74 



At first order: 



At second order: 



<7l = (5i) 



^3=(25i) 



<75 = {Si + 52) 



cr2 = {S2) 



^4 = (252) 



0-6 = {Si - S2) 



At third order: 



^7 = (35i) 

 as = {2Si + S2) 

 aii = {Si+2S2) 



as = (352) 

 ^10 = {2Si - S2) 



<''12 = (^i — 2^2) 



And at fourth order: 



^13=(45i) 



(Ti5 = (SS'i +52) 



ai7 = (25i+252: 



0-19 = (^i + 3^2) 



^14 = (452) 



uie = (35i - 52) 



C7i8 = (25i-252) 

 <720 = {Si — 352) 



It might be possible to include a larger number of intersecting free waves in order 

 to capture a broadly directional sea, but it would introduce additional complication 

 in distinguishing the effects of the individual waves, and will not be considered here. 

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

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

 determined by the free surface boundary conditions: the modified kinematic free 



