Wave Motion duc to a Submerged Obstacle. 526 
where A and B are given by (14), or by (16) since we suppose w small, Thus 
we have 
: 20-2 ye. 
R = 4rrpcat Lim eee 
(i /-\n i 
Lim { {«—(xo+ tu/e)} {1e-—(eo—tule)} 
= 4rrpeat Lim w {2aringe-**oF | 2% (w/c) + finite quantity } 
= Ang pate 4-20, ‘ (30) 
which is the same as the previous expression (19). 
Sphere. 
5. A sphere of radius a is at depth 7 below the’surface and is moving with 
uniform velocity ¢ parallel to the axis of z. The origin is in the free surface, 
the axis of z being drawn vertically upwards. As before, the first approxi- 
mation is a doublet D given by 
b = ca®x/ 27° ; r= PtP t (etsy (31) 
For the purpose of satisfying the conditions at the free surface we have 
js — tea 2 Nese Tole /(@+y?)} de;  z4f>0. (32) 
0 
This suggests at once suitable forms for the next approximation and for the 
free surface; the equations are similar to (6) of the previous case, and we 
obtain in the same way 
@ = D—D, +X, (33) 
where D, is a doublet at the image point (0, 0, /) and 
xo Sue 2 { Vo) e-U-2) dy | ce Joler/{ (w-+0u)?-+ 92} ] sin (<Vu) du, 
(34) 
The corresponding surface elevation is 
AS a |" Io {eV (ot+y?) ede 
—a3 [vo e—S kdk [pei Jo fey /{(a+ cu)?-+ 9} ]sin («Vu) du. (35) 
0 0 
The first term represents the effect of the doublets D and D,. It can be 
verified by approximate methods that the second term includes a main part 
like the well-known wave pattern for ship waves. Since the expression in 
(35) gives finite and continuous values for the surface elevation, it might be 
of interest to examine some points in detail; for instance, the elevation near 
the lines corresponding to the lines of cusps for a moving point disturbance, 
However, we pass now to the calculation of the resultant horizontal pressure 
125 
