NOTES ON THE THEORY OF HEAVING AND PITCHING 
with (16), it can be seen that in general it is only a fraction 
of the corresponding vertical force. 
In evaluating the pitching moment we take moments about 
the transverse axis Oy, assuming for simplicity that the 
centre of gravity G of the ship coincides with O. The 
pressures on the vertical sides will contribute to the total 
moment; but this part will be of the order of the horizontal 
force Fy multiplied by some fraction of the draught, and it 
can be seen to be negligible compared with the moment of the 
pressures on the flat bottom. We have then for the pitching 
moment 
i 4 x2? 
P=gprBe-*4 x(t ~ =) cos(pt+kx)dx 
-#L 
where 
— Posinp t, 
Pp = 48 prBLie-2r4 J, (7 LIX) (18) 
For a ship advancing through the waves, we have the same 
expressions, so far as this approximation goes, with 2 z/p 
the relative period of encounter; thus if A is the wave-length 
and v the corresponding wave velocity, and V is the speed 
of the ship, then 2 z/p = A/(v + V). 
On this theory, the equations for heaving and pitching on 
waves are, for this symmetrical model 
Mf+N,2+8pS6=Hycospt (19) 
I16+N,6+Wm0= — Posinpt (20) 
The forced oscillations are 
C= %ycos(pt — By); = — A sin (pt — fy) 
with 
bo = HolM{(p? — p2)? + k2 p2}*; ky = Ny/M: 
Oy = Poll {(p3 — v2)? + k3 p2}#; ky = NoJI; 
tan B, = ky p/(p? — p?); tan B, = ky pl(p3 — p?) . (21) 
the natural periods of unresisted heaving and pitching being 
2 mp, and 2 z/p, respectively. 
(3) Kesistance in Waves.—Let (I, m, n) be the direction- 
cosines of the outward-drawn normal at any point of the 
immersed surface of the ship. Then with the pressure in the 
undisturbed wave motion given as in (14), the resultant 
horizontal force backwards 
F= |[pras = ~gpkr||[ekssin(p r+ kx)dV_ (22) 
the latter integral being taken throughout the immersed 
volume of the ship at any instant. 
If we calculate this for the immersed volume Vo when the 
ship is held in its equilibrium position we obtain a purely 
periodic force Fo, such as was found in (17). Let the ship 
be in a slightly displaced position due to heaving and pitching, 
with the centre of gravity G raised a distance ¢ and with a 
pitch 6 about a transverse axis through G; we shall suppose 
G to be on the axis O z at a height c above O. Then, to the 
first order in € and @, it can be shown (R.16) that the horizontal 
force backwards in the displaced position is 
F=Fy—epkrE|]etssin(oe + kaynds 
~gpkr ol ebssin (ot + kx) {nx Ce= c)} dS (23) 
519 
where the integrals are taken over the equilibrium position 
of the immersed surface. 
The additional buoyancy and pitching moment, which were 
calculated for a special case in (16) and (18), are given in this 
more general form by 
H=—spr|[etcos(pit kx nds 
P= gpr|] etcos(pr + kn {I2 0) napa (24) 
Hence we may write the backward force as 
ep Oicl le moi? 
$ Pp DE 
De OE @») 
F= 
Fo — 
When calculated for any form of ship, H and P are in general 
of the form Hg sin (p t + a) and Ie sin (pt a a) respectively. 
The corresponding forced oscillations of heaving and pitching 
are then given by equations such as 
C = py Ho sin (p t + a — By) 
6 = py Posin (pt + a — Bs) 
4, and j2y being positive factors. 
Putting these expressions in (25) and taking mean values 
of the quadratic terms, we obtain a mean backward force 
on the ship 
R= pm, H2sinB, + 5k py P§sinB, . 
(26) 
(27) 
This is an essentially positive expression, so that this force is 
always a resistance. 
With ¢) and 0) the amplitudes of forced heaving and 
pitching respectively, this expression is equivalent to 
R = (aA) Hy So sin B, + (aIA) Po 9% sin By (28) 
where 8, and B, are the phase lags of the forced heaving and 
pitching behind the buoyancy and pitching moment 
respectively. 
References 
(1) Lewis, F. M. Trans. Soc. Nav. Arch. (New York), 
37, p. 1 (1929). 
(2) BRowNneE, MOULLIN and PERKINS. Proc. Camb. Phil. Soc., 
26, p. 258 (1930). 
(3) Brarp, R. Assoc. Mar. et Aero., 43, p. 231 (1939). 
(4) Baker, G. S. Trans. N.E. Coast Inst., 59, p. 23 (1942). 
(5) Kent, J. L., and Cuttanp, R. S. Trans. Inst. Eng. 
Ship. (Scotland), 84, p. 212 (1941). 
(6) Horn, F. Jahr. Schiff. Ges., 37, p. 153 (1936). 
(7) Kent, J. L., and CurLanp, R. 8. Trans. Inst. Nav. 
Arch., 78, p. 110 (1936). 
(8) Havetock, T. H. Phil. Mag., 29, p. 407 (1940). 
(9) ScHuLer, A. Zeit. f. Ang. Math. u Mech., 16, p. 65 
(1936). : 
(10) Krerrner. Trans. Inst. Nav. Arch., 80, p. 203 (1939). 
(11) Havetock, T. H. Phil. Mag., 33, p. 666 (1942). 
(12) Kent, J. L. Trans. Inst. Nav. Arch., 68, p. 104 (1926). 
(13) Kent, J.L. Trans. Inst. Eng. Ship. (Scotland), 76, p. 290 
1933). 
(14) Ae T.H. Proc. Roy. Soc., A 161, p. 299 (1937). 
(15) Havetock, T. H. Proc. Roy. Soc., A 175, p. 409 (1940). 
(16) Havetock, T.H. Phil. Mag., 33, p. 467 (1942). 
(17) SuveHiRo, K. Trans. Inst. Nav. Arch., 66, p. 60 (1924). 
(18) WaTANABE, Y. Trans. Inst. Nav. Arch., 80, p. 408 (1938). 
