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Institution of Naval Architects. 
THE COUPLING OF HEAVE AND PITCH DUE TO SPEED OF ADVANCE 
By PROFESSOR SiR THomMAS H. HAveLock, M.A., D.Sc., F.R.S. (Honorary Member and Associate 
Member of Council) 
Summary 
The object of the note is to discuss a particular type of coupling and to estimate its 
probable magnitude. 
The coupling effect is isolated by considering a specially simplified 
problem, namely a spheroid floating half-submerged in a uniform stream with surface 
conditions which preclude wave formation and without damping. This problem is solved 
completely; numerical calculations indicate that the alteration in resonance frequencies due 
to the coupling is likely to be negligible in ship problems. 
1. The chief cause of coupling between heave and 
pitch is lack of symmetry of the ship fore and aft, as 
for instance in the well-known hydrostatic coupling or 
in that due to damping. There is one type, which may 
be called hydrodynamic coupling due to speed of 
advance, which seems to exist even if the ship is sym- 
metrical fore and aft. This effect was introduced into 
the equations of motion of the ship by Haskind. In 
that work Haskind replaced the ship by the approxima- 
tion used in wave resistance theory, namely a source 
distribution over the longitudinal vertical section; 
further, the expressions were left in a complicated form 
and no indication was given of the relative importance 
of the terms in the equations. Recently Stoker and 
Peters® have made a systematic study of the general 
problem of the motion of a ship in a seaway, developing 
the equations in terms of a single parameter, namely the 
tatio of beam to length. In the equations of motion 
to the first order, they do not obtain any coupling terms 
of the type in question for a symmetrical ship. This 
might be expected as in their work the ratio of beam to 
draught is also supposed small; in fact their model 
approximates to a thin flat disc. Haskind’s work is also 
criticized as implying damped oscillations since the 
coupling terms occur as first order derivatives; but we 
shall see later that this criticism is unfounded as far as 
the coupling terms are concerned. This type of coupling 
has been the subject of discussion recently, for instance 
Weinblum,® and it seemed of sufficient interest to 
attempt to estimate its importance or otherwise. It is 
easy to see on general grounds that the coupling exists. 
If a floating solid, symmetrical fore and aft, is made to 
oscillate vertically in a uniform stream, the alteration in 
pressure is anti-symmetrical and so we get a couple 
acting on the solid; if it is given pitching oscillations 
the alteration in pressure is symmetrical and we get a 
heaving force. It also seems likely that the effect will 
be small, and that is confirmed by the present calcula- 
tions. 
In the theory of wave resistance for a ship advancing 
steadily in still water, a first approximation based on 
597 
the linearized free surface condition is in general a good 
approximation for ships of small beam/length ratio, and 
this remark applies even when the beam/draught ratio 
is not also small. But in attempting further approxima- 
tions it is difficult to know how far one may go without 
amending the free surface condition by including some 
approximation to finite wave theory. 
On the other hand, consider heaving and pitching of a 
ship at zero speed of advance. Here we do not need 
to restrict the relative dimensions of the ship, either the 
beam/length or the beam/draught ratio; the linearized 
free surface condition is adequate for a good first 
approximation, except no doubt for exceptionally large 
motions. 
Turning to heaving and pitching in waves with the 
ship advancing, one can see the difficulty of combining 
the general problem in a single calculation which will 
give useful results for the ship problem. The present 
unsatisfactory theory consists more or less in simply 
superposing the two sets of calculations; or if it is rather 
better than that, we are still left in doubt as to the 
validity and relative importance of the various terms in 
the equations of motion. On the other hand, if we limit 
ourselves to a rigorous development based on, say, a 
thin disc form, we may miss the important effects for 
the ship problem as regards heaving and pitching. 
However, the present note makes no attempt whatever 
on the general problem. The object is to isolate the 
particular type of coupling and if possible to estimate 
its importance. For this purpose we consider a specially 
simplified problem. It may be regarded as the opposite 
of the work just referred to; instead of taking a thin 
.disc and including the wave motion, we consider a form 
more like a ship but we exclude the wave motion com- 
pletely. The conditions may be visualized in this way. 
Imagine a solid floating in water and suppose the free 
water surface covered by a smooth rigid plane; the solid 
being assumed free to heave and pitch in a hole in this 
plane, the periods can be calculated. If there is a 
longitudinal uniform stream in the water, the oscilla- 
tions are coupled and the periods can be obtained. The 
