NOTES ON THE THEORY OF HEAVING AND PITCHING 
certainly most prominent when the period of encounter  _ the phase lags produced thereby; if there is no phase lag, 
is near one of the natural periods and, directly or 
indirectly, the phenomena are closely associated. The 
problem involves to some extent second-order terms 
and the analysis is therefore subject to correction by a 
more complete theory; but meantime we ignore the 
disturbance of the wave train by reflection and use the 
approximate equations for heaving and pitching as in 
the previous sections. The analysis is given in detail 
elsewhere (R.16) and a short account in the Appendix 
to the present paper. 
We calculate the force on the ship from the pressure in 
the undisturbed wave train; but, instead of taking the 
equilibrium position of the ship, we make the calculation 
for a displaced position, with a vertical displacement ¢ 
due to heave and a rotation 0 due to pitch. To the first 
order in ¢ and 6, the resultant force backwards is found 
to be (A.25) 
27,.0H 
eA aaeats Vt ean OE d 
In this, 2 z/p is the period of encounter with the waves 
and also the period of the forced oscillations; H and P 
are the buoyancy and pitching moment and are also of 
period 27/p. The first term Fo is the purely periodic 
horizontal force to which reference has been made earlier. 
Taking average values of the quadratic terms in the rest 
of (10) we obtain for the average steady resistance 
Re (a]A) Ho Co sin By + (aA) Po A sin Bo . (11) 
with Hop and Po the amplitudes of the buoyancy and 
pitching moment, f and 6 the amplitudes of the forced 
heaving and pitching, and f,; and f2 the phase lags of 
the oscillations. 
It is of interest to recall the history of the similar 
problem in rolling. In 1924 Suyehiro (R.17), experi- 
menting with a small model, measured a drifting force 
sideways on a ship when rolling in waves. The effect is 
small and probably is only appreciable in suitable con- 
ditions of forced rolling in resonance with the natural 
period of roll. Suyehiro himself ascribed the force to 
reflection of the waves by the side of the ship; however 
no calculations have been made of the magnitude of 
such an effect. In 1938 an alternative theory was put 
forward by Watanabé (R.18). Starting from the Kmloff 
equations, Watanabé deduced an expression for the 
drifting force involving the angle of roll and the phase 
lag between the roll and the actuating moment; applied 
to Suyehiro’s model, this expression gave a force of 
rather more than half the observed value. 
Returning to (11), consider the various factors when 
making numerical comparison with observed results. 
The values of Ho and Po have to be taken from such 
calculations of buoyancy and pitching moment as can 
be made for any given form. The amplitudes Co and 4% 
we shall take from observed results, assuming, as is 
necessary, that these are for forced oscillations. The 
most uncertain factors are the phase lags. It will be 
noticed that these are important in that on the present 
yiew the extra resistance arises from the damping and 
Dp - OP 
(10) 
517 
there is no resultant steady force. Reference has been 
made to the diagram given by Kent and Cutland (R.5) 
from which the phase lag for pitching might be deduced. 
It is not suitable for the present purpose, however the 
attempt may be made so as to obtain some idea of the 
magnitude of the resistance given by (11). 
If we take the results in waves of 490 ft. in length, the 
diagram shows that for zero speed of the model there 
was no appreciable phase lag. Hence, according to (11) 
there should be no resistance; and, in fact, the measured 
resistance under those conditions was very small. 
Incidentally the observed results also confirm the view 
that resistance due to wave reflection must be very small 
when the wave-length becomes greater than the length 
of the ship. — 
If we take next the same wave-length with a speed of 
8 knots for the ship, a rough estimate from the position 
of the wave trough gives a phase lag for pitching of 
about 12-5 deg. We shall assume the same value for 
heaving, and we take 8; = B2 = 12-5 deg. For Hp and 
Po we take the wall-sided ship with elliptical horizontal 
section which has been used in the earlier calculations. 
With L = 400 ft., B = 55 ft., d= 20 ft., A = 490 ft., 
and in waves of height 2r =5ft., we obtain from 
A. 16 and 18 
H,=358 tons; P, = 67,633 ft.-tons 
The observed measurements in 5-ft. waves give 
Co = 2-1 ft. and 0% =1-6 deg., approximately. With 
these values we get from (11) a resistance R = 3-66 tons, 
of which about 1 ton comes from the term in the heaving 
motion. From the given results in the same paper, the 
measured resistance for the 16-ft. model was 0-37 lb. 
or a resistance of 2:58 tons for the full-sized ship. The 
measure of agreement is perhaps as much as could be 
expected considering the uncertainty of the data and 
also that no special attempt has been made to calculate 
values for the particular model used in the experiments. 
It is not worth while adding further similar calculations 
at the present stage; but it may be said that the suggested 
theory is capable of giving results of the right order 
of magnitude. 
On the theoretical side, it is hoped that the various 
limitations and assumptions have been sufficiently 
indicated. On the experimental side, there is a lack of 
suitable data obtained under conditions approximating 
to the simplifications which have to be made before any 
calculations are possible; such experimental results 
would be a valuable and, indeed, essential aid in develop- 
ing and modifying any tentative theory of such a 
complex problem. 
Appendix 
(1) Damping in Smooth Water.—If a ship is making forced 
oscillations of heaving or pitching, we may calculate the wave 
motion by supposing each element of the ship’s surface to be 
the seat of an alternating source, say of strength mcos pt 
per unit area. Knowing the velocity potential of the distribu- 
