THE COUPLING OF HEAVE AND PITCH DUE TO SPEED OF ADVANCE 
results may be comparable with those for a ship 
advancing at slow speed; but in any case it seems likely 
that the coupling effects in the actual problem will be 
less than in this simplified case. The form of solid we 
consider is a prolate spheroid floating half-submerged; 
for this case the problem can be solved completely and 
the analysis is given in the Appendix. 
2. If his the heave and % the pitch, and U the stream 
velocity parallel to the axis of the spheroid, the equa- 
tions of motion for free oscillations are obtained in the 
form (25), 
(l+k2))Mh—pMUp+epSh=0 
(LK) I fp+qMUh+Mgmyp=0 
The first and third terms are in the usual notation for 
uncoupled heave and pitch; the second terms give the 
coupling effect. 2 and k’ are the virtual inertia coeffi- 
cients under the assumed water surface condition. 
k2, k’, p and q are positive numerical coefficients depend- 
ing only upon the length/beam ratio of the spheroid; 
explicit expressions are given in (18), (20), (22), and (23), 
from which numerical values can be calculated. 
If we write these equations in the form 
h—xUp+n2,h — 0 
b+ BUA + n2,4=0 
nm, and nz are the frequencies for uncoupled heaving and 
pitching, taking into account the virtual inertia. If we 
assume a periodic coupled oscillation of frequency p, 
we have 
pt — (ni + 1} + « B U2) p? + nin = 0 
Both roots of this equation in p2 are real and positive, 
and we have two simple undamped oscillations of, say, 
frequencies p, and p>. In each mode the heave and 
pitch differ in phase by 90 deg., and the motion alter- 
nates between heaving and pitching. Further, ifm, < n> 
and pj < pz, then we havé py <m and p2 > np; thus 
the coupling increases the separation between the reso- 
nance frequencies. This is a general effect of coupling 
terms; incidentally it may be remarked that for the 
coupling caused by damping, Korvin-Kroukovsky and 
Lewis“ observed that the resonance period for heaving 
was increased while that for pitching was diminished. 
3. To estimate the magnitude of the effect we take a 
numerical example. We choose a spheroid of length/ 
beam ratio equal to 10. The numerical values of the 
various coefficients were calculated with sufficient 
approximation for the present purpose. From (18) we 
find k2 = 2-42. This means an increase of about 
80 per cent in the heaving period as found without the 
added mass; no doubt this is rather large, but we have 
taken the extreme condition of a rigid water surface. 
Similarly from (20) we find k’ = 1-5, giving a corre- 
sponding increase of about 60 per cent over the basic 
pitching period. From (22) and (23) we obtain, approxi- 
mately, p = 1-16, g= 0-57. With f as the Froude 
number, we have U = f(2ga)}; and taking a 16-ft. 
598 
model as a definite example, that is, a= 8 ft., the 
equations of motion are 
h—T7-7fb + 17-657h =0 
~ +0-404 fh + 30-187 4 =0 
and the frequency equation is 
p* — (47-844 + 3-111 f2) p2 + 533-01 =0 
The periods of uncoupled heave and pitch are 
1-495 sec. and 1-144 sec. respectively. For f= 0-2, 
the coupled periods are 1-503 sec. and 1-138 sec. Even 
at a high speed f = 0-5, the periods are only altered to 
1-537 sec. and 1-112 sec. The curves in Fig. 1 show 
the variation in the coupled periods with increasing 
speed. At zero speed, the upper curve gives the period 
of uncoupled heave and the lower curve that of un- 
coupled pitch. The variation with increasing speed only 
becomes appreciable at very high speeds. 
im) 
1.6 
vu 
wd 
z 
a 
Q 
& 
al2 
Ke) 
2 4 fi 6 8 1.0 12 
FIG. 1—VARIATION OF RESONANCE PERIODS WITH SPEED 
4. Although the surface condition 0 4¢/0z=0 is a 
severe limitation as regards application to the ship 
problem, it was thought preferable to work out the 
simplified problem consistently on this basis. In the 
last section of the Appendix comparison is made with 
the work of Haskind. It appears that if we use tenta- 
tively rather mixed conditions with the oscillation 
potentials satisfying the condition ¢ = 0 at the surface, 
then the coefficients of the coupling terms approach 
numerical equality for a long spheroid for which 6/a is 
small; the coupling terms approximate to the values 
—14MU, and 4}MUh. For another numerical 
example, take k2 and &’ at their limiting values of unity 
for this surface condition and the equations (29) approxi- 
mate to 
2Mh—IMU¢+epSh=0 
21¢+4i1MUA+Mgmy=0 
For the 16-ft. model of the previous calculations these 
give a frequency equation 
pt — (67-92 + 2-514 f2) p2 + 1139 = 0 
For f = 0, the uncoupled periods of heave and pitch 
