NOTES ON THE THEORY OF HEAVING AND PITCHING 
effects. On Fig. | are also shown values extracted from 
model results given by Kent and Cutland (R.5); these 
results were for wave-lengths of 175, 350 and 490 ft., 
in waves of 5 ft. in height. It should be noted that no 
attempt has been made to fit this model beyond taking 
the main dimensions and displacement about the same. 
The points marked by a cross are for zero speed of 
advance, and they fit fairly well into the calculated curve. 
Points marked by a circle are for a speed of 8 knots. 
In the calculated curve for 8 knots we have used the 
same natural pitching period as for zero speed. There 
seems to be some evidence that the effective natural 
period increases with the speed. The large divergence 
MAXIMUM PITCH IN DEGREES 
6 7 8 
PERIOD OF ENCOUNTER IN SECONDS 
Fic. 1 
at the lowest period of encounter at 8 knots may be due 
to various causes; for one thing the calculated pitching 
moment Po is more subject to error at the smaller wave- 
lengths, and for another it is probable that the pitching 
in the smaller wave-lengths is not the simple forced 
pitching to which the calculations refer. 
Similar graphs could be made for heaving, but it 
should be remarked that observed maxima in long waves 
are generally greater than those given by calculation. 
This has been noted previously in regard to model 
results; it may be that the calculated buoyancy is more 
susceptible to change in wave-length or possibly that in 
long waves the damping is less—it might, for instance, 
be a better approximation in such cases to calculate the 
damping from the motion of the ship relative to the 
fluid motion in the wave train. Finally, in this brief 
review we may consider the phase lags for heave and 
pitch denoted by the angles B; and f>2 in (6). It is a 
simple matter, so far as the approximate equations are 
concerned, to determine the position of the ship in 
relation to the wave profile at maximum pitch or heave. 
This is, of course, a very important point. It has been 
examined by J. L. Kent in various papers; and, in 
particular, in Kent and Cutland (R.5) a diagram is given 
showing the wave crest and trough positions along the 
ship at maximum pitch. It is difficult to derive from this 
diagram results suitable for the present calculations. 
The model was not designed for the purpose; moreover, 
it is stated that successive pitches showed a periodic 
movement of the wave crest position backwards and 
forwards along the hull, the diagram giving mean 
515 
positions at the instant of lowest pitch. Referring to (6), 
there is maximum pitch with bows down when 
pt—fB2.=7/2. From A.13, it follows that the wave 
profile relative to the ship at that instant is given by 
¢ = —rsin(kx +2). Hence there is a trough at a 
distance } A — B2 A/2 7 ahead of amidships. 
If T, is the period of encounter and T, the natural 
period of pitch, and if damping were entirely neglected, 
we should have 8; = 0 for T,> T,, and B, = 7 for 
T, <T,. In the former case there is a trough 4A 
ahead of amidships and in the latter it is 4A astern of 
amidships. The damping smooths off this sudden 
change of phase; but whatever the damping we should 
have 6, = 7/2 for T, =T,. Hence there should be a 
trough at amidships, for a simple symmetrical model, 
at lowest pitch at the resonance period of encounter. 
In the diagram referred to above, there is a trough 
amidships for zero speed of advance at a wave-length 
of about 230 ft.; this corresponds to a period of en- 
counter of 6-77 sec., the natural period for the model 
being 6:2 sec. But, for various reasons, it is not possible 
to push the comparison so far as to determine the phase 
lags. The possible magnitude of surging effects, for 
instance, needs examination; and in various respects the 
theory is only a first approximation and requires 
amplification in conjunction with suitable experimental 
data. 
Resistance of a Ship among Waves 
A ship when moving through a regular train of waves 
is subject to an average steady resistance greater than 
that experienced at the same speed in smooth water. 
There are various obvious factors which may be supposed 
to contribute to this result: for instance, the disturbance 
of the wave motion by the surface of the ship, the 
alteration in the wave resistance due to interference with 
the wave train or due to altering attitude of the ship, 
or a more direct effect of the surging, heaving and 
pitching motions. 
If we consider only the first order approximate 
equations used in the previous sections, the regular wave 
train supplies an alternating addition to the resistance, 
such as that given in A.17; a more detailed examination 
of this periodic force may be found in R.14. 
In order to obtain an increased average resistance we 
have to take into account second order terms. When 
we are dealing with first order effects it is, generally, 
legitimate to consider factors separately and obtain the 
combined result by simple superposition; but this is not 
the case when we have to include second-order terms. 
Noting, however, that a complete theory including all 
second-order effects might well modify partial results, 
we shall examine two possible factors which lead to 
increased average resistance. 
Wave Reflection—The first possibility is the effect of 
reflection, or scattering, of the regular wave train by the 
surface of the ship, and this is undoubtedly a true con- 
tributing cause. It has been put forward recently as the 
sole basis of the extra resistance in a very interesting 
paper by Kreitner (R.10). The underlying hydro- 
