Kotik and Lurye 



From Eq. (2.21), we see that if Fij e'^^*^ is the complex steady state hydro- 

 dynamic force or moment on the body in the jth mode corresponding to the 

 steady state velocity e"^"^* in the ith mode, then 



Fij 



f i,.(r')e'-^'dr- = -H,. (2.22) 



where fij(t) is given by Eqs. (2.18) and (2.19) and the second equality in (2.22) 

 comes from Eq. (2.1). 



Kramer s-Kronig Relations 



If the integral in (2.22) converged suitably for all real a-, then it would be 

 an analytic function of a- in the half plane Im ct>o, vanishing as cr -»co, whence it 

 would follow that H^^ and H- ^ satisfy the Kramers-Kronig relations. Now, con- 

 struction of fij(T) from Eqs. (2.16) through (2.19) reveals that in fact, H^j is 

 the sum of two types of functions of a, such that the real and imaginary parts of 

 the first type satisfy the Kramers-Kronig relations, while the functions of the 

 second type are too singular either at cr = o or o- = m for the Kramers-Kronig 

 relations to hold. On the other hand, the functions of the second type depend 

 only on infinite frequency potentials and on the steady flow in the absence of 

 oscillations, and may therefore be regarded as easier to calculate. Thus it is 

 the less-known part of H. j that satisfies the Kramers-Kronig relations. 



Specifically, when i, j = 1, 2, 3 we find by substituting from Eqs. (2.16), 

 (2.17), and (2.18), into Eq. (2.22): 



s^ 



GO 



+ 1 S(t) e^'^^dT I fv^ . V0. n-;ij dS 



00 



s^ 



CO 



+ I S(t) i//j^( X, y, z, t) e^°^^dr I n -fx- dS 



CO 



+ J 3:^ '/'liCx, y, z,t) e^'^'^dr J J n -/Ij dS 

 s^ 



00 

 + 1 V^ • V0ii(x,y, z,r) e^'^^dr j rn-;ij dS i=l,2, 3 (2.23) 



j = 1, 2, 3 



412 



