where V- indicates the gradient in the (x. , y. , z. ) system, and $ = $(x. ,y.,z.,t) 

 is the velocity potential (viz., ^q plus a potential due to the presence and 

 motion of the structure). After some simplification, we find that the 

 boundary condition is 



+ IYq - a{z. + a. a.') + \ aQ cos 0.] sin X. [4] 



- [zn + a.r,(a sin 9.-6 cos 9.)] a-" on R. = a- 

 ■■0 0^ 1^ I'-'i 1 1 



. da^ 



where a- = . Second and higher order terms in the motion variables 



1 dz^ 



have been consistently dropped. 



Let $ = $Q + $1 . We substitute this relation into the last equation 



to obtain a boundary condition on $, . From Equations [2] and [3] we 



note that 



)A kz 



$Q = — ; — e COS (kx - wt) 



— e i cos fk(an cos 9. + R. cos A-) - cotl 



5] 



It follows that 



/^^*0 ' ^*M A kz. /r, , ^^ '. 3 v2 2> 

 a. = CO Ae i-^f-l+ka- a. +#k a. )cosX. 



+ 1^ k a- cos 3X.] sin (kaQ cos 0. - cot) 



- [(a-''+ J ka.) + l^ka- cos 2X.]cos (kaQ cos 6. -cot) 



+ ...} [6] 



where the omitted terms are of third and higher order in terms of the 

 spar radius a- and its derivative a.', or of second and higher order in 



