T. H. HAVELOCK 326 
in water at a given depth, suddenly started from rest with a given velocity 
and maintained at that speed. It has been found possible to make numerical 
calculations in this case, and the results illustrate various points of interest. 
2. Circular cylinder 
Take the origin O at the centre of the circular section, of radius a, at 
a depth f below the free surface, with Ox horizontal and Oy vertically 
upwards. Ifthe cylinder is given a small horizontal displacement c 57, from 
rest to rest, the velocity potential of the subsequent fluid motion is given 
by 0 
$ = 2ca°gt Sr | e-K2f—W sin («ar)sin(gttict) ict die, (1) 
0 
This is equation (12) of the paper already quoted (2), obtained there by 
a Fourier integral method; it can also be derived in the manner given later 
by Lamb (3) for the three-dimensional case. 
The velocity potential for continuous motion with variable velocity can 
be found by a direct integration. We consider first the simple case when 
the cylinder is suddenly started at time t = 0 and made to move with 
uniform velocity c. We obtain, noting that the origin is at the centre of 
the moving cylinder, 
ZS ca*x ca*x 
Pty wEOf—ye 
foo} 
t 
+ 2ca?g* {| dr | ePID sin{x(x-+-ct—cr)}sin{gtxt(t—7)}x? de. (2) 
0 0 
Deriving the surface elevation 7 from the relation 
CaP SCPE ON 
we obtain 
t (ce) 
7 = 2ca? [ az | e-*F sin{x(x-+-ct—cr) }cos{gixi(t—z) hc di. (4) 
0 0 
Hence we have 
co 
2a*f COSKX _ 
q = appt ina | ee "Fdie— 
0 
i [ (stench) ane nde 5) 
Ke+gx? KC—g*k* 
0 
where x) = g/c, and the principal values of the integrals are to be taken. 
