NOTES ON THE THEORY OF HEAVING AND PITCHING 
tion of sources, it is possible to calculate the average rate at 
which energy is being propagated outwards in the wave 
motion. It has been shown (R.11) that this mean rate of 
outflow of energy is given by 
27 
E = 2 7p (p3/g)| (P2 + Q2)d0 . 
0 
qd) 
where 
P+iQ= [> (x, y, z) epile.(@ + ix cos® +iysin0) dS (2) 
the integral in (2) being taken over the immersed surface of 
the ship in its equilibrium position. The axis O x is taken 
along the longitudinal axis of the water plane section with 
the positive direction from stern to bow, O y transversely at 
the midship section, and O z vertically upwards. We shall 
assume the source strength at each point to be such that 4 7 m 
is the amplitude of the normal component of velocity of the 
ship’s surface at each point, this being a reasonable approxima- 
tion in view of usual ship dimensions. A further simplification 
may be made by neglecting the distribution in the transverse 
axis O y, since the length 2 7 g/p2 is usually several times as 
large as the beam of the ship. 
Suppose the ship to be wall-sided, of uniform draught d, 
and with the horizontal sections ellipses of axes L and B. 
Let the ship be making forced heaving oscillations of 
amplitude £0. and period 2 7/p. The source strength over the 
flat bottom is of amplitude p C/4 7 per unit area; and we 
treat it as a line distribution at constant depth d along the 
central line, with strength proportional to the beam at each 
point. Hence from (2) we have 
AL 
A _pD& = 4] 4 x? 2 x/g. cos 
P+iQ= = Be va (1 _ EN eip?x/g.cos® d x 
—4L 
=$p (,BLe-rdie J, (q cos 8)/(q cos 8) 
where q = p? L/2 ¢. 
Hence from (1) the mean rate of propagation of energy 
outwards is given by 
oe Boe L2 (2 p5 e202 #/@ 
We now equate this to the mean value of N, (2, namely 
+p? N, 2, and we obtain 
GQ) 
1(q Cos Wee a0. (4) 
qcos 6 
N,; = (7/4 g) B2 L2 p3 e~ 2 P*d/e F, (5) 
where F, has been written for the integral in (4). 
This integral may be evaluated by quadrature using tables 
of Bessel functions. It was, however, found more satisfactory 
to calculate it from an equivalent series. It can be shown that 
sa (=l)™ (2m)! (2 m+2)! 
8 2 (m!)? {+i}? (m+2)!*4 
Similarly, if the ship is making forced pitching oscillations of 
angular amplitude 4, and period 2 z/p we have 
© 
Go". © 
aL 
4 x2 
P+ iQ =P Be “ra (1 =) eip?x/g.cos® qx (7) 
AL 
in which we have neglected the contribution of the vertical 
sides of the ship compared with the effect of the flat bottom. 
518 
The integral in (7) may be expressed in terms of Bessel func- 
tions, and we find 
P+iQ= a 0, BL2 e~ 2P*4I/e J, (q cos B)/(q cos 4) (8) 
Thus for pitching motion we have 
E= uae g BLS Pe cana ites? ha. 
qcos 8 
Equating this to 4 p? N, 62, we obtain 
= (mr p/16 g) B? L4 p3 e — 2r%dle F, (10) 
where F, is the integral in (9). This may be evaluated from 
the series 
ma (— 1I)"™(2m4+2)!(2m+ 4)! 
8 —yin\{ (m + 1)! (m+ 2)I\2 (m + 4)! 
ES Caper? Ci) 
(2) Buoyancy and Pitching Moment in Waves.—Suppose at 
first that the ship has zero speed of advance, and that the 
waves are moving directly towards it. The velocity potential 
of the fluid motion is 
b = (grip) ek2sin(pt+ kx). (12) 
with p? = gk; this corresponds to waves of elevation given by 
€=rcos(pt+kx) (13) 
the amplitude r being one-half the height measured from 
trough to crest, and the wave-length A being 2 7/k. The 
pressure p at any point is 
P) 
p=Po—epz+ pre. (14) 
The second term is the hydrostatic pressure whose effect is 
included in the equations of motion of the ship in smooth 
water. The third term 
a¢ 
Dae (15) 
is the additional pressure due to the undisturbed wave system. 
The resultant forces and couples are obtained, to this approxi- 
mation, by integrating this pressure, and its moment, over the 
immersed surface of the ship in its equilibrium position. 
With the same simplified model, we have for the additional 
buoyancy H, 
=gprekzcos(pt+kx). 
1L 
H = apres) 
SE, 
= Hy cos p t, 
4 x2)? 
los) cos(pt+kx)dx 
where 
(16) 
There is also a resultant horizontal force from the pressures 
on the vertical sides; measured in the negative direction of 
O x, from bow to stern, it is 
H, = +g prBAe274 J, (7 LIA) 
0 27 
18 =|42[teprBet cos(pt +4kLcos 6) cos 0d6 
d Jo 
=}4gprBA(l — e274) J, (@ L/A) sinp t 
(17) 
This force might be used as a similar first approximation in 
regard to the surging motion of the ship. By comparison 
