4> 



-ik A+^ ^ 



- X . f , (cos*. - sind), ) X 

 o b b b 



^/ 2 V 2 



-k / 1 + f, sinA, X -ik / 1+f, cos*, x 



= a^ e ° ^ "^ e ° ^ 1^ , (A-36) 



where a may be considered a real number. When this is introduced in 

 equation (A. 26) and only the real part is retained we obtain 



4> 



V 2 



-k / 1 + f, sintj), X 4a y 



n = a e cos (ujt -k / l + f, cosd), x) , (A-S?) 



+ + ^obb 



which shows the solution to be that of a sinusoidal wave propagating 

 in the positive x-direction with an exponentially decreasing amplitude. 

 In the same manner the wave solution of amplitude a_ in equation 

 (A-31) may be shown to represent a wave propagating in the negative 

 x-direction with an exponentially decreasing amplitude in the direction 

 of propagation. Thus, we have obtained a formal solution to the problem 

 of a long wave propagating over a rough bed. However, the solution 

 must be considered formal since it depends on the value of the 

 linearization factor f^ introduced through equation (A-23) . To obtain 

 the appropriate value of the constant f^, it must be related to the 

 true friction factor, f^^, and the wave characteristics. 



b. Application of Lorentz' Principle . To obtain an explicit 

 solution for the linearized friction factor, fj^, we use Lorentz' 

 principle of equivalent work. This principle is a useful tool for 

 obtaining approximate results from a set of linearized governing 

 equations in which the nonlinear term expressing the flow resistance 

 has been linearized. The principle (Ippen, 1966) states that the 

 average rate of energy dissipation calculated from the "true" nonlinear 

 friction term and that calculated from its linearized equivalent should 

 be the same. As shown by Kaj iura (1964), the instantaneous rate of 

 energy dissipation per unit bottom area may be approximated by 



E^ ^ Ut, . (A-38) 



u — b 



For a given area. A, and assuming a periodic motion of period, T, the 

 average rate of energy dissipation becomes: 



Ep = M dA {i 1^ Ut^ dt} , (A-39) 



