Kotik and Lurye 



Here <?!).( x, y, z) is a time-independent potential function satisfying the free 

 surface condition 



0i(x,y,O) = (2.11) 



and the boundary condition 



dn 



3n 



= -/x. -n on Sp , i = 1, 2, 3 (2.12a) 



(/I. _3xF)-n on S^ , i = 4, 5, 6 . (2.12b) 



0j .(x, y, z, t) is a potential function that satisfies the free surface condition, 

 Eq. (2.6), for t > 0, and the boundary condition 



-~ = -Vx(AixV„)-n on S^ , i=l,2, 3 (2.13a) 



^ = -Vx[(Mi.3>^^)-vJ-n on S„, i = 4,5,6 (2.13b) 



for t > ." 



The initial conditions on i>i- are 



0ji(x,y,O,O+) = (2.14) 



and 



^0ii(x,y,O,t),.o, = -g ^ <^i(''.y.z)z = o- (2.15) 



It can be verified by direct substitution that the function ^^(x, y, z, t) defined 

 by Eq. (2.10) does indeed satisfy Eqs. (2.6), (2.7), (2.8), and (2.9) when the func- 

 tions <^-(x,y,z) and 0j.(x,y,z,t) satisfy Eqs. (2.11) through (2.15). We recall 

 that Si appearing in Eq. (2.7) is given by Eq. (2.3) or (2.4). 



Duhamel's Principle 



We now suppose the body, initially at rest in the stream, to be given (at 

 t = ) a unit displacement in the ith mode. The fact that such a displacement 

 is not small is irrelevant. Let the potential corresponding to the unit displace- 

 ment be 4>^(x,y, z,t). Since in this case x.(t) = S(t), it follows from Eq. (2.10) 

 that 



'''Note that 0- has the dimensions of potential/velocity when i = 1, 2, 3 and poten- 

 tial X time when i = 4,5,6. ^^^ has the dimensions potential/length when 

 i = 1, 2, 3 and potential /angle - potential when i = 4, 5, 6. 



410 



