the reflection of a regular train of waves 
by the ship’s surface, (iii) a suggested theory which gives an 
extra resistance more closely associated with the heaving and 
pitching motions. 
No attempt is made to formulate a complete theory; the 
work is based, in the main, on the usual approximate first- 
order equations of motion and the hydrodynamical theory is 
that of potential fluid motion under gravity and neglecting 
viscosity. Details of mathematical analysis are given in an 
appendix, and the paper gives an account of the work together 
with numerical calculations and comparison with experi- 
mental data. 
Oscillations in Smooth Water 
The usual approximate equations for heaving and 
pitching in smooth water are 
ME+Ni6+g¢pSl=0 (1) 
16+N,6+Wm6=0 (2) 
In these equations € = upward displacement of the 
centre of gravity G, 0 = angle of pitch about the trans- 
verse axis through G measured positive with bows up, 
S = area of water plane section, W = g p V = displace- 
ment in the equilibrium position, m = longitudinal 
metacentric height. Further, it is assumed that the ship 
has a simple symmetrical form so that there is no 
coupling between heaving and pitching so far as first- 
order equations are concerned. Nj and Ng are coef- 
ficients which are considered later. 
Effective Mass and Moment of Inertia.—It has been 
observed that the periods of heave and pitch in still 
water are approximately equal, and it is easily seen 
how this arises. Suppose at first that we neglect the 
damping terms in (1) and (2), and also ignore the effect 
of the inertia of the surrounding water. Then (1) gives 
for the period of heaving 2 7 Va/a), where d = ee 
mean uniform draught. Turning to equation (2), the 
longitudinal metacentric height is of the order of the 
length of the ship and a usual first approximation is 
to take 
m=GM=BM=SKH/V=Kyd 
where k is the radius of gyration of the water plane 
section about the transverse axis. 
If K is the radius of gyration of the ship about the 
transverse axis for pitching it can be seen that, at least 
for uniform loading, K? differs from k? by a quantity 
of the order of the square of the ratio of draught to 
* Professor of Mathematics, King’s College, Newcastle-on- Tyne. 
512 
length; thus, except for special types of form or mass 
distribution, we may take K? as approximately equal 
to k?. Hence in (2), we have I = W k2/g and m = KJ, 
and the result is the same approximate period 2 7 //(d/g) 
for pitching as for heaving. 
For mean uniform draught ranging from 20 ft. to 
30 ft., this means a period of from 5 sec. to 6 sec. The 
natural periods for usual types of cargo ship generally 
range from 6 sec. to 7 sec. The difference arises from 
two causes, damping and the inertia of the water. Even 
with large damping the effect on the period is com- 
paratively small, and practically all the difference is due 
to the inertia of the surrounding water. 
The calculation of added mass for heaving usually 
proceeds on the assumption that we may replace the 
immersed volume of the ship by a double ship wholly 
immersed in an infinite liquid; this underlies the work of 
F. M. Lewis (R.17) and of Browne, Moullin and Perkins 
{R.2). There do not seem to be any similar calculations for 
rotation, or any with direct application to ship forms. 
One remark may be made about such calculations for 
a floating body. A complete solution, satisfying the 
condition of constant pressure at the free surface of the 
water, would include wave motion of the water. Neg- 
lecting gravity there are two alternative assumptions for 
the surface condition, that it is either a rigid plane surface 
or an open surface of constant pressure. We might take 
the condition at the free surface to be zero normal 
velocity or zero tangential velocity. The calculations on 
added mass have, taken the latter condition. It is of 
interest to note that the former condition, of a rigid 
plane boundary, has been used by Brard (R.3) in work on 
the corresponding inertia effects in the rolling of a ship. 
In my view, the choice of appropriate boundary con- 
dition depends not only on the mode of motion of the 
ship, but also upon whether its oscillations are of short 
period or of long period. However that may be, the 
inertia coefficients in the present problems are generally 
estimated by indirect methods, or in effect by comparing 
observed periods with those calculated without allowing 
for the inertia of the water. The only difficulty that 
arises is that often the stated periods have not been 
directly observed, but have themselves been deduced 
indirectly. There is, however, general agreement that 
a normal value for the added mass for heaving would be 
from 80 to 100 per cent of the displacement, with even 
more for broad, shallow forms; while for pitching the 
added moment of inertia might be normally 40 to 50 per 
cent of the moment of inertia of the ship—reference may 
t References at end of paper. 
