where 



and 



^r-^ T -M 



k .^"^if = k /s^ + f^ e ^ (A-81) 



o o 



As was the case in the solution for long waves over a rough bottom, 

 the above solution is formal only, since the appropriate value to be 

 assigned to the linearized resistance coefficient f has not been 

 determined. 



To obtain the appropriate value for f, Lorentz' principle is used 

 with the rate of dissipation per unit volume given by 



^D " " ^^d • (A-83) 



Thus, we obtain: 



f ^ U^ = aU^+ b|u|u^ , (A-84) 



where the spatial average is to be taken over the volume occupied by the 

 fluid. 



As a simple example we consider the velocity given by 



U = U cos ojt , (A-85) 







for which equation (A-84) yields: 



f = -(cc + I- 3 U ) . (A-86) 



oj^ 3tt o-^ 



In the discussion of long waves propagating over a rough bottom the result 

 corresponding to equation (A-86) was given by equation (A-46) . However, 

 although the friction factor, f^^, in equation (A-46) may be considered 

 known by applying Jonsson's (1966) empirical results, we have not 

 established a similar empirical relationship for the hydraulic properties, 

 a and 3, of a porous medium. 



120 



