d dU 



— [E(U + C g )] + S xx =0 (83) 



dx dx 



in which U is the horizontal flow speed taken positive in the shoreward 

 direction and S^ is the shoreward component of the radiation stress. The 

 return flow U is created to balance the mass of water thrown shoreward in 

 the crest of the breaking wave. Equation 83 will be evaluated at the end of 

 the tank near the wave board (called deep water) and at the break point. To 

 simplify the discussion the term S^ dU/dx will be omitted, although it is 

 non-negligible. Also, frictional losses in the tank, expected to be small, 

 are neglected. 



142. At the wave board, U must equal zero since the current cannot 

 penetrate through the board. In the open ocean, this is equivalent to U 

 being zero in deep water. With this consideration and the stated assumptions, 

 Equation 83 gives 



[E(-U + C g )] b = [EC g ] (84) 



in which the prime denotes conditions where return flow is present and the 

 minus sign in front of the magnitude U indicates flow in the seaward direc- 

 tion. If no return flow or only a very small flow is present, Equation 84 is 

 simply: 



[EC g ] b = [EC g ] (85) 



Equating the different breaking conditions (with and without a return flow) 

 having identical deepwater wave conditions yields 



[E(-U + C g )] b = [EC g ] b (86) 



Substitution of _the equation for wave energy density (Equation 39) into Equa- 

 tion 86 and elimination of constants on both sides of the resultant equation 

 yields the following expression: 



(H b ) 2 (C gb - U b ) = HgC gb (87) 



96 



