575 Forced Surface- Waves on Water. 
6. The same analysis may be applied to circular waves, 
and we limit consideration here to symmetry round the 
origin, The normal fluid velocity is supposed to be assigned 
over a vertical cylindrical surface ; for example, we take 
op 
— 5, —/ (2) cosat, for r=a. 5 0 0 (8) 
The velocity potential satisfies 
O° , 186 | 6 _ ; 
ar ite ae + 52 =0. o 6 o »o (6) 
The condition at the free surface is the same as before, and 
we assume the water to be deep. Elementary solutions of 
the required form are found in terms of suitable Bessel 
functions. The solution 
OaGr—owsONGap) 6 6 3 o o (a) 
represents diverging waves for large values of r ; while in 
the solution 
p=e'(k cos Kz—xkysin Kz) Ko(kr), . . (38) 
K(«r) tends exponentially to zero for large distances from 
the origin. 
Generalizing as before, we obtain the solution 
Hi) (or) (> 2a 
= 9 tat— Koz 0 0 — Koa, 1 eos eiat x 
p=2e Hi? (Kya) Vo f(a)e-"rdce = 
(Kk COS K2— Ky SIN K2) X 
x ue We Ko(«r) 5S (GCOS BBS HUE) Fa, (39) 
) Kai (xa) K+ Ke 
where the real part is to be taken. 
The surface elevation at a great distance from the origin 
is given by 
al 2 3 et(at—Kor-+a7) juices 
g ) Jo (Hot) =2Yo/(Ho2) Jo 
or, in real ‘eee: this gives 
G2 =2(—) f f(a@)iem wadiax 
C~ — T (a) e~*on da, (40) 
TWKor 
Jo (koa) sin (ot — —Kort| m+ 
ee a aa) eee 
Jo 2(ko@)+Yo0 2(xo@ 
310 
