332 T. H. HAVELOCK 
4. Three-dimensional motion 
Turning to the general problem, we take the origin O in the free surface, 
with Oz vertically upwards. Suppose a point source of strength m is 
suddenly created at the point (0,0, —f) and maintained for a short time 
5r. To satisfy the condition at the free surface for the initial motion, we 
take 
to=———, (23) 
with r? = x?+y?+ (2+f)?; 72 = a+y?4+ (z—f)?. 
The initial surface velocity found from (23) acting for a time 67 gives 
a surface elevation which can be put in the form 
i= LS | dé | e—*T cos(kar)K dk, (24) 
7 
=a 0 
with wo = xcos6+ysin 0. 
The velocity potential of the fluid motion at any subsequent time ¢ due 
to this initial displacement without velocity is 
= mg or i dé [ e*I+2 cos(Ka)sin(g!tx?)«? dk. (25) 
igen raat 
Consider now a source moving parallel to Ox at constant depth f, the 
strength m being a function of the time. Let 2 be measured from a moving 
origin vertically over the source, é from a fixed origin at the starting-point; 
and let s, be the €-coordinate of the source at any time t. Then we obtain, 
from (25), 
t 7 oe) 
= 0 ee | m(r) dr | dé | e-H+ cos(iear’)sin{giid (t=r) je! dr, 
1 2 au 
0 Ur 0 (26) 
with w’ = (€—s,)cos0+ysin 0. 
We may generalize this result for a solid body moving through the liquid. 
If the solid moves through an infinite liquid with unit velocity, we may 
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