expression describing the variation in Q In a diffraction zone must be 



o 



simple enough to allow an analytical solution. Otherwise, a numerical 



solution technique must be employed (Kraus and Harikai 1983, Kraus 1983, and 



Hanson and Kraus 1986) . If the incident breaking wave angle a is also a 



o 



function of the distance x , another term, eda /dx , must be added to the 



o 



right side of Equation 11. 



16. In summary, the assumptions which comprise the one-line model, in 

 which breaking waves are the dominant sand-moving process, are as follows: 



a. The beach profile moves parallel to itself fassumption of 

 equilibrium of the beach profile) . 



b. Longshore sand transport takes place uniformly over the beach 

 profile down to a depth D (depth of closure) . 



£. Details of the nearshore circulation are neglected. 



d. The longshore sand transport rate is proportional to the angle 

 of incidence of breaking wave crests to the shoreline. 



17. In addition, the following assumptions will be used to derive 

 analytical (closed-form) solutions of the one-line model (Equation 9) : 



a. The angle between the breaking wave crests and the shoreline is 

 small (small-angle approximation) . 



b. The angle of the shoreline with respect to the x-axis is small. 



18. In arriving at all solutions, it is tacitly assumed that sand is 

 always available for transport unless explicitly restricted by boundary and/or 

 initial conditions. 



Overview of Previous Analytical Work 



19. Pelnard-Considere (1956) was the first to employ mathematical 

 modeling as a method of describing shoreline evolution. He introduced the 

 one-line theory and verified its applicability with laboratory experiments. 

 Figure 3 shows a comparison between experimental results and the analytical 

 solution for the case of an updrift groin, as obtained by Pelnard-Considere. 

 Pelnard-Considere derived analytical solutions of Equation 9, the linearized 

 shoreline change equation, for three different boundary conditions: shoreline 

 evolution updrift of a groin (with and without bypassing) and release of an 

 instantaneous plane source of sand on the beach. 



20. Grijm (1961) studied delta formation from rivers discharging sand. 

 In the transport equation discussed in his article, the sand transport rate 



12 



