NOTES ON THE THEORY OF HEAVING AND PITCHING 
ment of 1-41. This is a very high degree of damping 
compared, for instance, with rolling. It seems probable 
that any numerical estimates have been deduced from 
resonance curves under forced heaving. The only 
published estimate appears to be that given by Horn (R.6). 
It is stated that the result of observations on various 
models gave an average value of 0-45 for the quantity 
NT/2 7M, in the present notation, or a logarithmic 
decrement of 1-41; it is also stated that the corresponding 
damping coefficient for pitching was of the same order. 
For pitching, calculations for the same model from 
A. 10 and 11 are shown in Table II. 
TABLE II 
N,/l 
SOM IUDAN 
cooooo 
BRUAMAWwWl 
NRK OW 
— 
For an appropriate value of I we use data from a 
model of Kent and Cutland (R.5), to which reference 
has already been made; this was a cargo ship 400 ft. 
x 55 ft. x 24 ft. of 11,332 tons, with a longitudinal 
GM = 467 ft. and a natural resisted pitching period 
of 6-2sec. Using I = T? W m/4 7, we get an effective 
moment of inertia I= 11-85 x 10°. It may be noted 
that this gives an effective radius of gyration of 0-31 L, 
which seems about the right value. With this value of I, 
the third column in Table II gives the values of N;/I. 
We notice the striking similarity in the values of N,/M 
and N>2/I in Tables I and II, with some interesting 
differences in detail. This agrees with the statement that 
the damping coefficients for heaving and pitching are 
of the same order. For a period of between 6 and 7 sec. 
Table II gives a logarithmic decrement of about 1-16. 
The form of model used for these calculations was 
chosen for simplicity to give the order of magnitude of 
the effect. The work could be carried out in detail for 
any form given by mathematical equations, with the 
corresponding source distribution over the surface; but 
such calculations are hardly worth while meantime, or 
at least not without corresponding experimental work 
on simplified forms specially arranged to test and develop 
the theory. 
Oscillations among Waves 
If, instead of being in smooth water, the ship is 
subject to the action of a regular train of waves, there 
are many new factors which should be taken into account: 
for instance, the disturbance of the wave train by reflec- 
tion from the ship, and the wave system produced by the 
forward motion of the ship. The hydrodynamic forces 
acting on the ship will no doubt affect the amount of 
damping and may alter the effective periods of pitch 
and heave. A first approximation involves neglecting 
514 
these complications and evaluating the forces on the ship 
from the pressures in the undisturbed train of waves. 
This was the simplification adopted by W. Froude in his 
theory of rolling, and it was also the basis of the well- 
known work of Kriloff on pitching and heaving. The 
conventional method is to suppose the ship held in its 
equilibrium position and to calculate the excess or defect 
of buoyancy and its moment from hydrostatic pressures due 
to the instantaneous position of the wave profile relative 
to the ship. We confine the discussion in this section to 
the first approximation, but we calculate the forces and 
couples directly from the pressure system in a regular 
train of simple harmonic waves. Reference may be 
made to A.§2, where results are obtained for the par- 
ticular model we are using, a wall-sided ship with 
elliptical horizontal section. For this model, equa- 
tions (1) and (2) for smooth water are replaced by 
ME+NiE +g pS6=Hocospt 
16+ No6+Wmd= — Posinpt . 
with Ho, Po given by A.16 and 18. 
(4) 
(5) 
The forced oscillations are then 
€ = {ocos (pt — B;); 8 = — Msin(pt —f2) (6) 
with fo, 9, B:, B2 given by A.21. 
In attempting any comparison with observed results, 
it must be remembered that the expressions have been 
obtained from a very simple form of model. In general, 
model results are for forms not readily adapted to 
mathematical calculation, and moreover there are other 
factors arising from lack of symmetry fore and aft; 
in particular, if the centre of buoyancy is fore or aft of 
the centre of flotation there is coupling between heaving 
and pitching. 
We make numerical calculations for the dimensions 
used in the previous section: L = 400 ft., B = 55 ft., 
d= 20 ft., and we take the wave height 2r = 5 ft. 
There are various possible methods of exhibiting the 
results. We choose that of graphing the total pitch 
2 @ on the period of encounter as a base, in each case 
for a given speed of the ship. Thus fora given period of 
encounter at a given speed we find the corresponding 
wave-length A, and then the value of Po from A.18. 
For the effective moment of inertia I we take the value 
used in the previous section and also the same natural 
period 6:2 sec. Further we obtain the corresponding 
value of N2 p/I from the data given in Table II. 
In Fig. 1 the two curves show the graphs for the ship 
at rest and for a speed of 8 knots. The double humps 
on these curves are of interest; they arise because not, 
only does the magnification factor have a maximum at 
resonance but the pitching moment Po has maxima 
depending upon the wave-length. This effect can be 
clearly seen in some curves of results from models; in 
particular, reference may be made to Kent (R.12), Fig. 3, 
and (R.13), Fig. 3. Of course in actual model results 
there would be no definite zeros of the pitching; the 
curves would be smoothed out by viscous and other 
