PART II: SOLUTIONS FOR SHORELINE EVOLUTION WITHOUT 

 COASTAL STRUCTURES 



General Formal Solution 



32. The basic differential equation to solve is Equation 9, together 

 with the associated initial and boundary conditions. An infinitely long beach 

 is assumed to be exposed to waves of constant height and period with wave 

 crests parallel to the x-axis (parallel to the trend of the shoreline) . The 

 shoreline will adjust to reach an equilibrium state in which the longshore 

 sand transport rate is equal at every point along the shoreline. Since the 

 wave crests are parallel to the x-axis, the equilibrium sand transport rate is 

 zero. An initially straight beach is thus the stable shoreline form in this 

 case. If the shoreline shape at time t = is described by a function 

 f(x) , the solution of Equation 9 is given by the following integral (Carslaw 

 and Jaeger 1959, p. 53) : 



^ , j( ) e -( x - C ) 2 /4et ^ (U) 



2 





for t > and -<» < x < oo . 



The shoreline position is denoted by y and is a function of x and t . 

 The quantity £ is a dummy integration variable. Consequently, the change in 

 both natural and manipulated beach forms can be determined if Equation 14 is 

 evaluated. Equation 14 may be interpreted as a superposition of an infinite 

 number of plane sources instantaneously released at t = . The source 

 located at point £ contributes an amount f(£)d£ to the system. Infinitely 

 far away from such a single source no effect on the shoreline position is 

 assumed (boundary condition) . Equation 14 is used to derive most of the solu- 

 tions dealing with various shoreline configurations in the following text. 



