The wave height is determined from the conservation of wave action (Jonsson 1990, 

 and others); 



_d_ 



dx 



(E(C + U)] 



= 



(6) 



where E is wave energy, C^^ is relative group velocity of the waves, x is wave 

 propagation direction, and co^ is relative angular frequency. Equation 6 assumes no 

 dissipation due to breaking or bottom friction. The subscript r represents variables 

 measured relative to the current, i.e., variables in a coordinate system moving with 

 the current. This one-dimensional formulation was developed under the assumption 

 of no refraction or diffraction, which is a reasonable assumption for normally 

 indicident waves in the idealized inlet. The wave energy is determined from linear 

 wave theory as: 



E = 1 pgH' 



(7) 



where H is wave height and p is water density. The relative angular frequency is 

 given by: 



co^ = ^gk taiihkh 



(8) 



Equation 8 is similar to Equation 2 for the situation of U=0, but its application is 

 different. Equation 8 is used to solve directly for (o^ with the value of A^ determined 

 from Equation 2. The relative group velocity is given by: 



1 to 



c=- — 



^^ 2 k 



1 + 



2kh 



sinhlkh 



(9) 



Applying Equation 6 between an offshore Region 1 where the current is negligible 

 and a Region 2 in the channel (which may have a different depth and a current) 

 gives: 



^EC' 



fE(C + U)] 



Sr 



«. 



(10) 



Solving for the wave height in Region 2 gives: 



16 



Chapter 4 Results 



