which may be written as the requirement that the Stokes' parameter 

 2 



a L 



+ 



o 



<< 2tt^ . (A-56) 



Thus, the limitations on the solutions obtained from the linearized 



set of governing equations are expressed as the inequalities given by 



equations (A-53) , (A-54), and (A-56). For a derivation of the appropriate 



governing equations when equation (A-56) is violated the reader is 

 referred to Peregrine (1972) . 



For a wave propagating over an uneven bottom a vertical velocity 

 may be imposed by the bottom boundary condition (eq. A-6) . So long as 

 the velocity obtained from this boundary condition is smaller than that 

 given by equation (A-48) the preceding limitations are applicable. 

 This in turn may be stated as a requirement that the bottom slope, 



tanB = ll^l , (A-57) 



s ' 9x' ' 



satisfy the inequality, 



tanB^ 



" < 1 . (A- 58) 



kh 



2. Long Waves in a Porous Medium . 



To derive the equations governing the propagation of waves in a 

 porous medium we consider an element as sketched in Figure A-2. 



In a porous medium of porosity, n, the discharge per unit area in 

 say the x-direction is given U = nUg where Ug is the seepage velocity, 

 i.e., the actual mean velocity of the pore fluid, and U is termed the 

 discharge velocity. With this definition it is seen that the discharge 

 velocity, (U,W), for an incompressible fluid and medium must satisfy 

 the continuity equation: 



or for a homogeneous medium, 



3U 8W 



33^ . 3/ = . (A-60) 



15 



