From the PoitosoPHicaL Maaazing, Ser. 7, vol. xxix. p. 407, 
April 1940. 
Waves produced by the Rolling of a Ship. 
By T. H. Havetock, F.R.S. 
1. Tue first part of the following paper deals with the surface waves 
produced by an elliptic cylinder, or a flat plate, submerged in water 
and performing small linear or rotational oscillations. The second 
part contains a short discussion of the energy dissipated in wave motion 
by a rolling ship, together with an estimate of the magnitude of this 
effect obtained from the preceding results. 
Submerged Elliptic Cylinder. 
2. The method adopted is to replace the oscillating body by some 
suitable distribution of sources and sinks or of doublets. Although 
the analysis could be extended to three-dimensional problems, we limit 
consideration at present to two-dimensional motion in a frictionless 
liquid. We begin with the solution for a horizontal doublet which was 
obtained for an oscillating circular cylinder (Havelock, 1917). Take 
the origin O in the free surface of deep water, Ox horizontal and Oy 
vertically upwards. Let there be a horizontal doublet of oscillating 
moment M cos ot at the point (0, —f) in the liquid. The velocity potential 
¢ is given by 
j= we eet Met | 
1 
GTO TS 6 S—Wsin pach, 5 2 (Il) 
0 gk—o*--t a 
with r,2=2?+(y+f)?. The real part of the expression is to be taken, 
and, further, the limiting value when the positive quantity py’ tends to 
zero; this latter process ensures that, at great distances from the origin, 
the motion will reduce to waves travelling outwards on either side. We 
may write (1) in the form 
@ e—K(Ff—Wsin Kx 4 
= — ett — ett Ve Met | = Pre Me resi (2 
p r2 ar ro ar 0 i K—Ky tip ( ) 
with Pe "+ (f—y)? and p—=p'o/g, RpSeel We 
The integral in (2) may be transformed by taking « to be a complex 
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