a. Linearization and Solution Technique. To obtain closed form 

 analytical solutions to the equations derived in the previous section 

 it is necessary to linearize these equations. To justify a linearization 

 of equations CA-20) and (A-21) we must impose the condition that 



|n| << h , CA-22] 



in which case h+n may be replaced by h. This condition implies that 

 the term U8U/Sx also may be omitted. Hence, the final task is to 

 linearize the bottom friction term. Since we will be looking for 

 periodic solutions to the linearized equations this is conveniently 

 done by taking 



V = -hT^ = —IT— ' . ^^-233 



in which fj^ is treated formally as a constant and oj is the radian 

 frequency of the periodic motion. 



Performing the above linearizations of the governing equations, 

 we obtain 



It - 37 ChU) - (A-24) 



and 



U' ^-^^ V" - ' ■ (A-25) 



To illustrate the solution of this set of equations we consider 

 the simple case of periodic waves of radian freqviency, to, propagating 

 over a horizontal bottom, i.e., h = hg = constant. To facilitate the 

 solution complex variables are introduced by defining 



U = Real {u e^'^^} 



n = Real { c e^*^^} , CA-26) 



where i = /^ and the amplitude functions, u and c> are complex 

 functions of x. The physical solution is given by the real part of 

 the solution as indicated by the notation Real { }. Introducing 

 equation (A-26) in equations (A-24) and (A-25) , these may be written 

 in terms of the amplitude functions: 



iwc + h 1^ = (A-27) 



o 8x ^ -^ 



109 



