where 



p = density 



t = time 



T = wave period 



d = water depth 



n = free- surface elevation 



g = gravity acceleration 



p = pressure 



V(u,v) = particle velocity 



z = vertical ordinate 



In the general case, linear or nonlinear, where the flow motion 

 can be expressed by a potential function 4)(x,z,t), the Bernoulli 

 equation yields 



-*t = g" " F " ^ ^'^ ^^^ 



and u = (})^ so that the energy flux becomes 



.t+T ^n 



'-d 



av T J, J , t"*'x 



dxdt (3) 



in which case (}> can be expressed at any order of approximation, such as 

 given by a Stokesian power series. Even though classical solutions for 

 cnoidal waves are irrotational , the potential function is not expressed 

 but rather the solution for (ri,u,v) is given; therefore, the energy flux 

 for cnoidal wave is determined from equation (1) where (V = u + w ] . 



The results of all the calculations pertinent to linear wave 

 theory and linear wave shoaling are given in Le Mehaute (1976) . 



Instead of expressing (f) at a first order of approximation as in 

 the linear wave theory, (|) is expressed at a higher order in equation 

 (3), the shoaling coefficient Kg = H/Hq becomes not only a function 

 of d/L or d/Lp but also a function of the deepwater wave steepness, 

 Hq/Lo- 



This calculation has been performed at a third order of approxi- 

 mation [Le Mehaute and Webb, 1964), and the fifth order of approximation 

 (Koh and Le Mehaute, 1966) based on the third-order solution and fifth- 

 order solution for a Stokesian wave as developed by Skjelbreia and 

 Hendrickson (1960). The first definition of Stokes for the phase velocity 



