A New Appraisal of Strip Theory 



By similar reasoning, z^, the coefficient of the heave velocity term is ap- 

 proximated by 



-J 



+ L/ 2 



N(x) dx (34) 



L/2 



where the integrand is the damping coefficient of the section and is calculated 

 by the Havelock-Holstein [4] formula 



N(x) = Ck)' Pe (35) 



e 



where A = ratio of the amplitude of the wave created by the oscillation of the 

 body to the amplitude of the oscillation of the body. 



With the exception of the restoring coefficients c, c, f , F which can be 

 evaluated on the basis of elementary hydrostatics, the remaining hydrod3niamic 

 derivatives of the equations of motion can be computed based on the knowledge 

 of added mass and damping coefficients of cylinders of various shapes. The 

 proposed expressions are summarized in Table 2 and compared to those devel- 

 oped by Korvin-Kroukovsky and his associates. Since the equations of Korvin- 

 Kroukovsky were developed with the origin of the body axes fixed at the center 

 of gravity of the ship, the new coefficients refer to the modified set of equations 

 in which xq is set equal to zero. Proper consideration was also given to the 

 different definition of the total vertical force existing between the two approaches. 

 Thus the expressions of Korvin-Kroukovsky have been corrected to allow for the 

 fact that the total force is to be taken positive downwards. 



Table 2 shows that the expressions for four of the newly proposed coeffi- 

 cients do not agree with those derived by Korvin-Kroukovsky. The differences 

 in the Korvin-Kroukovsky coefficients e(co^), B(cd^) , c and E(Wg) appear to be 

 mainly due to an erroneous time differentiation of a fixed body coordinate with 

 the result that: (a) a factor of 2 appears in the velocity dependent terms of 

 e(Wg) and B(c^g), and (b) a pseudo-three-dimensional term is introduced in co- 

 efficients e(a)g), B(a;^), C and E(ajg) . 



It would also appear that the introduction of terms dependent on the rate of 

 change of added mass over the ship length is inconsistent with the use of two- 

 dimensional theory. Despite these discrepancies however, it is expected that 

 the final values of these coefficients will not be seriously modified since it has 

 been shown by Jacobs et al. [5] that most of these terms which appear in the 

 Korvin-Kroukovsky approach but not in the new approach are numerically small. 

 It is hoped that in the near future these inconsistencies will be examined more 

 carefully and their implications assessed on the basis of experimental data. 



Since most of the coefficients of the equations of motion depend on the theo- 

 retically computed added mass and damping coefficients for two-dimensional 

 cylinders, this matter will next be considered in some detail. The first solution 

 of the potential problem of an infinite circular cylinder oscillating at zero for- 

 ward speed in an ideal fluid was given by Ursell [28] and his results for added 



333 



