Forces on an A.C.V. Executing an Unsteady Motion 



order terms, and then writing the remaining terms as a Taylor 

 expansion about the point z = 0. After dropping the higher order 

 terms again, we obtain simply 







z = 



f = o 



J t 



(9) 



The dynamic condition on the surface - the Bernoulli equa- 

 tion - in the stationary frame is 



1 ,, 2 ,2 2. 



<t>. +-5- (<i>t + <t> + 4> 



t 2 £ y z 



- f + -^ + Sf - l - 



10: 



where f is an arbitrary function of time, which may be put to zero 

 without loss of generality. Eq. (10) is now linearized to give 



[*.] ,-0 



+ JL + gf = 

 p 



11) 



The combined free surface condition is obtained from Eqs. 

 (9) and (11) by eliminating f : 



tt + g ^z 



z = 



1 s 



— P « 



12' 



The final boundary condition needed states that there should 

 be no flux through the water bed : 



4> 



z=-d 



= 



13' 



II. 2. The Potential 



The solution of this set of equations can be obtained by an 

 application of the double Fourier transform pair : 



F(w, u) = — 



00 00 



dy f(x, y) exp |-i(wx + uy) [ 



and 



f(x, y) = 



00 co 



f f 



— f dw / du 



•' J J 



F(w, u) exp |i(wx + uy) } , (14) 



and the Laplace transform pair : 



41 



