The damping terms of these equations of motion are given by the terms 

 linear in the velocities 4, t , and ijj. It should be noted that for a slender 

 body m -* 0, and thus the damping coefficients will be small, as was to be 

 expected. To solve these equations for the three unknown displacements 

 and their phases is a straightforward but tedious matter. For applications 

 in ranges not including a resonance frequency, it is much simpler to em- 

 ploy the undamped equations of motion, [31] to [33], and the resulting 

 displacennents, [34] to [36]. 



CALCULATIONS FOR THE CIRCULAR CYLINDER 



As a special case, we shall consider the circular cylinder R(z) = R 

 = constant. Then 



X = 1.0 







P = 1 r (z - zc)dz = - Ih - Z( 



1 r° 7 ? 



^2 = H J ^^" Zg^^^ = iH^ + Hzc+ Z( 



1 r' 



Qo(K) - H J_ 



Ql(K) = ^ C eK-(z-Zc)dz = le-K^--4-(l-e-^^)(l + Kzc) 



,0 

 -H 

 ,0 

 ■ H 

 .0 



e dz-— (1-e ) 



We shall assume, moreover, that the centers of buoyancy and gravity co- 

 incide, or zq = - H/2, so that the equations of motion are uncoupled and 

 there is no resonance in pitch or surge. 



Then 



h2 1 + e-K^ 1 - e"^^ 



17 



