THE EFFECT OF SPEED OF ADVANCE UPON THE DAMPING OF HEAVE AND PITCH 
By PRoFEssor Sir THomas H. HAvetock, M.A., D.Sc., F.R.S. (Honorary Member and Associate Member of Council). 
Summary 
Calculations are made for the damping coefficient for a specially simple case which may be 
taken to correspond approximately to a long narrow plank moving forward with velocity c and 
making forced pitching oscillations of frequency p. Curves are given for the variation of the 
damping moment with frequency at various speeds, the chief aim being to illustrate the effect 
of the critical condition when the parameter p c/g has the value 4. The results are discussed in 
reference to recent experimental work and the possibility of a steep rise and fall in the curve 
of damping near this critical point. 
The damping of the heave or pitch of a floating solid is mainly 
due to the energy lost in the wave motion produced by the 
oscillations. If the solid is at rest, apart from the oscillations, 
the problem can be formulated satisfactorily as a potential 
problem with the usual linearized condition at the free surface 
of the water. If the complete solution could be found in any 
given case, it could no doubt be also expressed in terms of some 
source distribution over the immersed surface of the solid. 
However, what is usually known as the source method of solution 
is an approximation which begins by assuming some simple 
source distribution and then adding the wave motion due to these 
pulsating sources; the method has obvious limitations on its 
application in general, but it has served to give interesting and 
useful results. If, in addition to the oscillations, the solid is 
moving forward with a constant speed of advance, the formula- 
tion as a potential problem with the linearized free surface con- 
dition is not satisfactory except in the limiting case when the solid 
is like a thin disc moving in its own plane. However, some pro- 
gress has been made by the approximate method of assuming 
some source distribution, and the calculations then require the 
wave motion due to a pulsating source advancing at constant 
speed. This problem has been examined by various writers and 
reference may be made in particular to Haskind,™ Brard,™ and 
Hanaoka.®) If p is the circular frequency of the pulsation and 
c the velocity of advance, it is known that the wave motion changes 
in character when the parameter pc/g = +4. It does not seem 
to have been pointed out explicitly that in fact some of the terms 
in the solution become infinite at this particular point. The 
object of the present paper is to examine this matter in some 
detail for a special case so as to see the effect of this mathematical 
infinity upon the damping for lower and higher values of the 
parameter. Consider for a moment a two-dimensional case, for 
instance a submerged circular cylinder making heaving oscilla- 
tions of frequency p and advancing with velocity c. At zero 
speed, there are two wave trains, one on each side of the cylinder. 
At speed c, if p c/g < 4, it can be shown that there are four wave 
trains, one in advance and three to the rear, the wave train in 
advance being that for which the group velocity is greater than 
the speed of advance. If the speed is increased, the amplitudes 
of two of these trains become infinite at the critical point when 
pc/g =4; and for higher values of the speed these two trains 
disappear, leaving only two wave trains both to the rear of the 
cylinder. The behaviour at the critical point clearly arises from 
a special, and interesting, case of resonance; and, as usual, the 
infinity could only be removed from the solution by introducing 
some frictional or other kind of dissipation. 
Turning to the three-dimensional case of a point source, one 
might hope that the infinity would disappear through integra- 
tion, but this is not the case; the solution contains integrals 
which are finite in general, but they become infinite at the critical 
value of the parameter. 
Calculations have been made by Haskind and by Hanaoka for 
the damping of a Michell-type of model with the source distribu- 
tion assumed to be in the vertical longitudinal plane; this assump- 
tion is the well-known approximation for wave resistance, and 
although it is of doubtful validity in general as regards the 
heaving or pitching oscillations it gives useful indications for 
simplified forms. Although the integrals used by Haskind 
become divergent at the critical value of the parameter, his curves 
do not show any infinity; possibly the range does not include the 
critical point. Hanaoka also gives a curve for the damping at 
various speeds; but the whole curve is explicitly for the value 
pclg = 0-6 and so is well beyond the critical point. 
Some recent experimental work by Golovato is of special 
interest. A model was made to perform heaving oscillations of 
given frequency while moving forward at some constant speed, 
and the forces and moment on the model were measured. In 
Fig. 13 of that paper the damping moment is shown in curves 
ona. base p(B/g)? for various values of the Froude number. 
A striking feature is the pronounced peaks at low values of the 
parameter. Golovato remarks: “‘The steep rise at low fre- 
quencies appears to coincide with a velocity-wave celerity ratio 
of 4 where the character of the waves generated by the oscillating 
body is known to change markedly.” This ratio is what we have 
denoted here by p c/g. It is curious that the curves for heaving 
do not seem to show the same effect, though one would expect 
the same cause to be operative for both heaving and pitching. 
The present calculations are for a simple line distribution of 
pulsating sources, but we can relate them to a possible physical 
problem. Suppose a long narrow plank, in a vertical plane, 
moving forward and at the same time making small pitching 
oscillations. Such a form, with pointed ends, is the most suitable 
for comparing wave resistance theory with experiment, and it 
might also be used similarly to test the approximate linear theory 
of heaving and pitching. However, even if it is not a practicable 
method experimentally, it is an appropriate form for the present 
state of theory. We may separate out the effects of the forward 
motion and the pitching; and we may assume the latter to be due 
to a simple source distribution over the flat submerged base of 
the plank, or for small enough beam to a distribution along the 
central line of the base. As numerical computation is rather 
lengthy in any case, we omit the pointed ends and reduce the 
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