Sir Thomas Havelock ® 
symmetrical with respect to Oz. For periodic motion of frequency o, the linearized 
condition for the velocity potential at the free surface is 
Kop +8 = 0; 2= 0; (1) 
with ky = o?/g. If there is a periodic singularity of order n at the point (0,0,f) in 
the water, we have the known solution 
— 1 nm (oo) 
Oz salle COs ain me cos atl pees Xo KnJ. (Ko) eked, (2) 
i 0 0 
valid for z+f>0. The principal value of the integral (2) is to be taken, and we 
have put 7} = 2+ y?+ (z—f)? = o?+ (z—f)?, and w, = (2—f)/r. If ry, 4, are polar 
co-ordinates referred to the image point (0,0, —f), the solution (2) can be expanded 
in the form 
@ Sead P,(1) nents) 5) n 1 P,_1(H2) ko 1| 
cosot—_r?+t Se) ret Tels eS ners segc tint n! Tf 
in n+ ss 
os is JP) wed, (3) 
nN. 0 K—Kg 
The principal value of the integral in (3) is 
— AVq(kyt) 0") 
K,(K@) dk, (4) 
2 (°K, cosk(z+f) )e«sin(Z +f) 
=|. K2+ Ke 
with the usual notation for the Bessel functions. We superpose on the motion given 
by (3) free symmetrical oscillations of frequency o so that as a >oo the motion 
approximates to circular waves travelling outwards. For this purpose we add to 
(3) the term 
_—1)r 
2! = TKR +S (Ky@) et) sin ot. (5) 
The motion as 7 >0o then approximates to 
b> ange)" ( z J’ sin ot w+inm) (6) 
. nm! \aky@. Oe eras 
In general as 7 +0, ¢ is of order w*; but from the expressions given in (2) and (3) 
it is possible to construct solutions in which ¢ is of order w~ or of higher order. 
These combinations of periodic singularities might be called wave-free singularities. 
They are given by 
o) a Ko P (41) 4 Frills) 
Ko Pi(Ms) _ (—1" nents) 
n+1 ret per rete 
wel eH cosot. (7) 
(aul) 
For instance, taking n = 1, the singularity {}« 97; 2P,(u,) +77 °P.(/4,)} cos ot at the 
point (0,0,f) gives a surface elevation proportional to (w?— 2f?) sin ot/(w? + f2)2 
For the particular application which is in view at present, we require the results 
when f is made zero. Thus from (7) we have wave-free solutions given by 
d ae {Ko Pen (Ht) , Pal) 
lon am antl cos ot, (8) 
with the origin O in the free surface. 
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