Ogilvie 



Item (1) is of no consequence if \(co) exists, for it then equals a^(co)/i.co. 

 Even if cl^(co) does not exist, our assumptions imply that a.^(co) does exist, and 

 so the applicability of (15) has been extended. 



Item (2) is also of no consequence, for we have just shown that the two ex- 

 pressions are equivalent. Also, M.^(ai)/iaj exists even when co = o , because of 

 the fact that 



J Mj^(T)dr = . 



Item (3) is not quite so easily disposed of. We note that, if the external 

 forces represented in (15) are well-behaved transients, then 



>[f.(co) + G.(a;)] -. 



as CO -* 



Also, wS(aj) = 0, for all . Therefore, if we multiply Eq. (15) (as now modified) 

 by CO and let w ^ o, we are left with: 



E ^3, i,(0) = Z "i'^ "^^ 



CO) 



This result is hardly unexpected, for it says simply that Cj^ must be zero if 

 a^(co) :|: 0, unless two or more a^(co) are non-zero in such a way that this sum 

 vanishes without the individual terms all vanishing. 



It is now evident that the above sum can be omitted from (B2) and thus from 

 the modified (15). However, the equation will not apply when w = o. With this 

 exception, Eq. (15) remains valid even when a^(ai) does not exist, provided only 

 that we replace ci^(co) by a^^(co)/ico. 



APPENDIX C 



Alternative Derivation of (17a) and (17b) 



Equations (17a) and (17b), relating the added mass and damping coefficients, 

 can be derived in a way which avoids inverting one of the transforms and using 

 the inversion to find the other transform. Thus it also avoids the double inte- 

 gration and the interchange of limiting operations (which were not proved valid) 

 in the derivation of (17a). The proof which follows is not entirely rigorous 

 either, but it shows some of the physical bases on which the final formulas 

 stand. 



The kernels K . ^( t ) in the convolution integrals all have the property that 

 K(t) = for t < 0. (I shall omit the subscripts hereafter.) Furthermore, they 

 approach zero as t -» co. Therefore the Fourier transform of K(t) can be 

 written: 



78 



