Kotik and Lurye 



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F(t) = Tpp_;(oo) §(t) + — rp \ p^(o-) sin crt do- (3.3) 



CO 



F(t) = Tpp;(co) S(t) + ^ rp \ [Ap;(a)] cos at da (3.4) 



(S(t) has dimensions T'M- Note that the S -function acceleration of the body 

 produces a s -function hydrodynamic force having strength proportional to the 

 added- mass parameter at infinite frequency. Heave at infinite frequency is uni- 

 form translation of the double body in an infinite fluid. Note also that the two 

 integrals in (3.3) and (3.4) are equal. This implies that 



00 



r Ap;Ca)da = , (3.5) 







a useful fact which does not seem to have been observed previously. 



The relations Eqs. (3.2)- (3.4) are useful for direct calculation, when p^(a) 

 and/or p^C^) are known, exactly or approximately, and for finding asymptotic 

 expansions as t->0, as. For example, to find F(t) as t^co, we first write 



I 00 00 



r ;, s • , (. s COS at f COS at ^ r /^ o j 



I p^(a) sin atda = - p^(a) — p- + J — p- — [p^(a)] da 



= ^^o(i). (3.6) 



Now as stated in [5], for the heaving motion of a cylinder of arbitrary section, 



p;,(o) = (2a) Vr, (3.7) 



where 2a = width at the free surface and r is the submerged volume per unit 

 length, so that for such a cylinder, we have from Eq. (3.3) 



F(t) -^rp^ = ^ .s t-^^. (3.8) 



^ ^ 77 TTt 77t 



This hydrodynamic force per unit length exerted by the body on the fluid is 

 downward if the S -function acceleration is downward. 



For an arbitrary heaving three-dimensional body we have, as noted in [5], 



p'(a-) = p (Ka) = bjKa+o(Ka), (3.9) 



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