The overall importance of the accuracy of specified wave heights on the 

 resulting nearshore circulation and rip current models should not be under- 

 estimated. 



3. Analytic Solutions . 



Bowen (1967, 1969b) was the first to analytically solve the two-dimen- 

 sional motion equations (Eulerian form, eqs. 98, 99, and 100). Of interest 

 at that time was theoretical proof that nearshore circulations and rip cur- 

 rents could be produced by variations in breaker height along the coast. 

 Steady regular waves of normal incidence on a plane beach with a periodic 

 wave height variation alongshore were specified. Convective acceleration 

 and wind stresses were neglected. In a first approximation, by also neglec- 

 ting the lateral mixing stresses, an analytic solution was obtained in terms 

 of the mass transport stream function, ^. As a second approximate solution, 

 only convective acceleration, radiation stress, and lateral mixing terms 

 were considered. This case produced a nonlinear problem requiring numerical 

 integration methods, employing successive, overrelation procedures. In both 

 cases, oversimplified bed shear and eddy viscosity models reduced the ac- 

 curacy but not the general, qualitative nature of the results (Fig. 33). 



The vertical axis in terms of wavelength of the alongshore wave height 

 variation reveals that in both cases, rip currents form in regions of low 

 wave height. The rip currents also strengthen (narrow streamlines) as the 

 eddy viscosity decreases (increasing Reynolds number). _ It is not clear what 

 Re = means (Fig. 33, b) since by definition. Re = UL/v . Also these same 

 figures show the maximum current outside the breaker line. The solution with 

 only bottom friction (Fig. 33, a) is more realistic in this regard by showing 

 the strongest longshore currents in the surf zone (left of the breaker line) . 

 Since specification of the alongshore wavelength for alongshore wave height 

 variation fixed the circulation cells, Bowen solved a forced circulation pro- 

 blem. However, these shortcomings do not diminish the pioneering importance 

 of Bowen' s efforts. 



Sonu (1972) repeated the theoretical analysis with only bed shear stress 

 for the case when wave height variation alongshore is created by bottom topo- 

 graphy. Here, rip currents returned offshore where water depths were large, 

 as expected. Other solutions followed for different initial or boundary con- 

 ditions and by making assumptions that were analytically tractable. O'Rourke 

 and LeBlond (1972) investigated the circulation currents induced by oblique 

 waves in a semicircular bay. They neglected the convective acceleration and 

 lateral stress terms. Similar analyses were ];-eported by Noda (1972a, 1972b, 

 and 1974). 



LeBlond and Tang (1974) also recognized that the resulting current pat- 

 terns interact with the wave heights initially specified. Bowen 's work was 

 extended to include wave-current interaction effects. More importantly, 

 LeBlond and Tang also sought to solve a free oscillation problem and hypo- 

 thesized that the wave-current feedback mechanism would create a preferential 

 spacing for rip currents. But, as discussed in Chapter 2, it became necessary 

 to invoke an additional condition in their eigenvalue formulation which turned 



120 



