Current Progress in the Slender Body Theory 



M^.(oj) = Re 



(-ia;)2 



(3.3) 



in phase with the accelerations (-iw)^ ^. , and "damping coefficients" 



B^.(co) = Re 



C- -(a;) 

 1 j ^ ' 



(-iw) 



(3.4) 



in phase with the velocities (-iw) l,.. 



Thus there are three interpretations of the linearity Eq. (3.1). When (3.1) 

 is to be viewed as an operator equation we write C^j = CijC^) but reserve the 

 combination "-iw" to mean the operator "B/Bt." The second interpretation 

 views Cij(aj) as a transfer function with 6J as a (complex) Fourier transform 

 variable, while the third views Cij(w) as the frequency response for real w. 

 The following analysis may be given any of the three interpretations although it 

 is mainly expressed in the language of the second of them; that is, we seek a set 

 of 36 complex valued transfer functions Cij(&j) of the complex variable w. How- 

 ever, in practice one need calculate the Cij only for real w, so that added 

 masses and damping coefficients would be obtained directly, as in the third in- 

 terpretation. 



Evaluation of the Velocity Potential Near the Ship 



Firstly let us linearize with respect to the amplitudes Cj(t) of motion, 

 which are assumed to be small of order a, for some small parameter a which 

 measures the general size of the motions. Thus we expand the velocity poten- 

 tial in the form 



(0) 



(x,y, z) + 



(1) 



(x,y,z,t) + 



(2) 



(x, y, z, t) + 



(3.5) 



where 



(0) 



is the steady flow due to a uniform stream u past the ship fixed in 



its equilibrium position, while 0( i) = 0(a) is the first approximation to the un- 

 steady potential for small oscillations of order a about this position. Further 

 terms 0( 2) ... describe non-linear effects due to not-so-small oscillations and 

 will not be investigated in this paper. 



Now if the ship is slender, each of the potentials 0( o ) » '?^( i ) > • • • ^^V ^^ 

 further expanded in terms of the slenderness parameter e, in a manner typified 

 by the expansion of the steady term cp^ ^ . which has been obtained previously 

 (Tuck, 1964). Thus near the ship we can write 



(0) 



= Ux + 



(WALL) (FS) 



\o) (x,y,z) + f^o^ (x) 



+ 0(e^ log^ e) , 



(3.6) 



where the term "Ux" represents the free stream and is of zero order in e, 

 while the contents of the square brackets are of order e^ log e and represent 

 the steady disturbance to the stream due to the presence of a fixed ship. This 



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