Current Progress in the Slender Body Theory 



-p[-ia;0(j^ + Vc/)^0) • V0(ij .• (3.22) 



(1) 



Equations (3.21) and (3.22) give the steady and linear unsteady pressure fields 

 for an arbitrary body. Now if the body is slender, both pressures may be con- 

 sistently approximated in the form 



(WALL) (FS) , . 



'(0) = P(0) (^'V'^) + P(0) (""^ ^•^•^•^^ 



(1) 



( WALL) (FS) /„ „^\ 



pji) \x,y,z,t) + p;j/(x,t) , (3.24) 



as was done for the potentials. We shall not write down the "wall" pressures, 

 which are not required for the present analysis; the free surface contributions 

 are 



P^o^ = -P^<1?'(-) (3.25) 



p[7,''= -p(-i^+u^)f;n\x,t). (3.26) 



These formulas give the free surface dependent part of the pressure everywhere 

 in the field of flow. In particular from (3.14) we can express the unsteady pres- 

 sure field as a sum of contributions from each mode of motion, with appropriate 

 transfer functions. 



In order to find the forces on the ship we require the pressure on the in- 

 stantaneous hull surface. This is obtained by evaluating the pressure on the 

 equilibrium hull surface and adding a correction term to account for the dis- 

 placement of the ship in the non-uniform steady flow field. Thus if p^ j ^ now 

 denotes the unsteady pressure evaluated on the equilibrium hull surface, then 

 the unsteady pressure on the actual hull is 



P(i) + ^ • ?P(0) + 0^^^) ' 



where g. is the vector displacement of the hull at any point. In particular the 

 correction to the FS part of the pressure is 



( FS) 3 ( FS) 



a • VP(O) (X) = ^^_p^^^ (X) 



= -^iUfJo'^"(x) , (3.27) 



where ^i(t) is the surge displacement (note from (A2.1) that the x component 

 of a also contains terms involving the pitch and yaw angles ^^(t) and ^ g(t), but 

 these contributions are negligibly small in e compared with other retained 

 terms). Thus the only contribution from this correction is in surge excited mo- 

 tion, and we can write for the pressure on the actual hull 



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