gravity, and the bottom shear stress term introduced in the derivation 

 of equations (A-24) and (A-25) in Appendix A has been omitted. 



For the flow within the porous structure, the linearized governing 

 equations are derived in Appendix A, equations (A-74) and (A-75), and 

 may in the present context be written as: 



n ^ + h ^ = (continuity) (3) 



)t o 3x 



and 



~^rr+g^+ f— U= (conservation of momentum) , (4) 

 n 5t '^ 3x n ^ -^ 



in which co is the radian frequency, 27r/T, of the periodic wave motion, 

 n is the porosity of the porous medium, U is the horizontal discharge 

 velocity, i.e., equivalent to the velocity variable used in equations 

 (1) and (2) , S is a factor expressing formally the effect of unsteady 

 motion (see App. A) 



S = 1 + k:(1 -n), (5) 



where k is an added mass coefficient. With < expected to be of the 

 order <_ k ^ 0.5, equation (5) shows that 1 ^ S < 1.5. The nondimen- 

 sional friction factor, f, arising from the linearization of the flow 

 resistance is related to the flow resistance, which more realistically 

 is given by a Dupuit-Forchheimer relationship through 



f - = a + b|u| (6) 



in which the hydraulic properties of the porous medium are expressed 

 by the coefficients a and B. The coefficient a expresses the laminar 

 flow resistance, which is linear in the velocity. The turbulent flow 

 resistance which is quadratic in the velocity, is expressed by the 

 coefficient 8. The friction factor is regarded as constant, i.e., 

 independent of x and t, in the following. 



With the equations being linear, complex variables may be used. 

 Thus, looking for a periodic solution of radian frequency, to, we may 

 take 



n - Real {c(x)e^'^^} (7) 



and 



U = Real {u(x)e^"^} (8) 



