Even though the initial condition is different from the previous one, the 

 solutions tend to be similar as time increases and are, therefore, both 

 applicable to the problem of shoreline sand dumping. 



Also of interest is the solution, proposed by Larras (1957] , of a 

 beach equilibrium shape between two headlands or groins described by the 

 equation: 



9Q 

 9s 



where s is the distance along the shoreline. This" indicates no sand 

 transport along shoreline configuration and, therefore, yields an 

 equilibrium to obtain: 



7a 

 ds = L cos —T- da (where L is a proportionality constant) , 



which gives 



„ r . 11a 11 . 3a "I 



^ = ^ r^"^ "^ """"^l (22) 



^ r 11a 11 3a "1 

 y = -R l^cos ^ + ^ cos -^ J 



Equation (22) defines a hypocycloidal form as might be found between two 

 headlands (see Fig. 10) . R is a parameter which is related to the 

 relative curvature of the shoreline. When R ^- <» , a straight shoreline 

 solution is obtained. 



Another family of solutions was given by Grijm (1960, 1964). In 

 these two publications, Grijm used the most commonly accepted expression 

 for dependence of longshore transport on angle, f(a) = sin 2a , and 

 applied the theory to cases where the angle of incidence, a , is not 

 necessarily small. Subsequently, he established the kind of shoreline 

 which can exist mathematically under steady-state conditions. 



Even though the theoretical approach obeys the same physical assump- 

 tion as the previous theory (except for the allowable range for the 

 angle of incidence), his mathematical formulation is not as simple. The 

 shoreline is defined with respect to a polar coordinate axis. The con- 

 tinuity equation is solved in parametric form, which is integrated 

 either by computer or by graphical methods. Details of Grijm' s compu- 

 tations are not available. 



26 



