N 

 (Kq + NMrq) -y = - pgkaQATQ y sin 6- cos (kaQ cos 0. - cot) [ 1 8f ]' 



i=l 



Of course these equations can be solved very easily by substitution 

 of xq = Ci sin(cot + 6, ), yQ = Ct sin(cot + ^-i), etc. The results will not 

 be written because no additional perspicuity seems to follow. 



VISCOUS DAMPING 



All damping forces introduced so far correspond to the energy lost 

 through radiation of surface waves. In addition, energy -will be lost through 

 the mechanism of viscosity. The viscous damping forces, in general, will 

 be of second order in the motion variables. As an example, suppose that a 

 right circular cylinder translates in a direction perpendicular to its axis . 

 The viscous drag is proportional to the velocity squared, and so is negli- 

 gible by the standards already assumed. 



If the cylinder has an axial motion, ho'wever, the viscous force -will 

 be linear in the velocity. For example, if a right circular cylinder of 

 radius a has an axial velocity Re {We } -Wq cos (cot + £ ), then we can 

 see easily from elementary fluid mechanics that the velocity anywhere in 

 the fluid is 



w(r,t) = Re-I ^ '- We^"^ 



K 



= Wq Re 



[ker \/— r + i kei V — r 

 V V J 



[kerV— a + ikeiN/— a 

 V V J 



i(Gjt + e ) 



where Kq(x) is the modified Bessel function of argument x and order 

 zero, and the second expression gives Kq(x) in the Kelvin representation 

 of its real and imaginary parts. The only velocity connponent is that which 



26 



