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THE FORCES ON A SUBMERGED BODY MOVING UNDER WAVES 
By PROFESSOR Sik THOMAS H. Havetock, M.A., D.Sc., F.R.S. (Honorary Member and Associate Member of Council) 
Part I. MOTION NORMAL TO THE WAVE CRESTS 
Summary 
A theoretical investigation of the forces and moments on a submerged spheroid moving 
close to the surface under waves. 
Expressions are obtained for the surging force, heaving 
force and pitching moment taking into account the speed of advance and also the disturbance 
of the wave train by the solid; graphs are given to show the variation of these quantities with 
the speed of advance and with the wavelength. 
1. Introduction 
The theory of the forces on a submerged spheroid 
moving through smooth water was examined in a 
previous paper (Ref. 1), and has been discussed in detail 
recently by Wigley in these TRANSACTIONS (Ref. 2). An 
interesting and important extension is the same problem 
when the spheroid is moving steadily either with or 
against a regular train of transverse waves. In con- 
sidering the similar problem for a surface ship it is usual 
to assume the pressure on the ship to be that due to 
the undisturbed wave train, as, for example, in the 
so-called “Smith effect” or as in the classical theory of 
the motion of ships among waves as developed by 
Froude, Kriloff and others; broadly speaking, this is 
equivalent to neglecting the various virtual inertia 
coefficients of the ship. Moreover, the effect of the 
speed of advance of the ship is assumed to be simply 
an alteration of the frequency of encounter with the 
waves. A more adequate theory for surface ships 
presents great difficulties; however, for various reasons, 
it is possible to carry the theory further for a wholly 
submerged body uncer certain conditions, and the 
present paper deals with the motion of a submerged 
prolate spheroid. The mathematical analysis is given 
in Sections 2, 3, 4, and the notation and main results 
are summarized in Section 5. General remarks are made 
in Section 6, together with graphs for the force and the 
moment coefficients; a point of special interest is the 
effect of speed of advance and the difference between 
moving against the waves or with the waves. 
2. Velocity Potential 
A prolate spheroid, of major axis 2 a and eccentricity e, 
is moving axially under water with velocity V parallel 
to the surface and there is a regular train of transverse 
waves, of wavelength 2 z/« and wave velocity c, moving 
in the opposite direction; the axis is at a depth d below 
the surface. It is convenient to reduce the spheroid to 
relative rest by superposing a uniform stream V in the 
opposite direction. We now take fixed axes with the 
origin O at the centre of the spheroid, Ox axially, 
590 
O » transversely, and O z vertically upwards. We begin 
with the velocity potential for a spheroid in a uniform 
stream (as given for instance in Ref. 3), 
$= Vx —aeVPi (pe) Qi (€)/Qi1 (60) (1) 
when the dot denotes differentiation, and the spheroidal 
co-ordinates ¢, #, @ are given by 
3 = Gen ce 
y = ae(l — pp?) (2 — 1)t sin w; 
z=ae(1 — p2)* (@ — 1)? cos w (2) 
In these co-ordinates the spheroid is given by 
€ = () = l/e. We add the velocity potential ¢; giving 
the assigned train of waves moving in the negative 
direction of O x on the surface of the stream; it is easily 
verified that 
di = —hce"4**7 cos(k x + ot + a) (3) 
with « an arbitrary phase angle, gives a wave train on 
the surface with elevation 
=hsin(kx +ot+a) (4) 
provided 
o=K«(V+0o); c2= glk 
We now take 
f = Vx —ae VP; (1) Qu (2)/Qi (So) + gi + bo - ©) 
with 4; given by (3). ¢2 represents the disturbance of 
the wave-train by the spheroid, and is to be determined 
so that the normal fluid velocity is zero over the spheroid; 
thus we must have 
d (gi + G2)/9F=0; C= lo (6) 
We should also add a further term to (5) to represent 
the steady wave pattern produced by the spheroid 
itself, which would be determined as in Ref. 1 for motion 
through smooth water; to a first approximation the forces 
due to the transverse waves would be simply additive. 
Meantime we shall assume the conditions to be such 
that the former forces are small compared with the 
latter, an assumption which can be checked by calcula- 
tion from the results given here in Section 5 and those 
given in Refs. 1 and 2. 
