Q = Q K. tan a, 



O 1 D 



S tan a, < 1.23 

 b 



(12) 



Q = Q, 



1.23 < tan a. 



(13) 



where K and K„ are constants. From these equations as a starting point, 

 the growth of river deltas was studied. 



23. Bakker (1969) extends the one-line theory to include two lines to 

 describe beach planform change. The beach profile is divided into two parts, 

 one relating to shoreline movement and one to movement of an offshore contour 

 (see Figure 4) . The two-line theory provides a better description of sand 



1 



Groin -y jfir- 



Figure 4. Definition sketch for the two- 

 line theory (after Bakker 1968) 



movement downdrift of a long groin since it describes representative changes 

 in the contours seaward of the groin head. Near structures such as groins, 

 offshore contours may have a different shape from the shoreline. The two 

 lines in the model are represented by a system of two differential equations 

 which are coupled through a term describing cross-shore transport. According 

 to Bakker (1969), the cross-shore transport rate depends on the steepness of 

 the beach profile; a steep profile implies offshore sand transport; and gently 

 sloping profile implies onshore sand transport. Analytical solutions of the 

 two-line theory are not included in the present report. However, an example 

 of a two-line theory solution for a groin system is shown in Figure 5. The 

 solution describes the stationary form of the shoreline for various groin 

 spacings given in multiples of a nondimensional groin length L 



24. The two-line theory is further developed in Bakker, Klein-Breteler, 

 and Roos (1971) in which diffraction behind a groin is treated. In this case, 

 it became necessary to numerically solve the governing equations. Expressions 

 for the coastal constant (diffusion coefficient e) for the one- and two-line 



14 



