Ogilvie 



apparent incident stream in the observation reference frame. The 

 latter has the velocity potential Ux', and so the complete potential 

 is: 



^(x',y',z',t) = Ux' - (U cos 65+ icoljsin ^5) <|),(x,y ,z) 



+ (- U sin ^g+ ic43 cos ^5)4)3(x,y,z) + icolgcfjglx.y ,z) 



(2-77) 

 ^ Ux'- U<f>,(x,y,z) +[ico«j>3(x,y,z)]i3(t) 



+ [ia)#5(x,y,z) - U<<»3(x, y ,z)] ^t) . (2-77') 



The potential ^ has been defined basically in terms of the inertial 

 reference frame, although most of the right-hand side here is ex- 

 pressed in terms of the body- fixed system. Note that not only the 

 incident stream is defined in terms of primed coordinates, but also 

 the body motion is really defined in those coordinates as well; in 

 particular, heave motion is a translation of the body along an axis 

 fixed with respect to the fluid at infinity. 



The Bernoulli equation must be used for computing the pres- 

 sure: 



-£ = 



<}>, + -i («!»,. +<^y. +4>?). 



The linear approximations of the derivatives here are as follows: 



(|>t = [(iu))%3+(ia)U)^je3(t) 



+ [(io)%5- (icoU)<j>3+ (iu)U)(z4»,^ - xc|,,^)]e5(t); 



4»^. = U[ 1 - ^,J + [ iw<}>3j k^{t) + [ ico4>5^ - U<j>3^ - U4>,J IgCt); 



<|>y. = - U[ (}>,^] + [ iu,<j»3^ i^{t) + [ iu,<)»5^ - U<j)3y] i^{t)', 



<«>,. = - U[ i^.J + [ ico4J>3j e3{t) + [ ico4»5^ - U*3^ + ^\] e5(t). 



Some simplification has been done through the dropping of quadratic 

 terms in ^j . Substituting these expressions into the Bernoulli 

 equation and simplifying somewhat, one finds that: 



726 



