and 



n 3t ^ 8x n 



where the flow resistance term has been linearized by introducing the 

 dimensionless factor, f, defined by 



a + 6 U , (A-76) 



in which w is the radian frequency of the wave motion, which is assumed 

 periodic. 



The solution technique is that used in the solution of the 

 linearized equations governing the propagation of long waves over a rough 

 bottom, i.e., we take 



n = Real ice } 



U = Real {u e^"'''^} . [A-77) 



For the case of constant depth h = h introducing equation (A-77) in 

 the governing equations leads to 



u - - ■ g". |1 , (A-78) 



laj(S-if) 3x 



and the equation governing the amplitude function 



2 2 



^+ ^ (S-if)c = CA-79] 



8x ^ o 



Both of these equations are seen to be similar to those discussed 

 in Appendix A.l.a and the results obtained there are readily generalized 

 to give the solution for a progressive wave in a porous medium. The 

 general solution consisting of waves propagating in the positive and 

 negative x-direction is found to be given by 



C = a e"^^ + a e^^^ , (A-80) 



19 



