Nonsinusoidal Osailtations of a Cylinder in a Free Surface 



M(\,a) = M(>.,Tr - a), (B-9) 



and In place of Eq. (B-4) and (B-6) we require 



M— as X.^00 in -iT<a<0. (B-iO) 



The solution of this problem is 



- , ,. . cos Zma , ^ i sin (2m - 1)0- 



M^(\,Qr) = + K,a< ■ -—-: 



"» XZ"™ 'I (2m- l)\2m.| 



00 



Y (2n + l)a2n*i sin (2m + 2n + l)a \ ,„ . .. 



- L 2m + 2n + i x^'"*^"^! / ^^-^^' 



where m is a positive integer. M^ is often called the multipole 

 of order m. Although this expression for M^ trivially satisfies 

 Eq, (B-6) in the ^-plane,the expression above still does not repre- 

 sent the outgoing plane waves. To satisfy this radiation condition 

 we introduce a source function MQ(x,y) which satisfies all the 

 required conditions except the boundary condition on the body. The 

 expression for the function Mq Is 



/*00 



where + means that a Cauchy principal value Is to be used. There- 

 fore we represent our solution as 

 00 



^, = ^ (b„+jcjM„(x(X,a),y(\,a))e-^'» (B-13) 



m=0 

 where b^ and c^ are the unknown strengths of the singularities, 

 q is the phase difference between the naotions of the body and the 

 fluid, bo = Q = source strength, and Cq = 0. The unknown constants 

 ^m> ^m> ^^^ ^ ^^® ^° ^® determined from the boundary condition on 

 the body given by Eq. (B-3). 



We Introduce the stream function ^1 which Is the harmonic 

 conjugate of the velocity potential ^|. The Cauchy- Riemann relation 

 gives d<p./Bn = 8^, /8s along the contour of the cylinder where s Is 

 the arc length of the contour In the counter-clockwise direction. The 

 boundary condition for ^| on the cylinder can be shown to be 



^.K'Vo) = - b^i^- <^-^^^ 



The expression for ^| In terms of the harmonic conjugates of M^ 



945 



