Current Progress in the Slender Body Theory 



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0.8 



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0.6 



0.5 



0.4 



FIRST ORDER THEORY 



SECOND ORDER THEORY 



O EXPERIMENT 



6.0 



Fig. 5 - Pitch response from second order theory 

 compared with first order theory and experiments 



In deriving the second order theory, the fundamental result we use is that 

 any velocity potential 4> representing a regular disturbance of the fluid by the 

 ship can, near the ship, be written in the form 



g!)(x, y, z) 



*ALL)(x,y,z) + f^^^Vx) 



(1.1) 



where 0^ **^^^ is the potential for the identical problem but with the free sur- 

 face replaced by a rigid surface or "wall" (which is, by reflection, the problem 

 for a double body consisting of the ship hull plus its image above the free sur- 

 face in an infinite fluid). The fvinction f<^^^(x) contains all the free surface 

 effects (and in particular is dependent on the acceleration of gravity whereas 

 ^( WALL) ^g j^Q^)^ 2C[v6. is defined by an integral transform of the form 



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