Understanding and Prediction of Ship Motions 



If 



03 



(t) dr 



is absolutely convergent, then the Riemann-Lebesgue lemma* says that 



CO CO 



lim K-.(t) sin cor dr - lim K-, (r) cos wr dr = . 



Then, from (16), it is evident that 



so that we know the constant term bj^ if we know b*^(aj) for all co (as assumed). 



I The inverse of the cosine transform is given by: 



Kjk(t) = Tij [blk(^) - bju] cos -t d. 



Then the added mass is: 



00 CO 



M-k(^) = Mjk - 7^ I sin a)t rb*k(^') - bjk] cos a;'t do;' dt . 



'- 



The last formula would be rather awkward for purposes of computations, 

 and indeed a much simpler formula is possible. For the moment, let the upper 

 limit of the outer integral be a large positive number, M, and interchange the 

 order of integrations. Then we can use the Riemann-LeJDesgue lemma again in 



letting M ^ CO, finding: 



Mjk(^)-^jk = - ^ ^Ji™ J ^^'[''jk('^')-bjk] J^ sin a;t cos a;'t dt 



1 1- I r \'u* ^ '^ K 1 [ c°s (c^' + oj)U 1 



= — lim J b;,(w)-b;, —. - — r- — 



77a; M-»colJ L^ -"JL CO + oj co + a 



cos (oj' - cj) M 1 



do)' 



- [ [b;k(-')-^k] 



= ^f[b:kC-')-bjk] 



dco' 



co' - 60 co' 



1 , , 2 f r, * , ,, , n do;' 



doj =—4- b-,(c<J)-b-, ■ 



+ ^J ^ Jo L ^^ J^J a;'2-a;2 



(17a) 



*See, for example, Whittaker and Watson (1927), p. 17Z. 



41 



