Images in Some Problems of Surface Waves. 270 
In order to keep the various integrals convergent and so to obtain a definite 
result, we adopt the usual device of a small frictional force proportional to 
velocity and in the limit make the frictional coefficient uw’ tend to zero ; further, 
we neglect the square of the fluid velocity at the free surface. 
If y is the surface elevation, the pressure equation gives the condition at the 
free surface, 
- — gn + pd =const., (2) 
we have also, at the free surface, 
@ 0 
Ove Se 0b (3) 
And as we are dealing with the fluid motion which has attained a steady state 
relative to the moving axes, these conditions give, in terms of the velocity 
potential, 
ey = pl () 
to be satisfied at y = 0. Here we have put x = g/c? and B= ple. 
We now assume the solut on to be given by 
ge — iMe= | 
20 
POD Ge AL | F (ic) 74 dic. (5) 
0 
0 
The first term represents the doublet (1) in an equivalent form, valid for 
y + f>0. The function F(x) can now be determined by means of (4), and 
this gives 
F(x) = iMe® (1 4 jal (6) 
Hence the velocity potential of the image system is 
co 20 ikx—« (f—y) 
eK IV de + YineoMe™ | eee ee (7) 
ime | - 
0 K—kKo + tu 
0 
By comparison with (1) and the first term in (5), it is easily seen that the 
first term in (7) is the velocity potential in the liquid due to an isolated doublet 
at the image point (0, f), of moment M with its axis making an angle z — « 
with Oz. 
To interpret the second term in (7) we put 
4 = | eT UP +i (KK) dp, uw > QO. (8) 
kK — ko + aes 0 
267 
