Wawe Motion due to a Submerged Obstacle. 524 
We transform these integrals by contour integration in the plane of a 
complex variable «, treating separately the cases of w positive and x negative ; 
after making w zero in the final results we obtain 
2a*f Beerec =m cosmf—Ko sin mf ; 
7= +f? + Ara?Koe Kof SIN K9v+ 2a? Ko \. > wan” e™ dmv 5 BOS 0, 
2a? f 2. (7 mcosmf—xo sin mf _,,, 
= —<—$<$<$<$ << pm im, 0. 18 
n Fame 5 + 2a [ aplnape e mM ; “> (18) 
These agree with Lamb’s results for the circular cylinder in a uniform stream. 
The wave resistance R is derived from the regular waves in the rear, by 
considering the rate of increase of energy and taking into account the 
propagation of energy in a regular train ; we have 
R = i9p (amplitude)? = 47?¢patio?e—PF, (19) 
4, We have now to obtain the resistance R by direct summation of the 
horizontal component of fluid pressure on the cylinder. It is clearly 
necessary to proceed to a further stage with the velocity potential, since we 
have assumed so far that the surface effect is negligible in the neighbourhood 
of the cylinder. If we write (13) as 
@¢=D+xX, (20) 
the doublet D is the first approximation, satisfying the boundary conditions 
on the cylinder; X, is the image of the doublet in the free surface, found by 
satisfying the conditions there. The next step is to find Xe, the image of Xy 
in the cylinder, ignoring then the effect of X2 at the free surface. It follows 
that Xz is the image of X, in the cylinder, found as if the cylinder were at 
rest in a field defined by X;. Taking polar co-ordinates with the origin at the 
centre of the circular section of the cylinder, we have 
x= 7cosé; ytf = rsin@; (21) 
also the conditions for Xz are that it should be a potential function, the 
components of velocity must vanish as 7 becomes infinite, and 
0(Xi+X.2)/or=0, forr=a. (22) 
But from (13), Xi consists of a summation of terms of the form 
We obtain Xz by replacing each term by the expressions 
eee A COS ‘ 
Ge GEO en (xa? cos 8/7), 
and the above conditions for Xj are then satisfied. This process amounts 
simply to inversion; we may think of X as due to a line distribution of 
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