Ship Waves. 3 
where ow’ = (x —h)cos 0+ ysin 0. We have omitted here the symbol for 
the limiting value as p. is made to vanish, but that is always to be understood. 
It is assumed that the integrals are convergent. From a physical point of 
view it is easily seen that divergent or indeterminate integrals may arise if 
the distribution contains finite sources or sinks which extend up to the free 
surface of the water. From the method of obtaining the velocity potential 
(1), we see that the appropriate form of (4) in such cases will be found by taking 
the integration with respect to the depth f to extend from a positive quantity 
d to mfinity and then considering the limiting value as d is made to vanish. 
We may note another form for (4) which is obtained by integrating by parts 
with respect to . Provided M is continuous in this variable and is zero at 
the two limits, we have 
‘oO eat bica’ 
& [i ail y 
eee Seen AC, g Od yo ere (5 
5 =| oh if | _see \ Kk — Ky Sec? 6 + iy sec 0” oan) 
Further, the normal component of fluid velocity at any point of the vertical 
plane y = 0 is equal to 
2 OM/oh. (6) 
Hence from (5) we may obtain the surface elevation for any assigned dis- 
tribution of normal fluid velocity over this plane. 
3. Consider first a simple line distribution of constant moment M per unit 
length on the z-axis, extending from the free surface to an infinite depth. 
Here we shall have to suppose first that the distribution extends up to a depth 
d below the surface, and then take the limit as d is made small. 
Integrating with respect to f, we obtain 
7 eo) —Kd tine 
t= | a0 | 2a a (7) 
C15) 0 0 K — Kg sec? 0 +- 2y sec 0 
Tn the integrand we write 
K a Ky Sec? 9 
K — Ky sec? § + in sec 6 K — Ky sec? § + iu sec 0’ 
(8) 
omitting terms which will give no contribution in the limit when p is made 
zero. The integrations in 0 and « in (7) corresponding to the first term on the 
right of (8) give the value 2r/r(d? + 22 + y?)#. Hence, putting d = 0, the 
contribution of this part to the surface elevation is 2M/u(a? + y2)}. Taking 
the second part of (8), the corresponding integral in (7) remains convergent 
349 
