Paul ling 



through the free water surface, the pressure and its derivatives 

 vanish for that part of the member's surface or volume above the 

 free surface. 



The evaluation of both of these integrals is a straightforward 

 but rather tedious process, yielding the components of force in the 

 ^"H^-directions , and the moments about these axes. We first per- 

 form the indicated differentiation of the two terms in the pressure 

 equation with respect to ^, T|, and ^, then_substitute for ^, t|, 

 and ^, their values transforme d to the o xy z-coordinates. The 

 element of volume is given in oxyz by A dx where A is the 

 constant cross sectional area of the cylinder. Since the cross 

 sectional dimensions are assumed small compared to the wave 

 length, the integrand may be assumed constant over A and, conse- 

 quently, the volume integrals are reduced to one-dimensional 

 integrals in x to be evaluated over the length of the cylindrical 

 member. 



If the member is completely immersed, the evaluation of 

 this integral yields two terms corresponding to the two terms in 

 Eq. (9) for the pressure. The first term is the static buoyancy 

 force or moment and the second is a time -dependent "variable 

 buoyancy" corresponding to the variation in effective weight density 

 of the fluid as a result of the wave motion. 



If the member projects through the free surface, the region 

 of integration must be dealt with in two parts. The first part is the 

 constant volunae of the member below the mean water surface, and 

 the second is the time varying part of the volume of the member lying 

 between this mean waterline and the instantaneous water surface. 

 Evaluation of the integrals over the first part of the volume, i.e. , 

 that part below the mean waterline, yields a result identical with that 

 for a completely submerged member. In evaluating the integral over 

 the second part of the volume, we note that this volume is of the same 

 small order of magnitude as the wave amplitude. Since the velocity 

 potential, (8), and therefore the second term in the pressure, (9), 

 is of this same small order of magnitude, only the first term in the 

 pressure, i.e. , the hydrostatic part, has a linear contribution to 

 this part of the integral. 



In evaluating these integrails , (11), for the pontoon or point 

 volume, we note that the assumed small dimensions of the pontoon 

 imply that the integrands are constant over its volume. Therefore, 

 the integration reduces to taking the product of this volume with the 

 appropriate derivatives of the pressure expression. 



The first parts of the last two terms in (7), i.e. , the drag 

 and added mass forces associated with wave induced water motion, 

 are each computed in a similar manner. Let Uj and Uj represent 

 the components of fluid velocity and accele^atj^on in o^t|^, with Uj 

 and u, the corresponding components in oxyz. A coordinate 



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