NOTES ON THE THEORY OF HEAVING AND PITCHING 
be made, for instance, to G. S. Baker (R.4). We may 
examine this in a few cases from the point of view of 
the same approximate basic period 2 7 »/(d/g) for both 
heaving and pitching. 
With data from Kent andCutland(R.5) for a cargo ship 
of 400 ft. x 55 ft. x 24 ft., we take the mean uniform 
draught as 21-5 ft. This gives a basic period of 5-13 sec. 
The natural resisted periods of pitch and heave are given 
as 6:20 and 7-42 sec. respectively; taking the ratio of 
each of these to the basic period and squaring, we get 
the corresponding added moment of inertia and added 
mass, namely about 46 per cent and 100 per cent 
respectively. 
Similarly, from the details given for the motor ship 
San Francisco (R.6) with a mean draught of 22 ft. the 
basic period is 5-19 sec. The observed periods of pitch 
and heave are given as 6:51 and 7:34 sec.; and we deduce 
corresponding inertia increments of 57 and 100 per cent. 
For a different type, a fast ship 400 ft. « 48 ft. x 13 ft., 
we have data taken from Kent and Cutland (R.7). The 
mean uniform draught of 10-5 ft. gives a basic period 
of 3-59 sec. From resonance effects in rough water 
the natural resisted periods of pitching and heaving were 
assumed to be approximately 5-4and 5-8 sec. Accepting 
these values, we get an increase of moment of inertia of 
about 125 per cent, and of mass of about 160 per cent. 
These values seem too high, though increased values 
would naturally be expected from the greater ratio of 
beam to draught. 
Leaving aside the approximation in using the same 
basic period for both pitching and heaving, the total 
effective mass and moment of inertia can, of course, be 
calculated if we know the requisite data and the observed 
periods; for from (1) and (2) we have M = g pS 13/4 x 
and I= m W T2/4 7?. 
Damping.—We consider now the second term in 
equations (1) and (2), representing the damping of the 
natural oscillations. This arises partly from frictional 
effects and partly from energy lost in the wave motion 
produced by the oscillation. In order to evaluate the 
latter contribution we ignore for the present all effects 
due to viscosity. In the problem of rolling the associa- 
tion of damping with wave motion has been familiar 
since the time of W. Froude. Some recent calcula- 
tions (R.8) have shown that it is certainly capable of 
accounting for a large proportion of the observed 
damping for a ship with zero speed of advance. The 
rolling problem is simpler than that of heaving and 
pitching in that the damping is small; on the other hand, 
it is more difficult to calculate the wave motion directly 
in terms of the form of the ship. 
For damping due to heaving, reference may be made 
to some small-scale experimental studies. In particular, 
Schuler (R.9) examined the waves produced by a prism 
making vertical oscillations, and, among other results, 
deduced that the damping was due to wave motion, 
viscous and other damping being negligible in com- 
parison. In theapplication to ship motion, Kreitner (R.10) 
has emphasized the importance of this kind of damping 
in heaving and pitching. 
513 
Calculations of the magnitude of this effect have been 
given in a recent paper (R.11), and also in the Appendix 
to the present notes. 
Suppose the ship is acted on by a periodic force, 
say Ho cos pt, so that it is making forced heaving of 
period 27/p. We could write the equation of motion 
in the form 
Mf+ ¢pS6=X+ Hocospt (3) 
where we consider X as the vertical resultant of the 
additional fluid pressures due to the wave motion. The 
assumption is that if X could be calculated it would be 
a resistance proportional to the velocity ¢ and could be 
transferred to the other side of the equation and be the 
term N, ¢ as in equation (1). Meantime we can only 
evaluate N, by indirect methods. The impressed force 
Ho cos p t does work at a rate just sufficient to maintain 
the forced oscillations; if the latter are of amplitude fo, 
this mean rate of work is }p?N; @. This is equated 
to the mean rate at which energy is propagated outwards 
in the wave motion, and so we obtain an expression 
for N;. To determine the wave motion we replace the 
ship by a suitable distribution of alternating sources 
over its surface and hence deduce an expression for the 
mean rate of outflow of energy (A.1 and 2).* The same 
argument applies to the pitching motion with reference 
to the forced oscillations, and we derive an expression 
for the corresponding factor N2. Calculations have been 
made for a simplified form of ship; wall-sided, of 
length L, beam B, of constant draught d, the horizontal 
sections being the same and elliptical in shape. The 
expressions for N, and N2 are given in A. 5, 6, 10 and 11. 
For numerical values we take L = 400 ft., B = 55 ft., 
d= 20 ft.; these dimensions giving a rough corre- 
spondence with a cargo ship of about 10,000 tons dis- 
placement. Calculations for N; from A.5 and 6 have 
been made for six different values of the period T = 2 m/p 
and the results are shown in Table I in |b.-ft.-sec. units, 
the Ib. being the unit of force. 
TABLE I 
N, x 10-6 
SCOAmMANAMN 
— 
For the values of N,/M in the third column, we 
have assumed an added mass of 90 per cent and have 
taken the effective mass M to be 19,000 tons. If the 
ship were heaving in a natural damped motion of 
period T, the logarithmic decrement of the motion 
would be given by N,1T/2M. If, for instance, the 
natural period is 7 sec., then taking the corresponding 
value from Table I, we should get a logarithmic decre- 
* A. refers to the appendix, and R. to the list of references. 
