The shoreline may be deduced at any time, t , by a homothetic trans- 

 formation about the oy axis from the knowledge of the shoreline at a 

 given time, t , and also by applying the simple relationship (see Fig. 3) 



AD AC 



(17) 



ft - 0.38t^l ^^'^ \t^ - 0.38t 1 



The theory of Pelnard-Considere has been verified in laboratory ex- 

 periments with fairly good accuracy. The steady-state littoral drift, 

 Q , was obtained experimentally from preliminary calibration over a 

 straight shoreline. The results of these experiments are shown in Figs. 

 6 and 7. However, the shoreline predicted by theory is not expected 

 to be valid downdrift of the groin because of the influence of wave 

 diffraction around the groin tip. Some sand begins to bypass the groin 

 by suspension before t = t (see Fig. 5). Also, different boundary 

 conditions apply to different contour lines since the deeper contour 

 lines reach the end of the groin before the contour lines which are near 

 the shoreline, which implies the one-dimensional theory is no longer 

 entirely satisfactory. 



Subsequently, Lepetit (1972) also conducted laboratory experiments 

 which verify the results of a numerical scheme based on the theory of 



Pelnard-Considere. He used the law, Q°: sin a Vcosa . Lepetit's ex- 

 periments were carried out with a very small angle between wave crest 

 and shoreline. 



1 . Refinement and Extensions of the Pelnard-Considere Model . 



After Pelnard-Considere ' s contribution, the mathematical formula- 

 tion of shoreline evolution has proceeded at a slow pace. The first 

 refinements came in improving the longshore transport rate (littoral 

 drift) formula, in particular, modifying the expression relating sedi- 

 ment transport to incident wave angle. 



Based on results from laboratory experiments performed by Sauvage 



7a 

 and Vincent (1954) , Larras (1957) introduced the function f (a) = sin -j- , 



also used by Le Mehaute and Brebner (1961) . New theoretical forms of 



shoreline evolution are determined as solutions of the diffusion equa- 



7a 

 tion. Introduction of the relationship f(a) = sin — instead of tana, 



allows obtention of solutions valid for larger wave angles. 



Of particular interest are the cases of shoreline undulations, since 

 assuming linear superposition, any form of shoreline may be approximated 

 by a Fourier series. The solution of the diffusion equation is then of 

 the form: 



22 



