q o 



iMi -Yo^+ocy; 1 ) 



(86) 



d$ = q ds 



d* 



q Q hd t 



(87) 



Omitting higher order terms in Equation (86) and integrating, we obtain 



5 = Y 



r ds r dt 



~2 * n = Y "2 



J q o J q 



(88) 



Since Yrv/q n corresponds to the local wave number, the new coordinates mean 

 the streamline coordinates with scales vary proportionally to the local 

 wave length. The above result takes account of the distortion of the wave 

 pattern due to the nonuniform base flow velocities near the hull. The 

 boundary condition on the hull surface is satisfied in the case of 

 symmetric flow, because the velocities u and v are tangential to the hull 

 surface at z' = and are significant only in a thin layer of thickness 

 0(y n ) near the free surface. Though the theory developed here looks like 

 a reasonable representation of the actual phenomena, no numerical results 



have been presented so far. It should be noted that a purely numerical 



20 

 method has been employed in a similar boundary value problem by Dawson 



and the result shows a plausible agreement with experiments. 



HYDRODYNAMIC FORCES ON OSCILLATING 

 SLENDER SHIPS 



Application of the Slender Body 

 Theory 



As mentioned in the preceding chapter, the possibility of mathematical 



analysis of the fluid motion around a ship hull depends substantially on 



35 



