small windows, in each window there may be only one or two free waves dominating 

 the motion. 



Taking advantage of the local nature of the approach, only one or two intersect- 

 ing free waves are considered in each window. While unlikely to capture the full 

 complexity of a broadly directional sea, this approach should be able to capture the 

 dominant modes in each window. The direction, frequency, and amplitude of the 

 dominant modes will vary substantially from window to window as they do in the 

 two dimensional approach, having an integral effect that includes a large variation in 

 directionality. 



The two wave method is expected to be particularly effective for the case of 

 standing waves or short crested waves, as would be found near a reflecting surface 

 when the incident wave field is almost unidirectional, as it often is in shallow water. 



Expanding Eq. 4.11 to fourth order, the potential function for two intersecting 

 waves is: 



20 



4>{x,y,z,t) = V\x + UyV ^^Ai(£{KihJ<,z)smai (4.18) 



where: 



cosh K,{h + z) da^\da^^ 



cosh hih V ox Oy 



Si = ik:r,lX + ky^iy - LJit + Qi) 

 S2 = {kx,2X + ky^2y — ^2^ + 02) 



At first order: 



CTi = (^i) cr2 = {S2) 



At second order: 



^3 = (25l) (74={2S2) 



0-5 - {Si + S2) (Tq = {Si - S2) 



