Lee 



-J 



^ iili-P^ dp - JTre'^'^^o 3in K,x Y (51) 



Since this equation is valid for all values of a in the interval 

 (- ir/Z , 0) on the circle of radius X.o> in principle we can take an 

 infinite number of or's to set up an infinite system of a linear alge- 

 braic equation which can then be solved for b|^^Q,^ and c,^^/Q,^. 

 In practice, the infinite series is truncated, so only a finite number 

 of these unknowns is sought; this finite number is equal to the number 

 of chosen of's. The proof for the convergence of such a truncation 

 scheme was given by Ursell [ 1949] . Once the values of a number N 

 of b^ /Qj^ and c^j^/q,^ (i.e. m= 1,2,...N) is known the values of 

 q and Q^ are readily obtained from Eq. (50) by 



^k k 



q^ = tan-'|:i^2i4^, (52a) 



I Rej A j 



where 



A = 



Q^= |A|, (52b) 



BwU:.,y^) 



S(Tfr"^)^*-<^-"'^ 



m=0 



Thus we finnaly can express the solutions of the velocity potentials 



Qk by 



V>^(x,y) = G^(x.y) + W^(x,y) for k = 5,6. (53) 



V. PRESSURE, FORCE, AND WAVE 



5. 1 Pressure on the Cylinder Surface 



If we expand P which denotes the pressures on the cylinder 

 in the same way as $ was expanded in Eq. (17), we find that 



P(xo,yo +y(t),t) = e{Pi(xo,yo + y(t))e"^'^'* +Pjje'^'^'*l 

 ./ -j2<r,t -jZo-jt -j(o-, + o-2)t 



+ P e 

 VI 



922 



