Current Progress in the Slender Body Theory 



where m^ .(x) is the two-dimensional added mass coefficient with a rigid free 

 surface condition.* 



We can now write down the heave force F3 and pitch moment F^ from Eqs. 

 (2.11), (2.12), (2.17), (2.19) and (2.20), using Eqs. (Al.3-6) of Appendix I to 

 evaluate the surface integrals. Thus it follows that 



= -We 



2 I X - <f I 

 - ±c.^pe-'"^' j /A\B(x)r log L sgn(x-a 



^^[B(a (^3- -^5 -iAe^'^'^jJdfdx 



+ Yo(K|x-^l) - 2iJo(K|x-^|)} d^dx 

 + ipgAe-i"* I (-^] [B(x) -KS(x)] e^'^'^dx + 0( e') , (2.21) 



where the last term represents the slender body approximation of the Froude- 

 Krylov exciting force and moment, to second order in e. 



The total force and moment will include the hydrodynamic components, 

 represented by (2.21), plus the conventional hydrostatic restoring force and 

 moment (of first order in e) and the inertial force and moment of the ship's own 

 mass (of second order in e). Setting the sum of these equal to zero yields a 

 consistent set of equations of motion, accurate to second order in e . We note 

 that the first order contributions include only the hydrostatic terms plus the 

 first order Froude-Krylov contributions. The solution of this first order sys- 

 tem was illustrated in Figs. 1 and 2. There are various second order contribu- 

 tions in Eq. (2.21), each of which is interesting by itself. The first integral 

 gives the "strip theory wall forces" involving the stripwise zero frequency 

 added mass times the relative acceleration, including the incident wave height. 

 The first double integral gives the corresponding "wall" three-dimensional cor- 

 rection to the added mass and exciting force. The second double integral con- 

 tains the free surface effects, including an added mass contribution from the 

 real part of the kernel Hg + Y^, , and a damping contribution from the imaginary 

 term -2iJn. Note that in all cases the relative displacement i, - x^ - iAe^'"" 



*This is not a unique definition by itself. We may say that mjCx) is the coefficient 

 of the force associated with the pressure icLip<i>^^^'> e'^"^, and (p^^^^ must be of 

 the form ^(20) ^^ log( y^ + ^^)/L2 as y^ + z^ ^co. 



143 



