125 



At first order: 



At second order: 



<T3 = (25i) 



as = {Si + S2) 



02 = {S2) 



C74 = {2S2) 



cr& = {Si - S2] 



At third order: 



<r7 = {SSr) 

 ^9 = (25i + ^2) 

 an = {Si + 2S2) 



as = {SS2) 

 (Jio = (25i — S'2) 

 ai2 = {Si — 2^2) 



And at fourth order: 



<T13 = (45i) 



ai5 = (35i + S2) 

 ai7 = {2Si + 2S2] 



<:^19 = (-Si + 352) 



<7l4 = (452) 



<7l6 = (35i — ^2) 



<7i8 = (25i-252] 



<7"20 = {Si — SS2) 



Both of these potential functions solve the field equation (Laplace, Eq. 4.2) and the 

 bottom boundary condition (Eq. 4.3) exactly. The rest of the solution is determined 

 by the free surface boundary conditions: the modified free surface boundary condition 



2 du dv dw 



+ u -^r- + uv^r + '^''^^r- 



ox ox ox 



du 2 ^^ ^^ 



oy oy oy 



du dv 2 ^^ 



dz dz dz 



(6.3) 



at 



