as given above, along with the energy equation for steady-state conditions, 

 are expressed in terms of an equilibrium state plus a small perturbation of 

 the steady state. The four dependent variables, ^, n, h and E, are often 

 employed. After normalization of the resulting perturbation equations, they 

 are subjected to small disturbances in each of the four perturbation depen- 

 dent variables involved. These perturbation variables are expressed as com- 

 ponents of a Fourier series periodic in wave number in the longshore direc- 

 tion and in time. The eigenvalues of the resulting system of equations are 

 determined after specification of sufficient boundary conditions. The solu- 

 tion techniques at this stage are quite involved and normally involve poly- 

 nomial expansion methods and numerical methods to determine the eigenvalues. 

 The resulting eigenvalues \ are related to the eigenvalue wave numbers of 

 rip current spacing 2tt/A, i.e., lower eigenvalues mean larger wave numbers. 

 Wave numbers normalized by the surf zone width, x, are usually employed. 

 Consequently, 



2tt 

 Ax^ = :j^ — x^ = a dimensionless eigenvalue wave number (111) 



r 



where L is the rip current spacing. The lowest possible eigenvalues obtain- 

 able (or an infinite number of possibilities) are of principal interest since 

 they represent the fundamental modes of instability (and largest rip current 

 spacings) possible in the surf zone. Parameter studies are then conducted to 

 determine the relationships between the independent variables involved (e.g., 

 tan 3, Yj Cj., etc.) and the eigenvalue wave numbers that result. Because of 

 the mathematical complexities, only regular waves of normal incidence on plane 

 beaches are usually involved. Also the lateral mixing stresses are neglected. 



Under these conditions, Dalrymple and Lozano (1978) showed that 



Xx^ = J- + 2.8 (112) 



where 



\ = -^ (113) 



f '= a Darcy-Weisbach type friction coefficient 

 = 8C^ (as defined in eq. 52) 



and A^ lumps the key independent parameters. They included wave-current re- 

 fraction by rip currents in their analysis and showed that by doing so no 

 extra conditions were needed to predict rip current spacing. 



As pointed out by Dalrymple (1978), all such free-oscillation eigenvalue 

 solutions rely heavily upon y and how energy dissipation is modeled in the 

 surf zone. It is tacitedly assumed that the form for y (constant or x-depen- 

 dent) holds for all orders of perturbation. Miller (1977) and Miller and 



122 



