NOTES ON THE THEORY OF HEAVING AND PITCHING 
dynamical theory has been examined in a recent paper 
(R.15) to which reference may be made for details of the 
analysis. The ship problem is the reflection of the 
wave train by the ship, which is itself free to move and 
does take part to some extent in the motion of the 
surrounding water; it is in fact the dynamical problem 
of the motion of the complete system of ship and water. 
Leaving this on one side we consider the forces on a fixed 
obstacle in waves. The fundamental case is that of a 
regular train of waves incident normally upon a fixed 
vertical plane, which we may take of infinite draught. 
There is perfect reflection of the waves; if r is the ampli- 
tude (half wave height) of the incident train, there is an 
oscillation of amplitude 2r at the plane. The usual 
first order theory for waves of small height gives a 
periodic force of (g prAjfm)cospt for the additional 
force per unit width of the plane, A being the wave- 
length and 2 7/p the corresponding period. Carrying 
the theory to second-order terms, the result of the 
analysis is to give an additional average steady force on 
the plane amounting to 4g p77 per unit width. If the 
waves are incident at an angle « to the plane, the cor- 
responding average force is 4gpr?sin?a per unit 
width. An interesting problem would be the reflection 
of waves by a vertical cylinder of elliptical cross section, 
like the model used in the previous calculations of this 
paper; it is possible to obtain an analytical solution, 
but the functions involved have not been tabulated 
sufficiently to allow of numerical results. The correspond- 
ing work has been carried out for a vertical cylinder of 
circular cross-section, giving the variation of amplitude 
round the cylinder and the resultant steady force and 
the dependence of both these quantities on the wave- 
length. When the wave-length is small compared with 
the diameter of the cylinder, the resultant steady force 
approximates to the value 3gpr*a, where a is the 
radius. An interesting result shown by these calcula- 
tions is that this limiting value is practically attained so 
long as the wave-length is not greater than the diameter. 
We may obtain this limiting value by making an extreme 
assumption. Imagine the waves to be completely 
reflected by the front half of the cylinder, leaving smooth 
undisturbed water round the rear half. Then treat each 
element of the front half as if it were part of an infinite 
plane upon which the waves are incident at an angle «. 
On this assumption we should have for the resultant force 
a 
R=t¢p sin? ady (7) 
(44 
taken over the transverse diameter of the cylinder; and 
this gives the result 3 g pr’a. 
This suggests a similar expression for a vertical 
cylinder of any horizontal cross-section. With the 
extreme assumption of reflection round the front half 
and smooth water round the rest, it appears that the 
steady average force due to wave reflection should not 
exceed the amount 
R=t¢gprBsin?a (8) 
where B is the maximum beam, and the last factor is 
516 
the mean value of sin?« with respect to the beam, 
a being the angle which the tangent at any point makes 
with the fore- and aft- central axis. 
Kreitner (R.10) gives an expression which, in the 
present notation, is 
R=gprBsinag (9) 
In deriving this, it is apparently assumed that the 
average pressure on a plane can be calculated from the 
instantaneous value of the hydrostatic pressure due to 
the elevation of the water surface. When numerical 
values are obtained for ship forms, the general result is 
that the expression (8) gives about one-quarter or 
one-fifth of the value given by (9). 
If we take the elliptical model used in the previous 
sections, an expression for the mean value of sin? « can 
be readily obtained; with L = 400 ft., B = 55 ft., the 
value of this factor is 0-183. In waves of 5 ft. in height, 
(8) then gives an extra resistance of about 0-9 ton. 
With a normal ship form with moderate bow angle, the 
mean value of sin? « would be about 0-1, reducing the 
extra resistance by this calculation to about 4 ton. The 
observed extra resistance for a ship of that type would be, 
on the average, about 2% tons. 
It should be noted again that the expression (8) is put 
forward only as an outside limit for a fixed obstacle of 
great draught. In the actual problem the ship is free 
to respond to the wave motion; further, unless the 
wave-length is very much less than the length of the ship, 
the finite draught of the ship seems likely to reduce the 
amount of the reflection effect. The general conclusion, 
so far as the present calculations go, is that, while wave 
reflection is a true contributory cause and must be in- 
cluded in a complete theory, it is only capable of 
accounting for a fraction of the observed extra resistance; 
we must, however, add the reservation that forward 
motion of the ship through the waves might modify 
that conclusion. 
A possible application of the formula (8) would be to 
determine the mean pull on the mooring rope of a ship 
subjected to waves which are short in comparison with 
the length of the ship. This has been investigated by 
Kent and Cutland (R.5) and details of the comparison 
with model results will be found in that paper. The 
experimental conditions most nearly approximating to 
the theoretical assumptions were for a 16-ft. model 
moored in waves of 7 ft. in length; the height of the 
waves was given values ranging from 0-12 ft. to 0-32 ft. 
It was found that, on the average, the value calculated 
from (8) was about 56 per cent of the observed mean pull. 
Resistance associated with Heaving and Pitching.— 
Another possibility is suggested by the consideration that 
first-order effects which in themselves are purely periodic 
may, through phase differences, give rise to a steady 
additional resistance. Such a theory would associate 
the resistance directly with the oscillations of surging 
heaving and pitching—though it is probable that the 
first of these plays only a minor part. There are different 
views of the extent to which the resistance depends upon 
the heaving and pitching motions; but the effect is 
