Governing Equation for Shoreline Change 



107. The equation governing shoreline change is formulated by conserva- 

 tion of sand volume. Consider a right-handed Cartesian coordinate system in 

 which the y-axis points offshore and the x-axis is oriented parallel to the 

 trend of the coast (Figure 6). The quantity y* thus denotes shoreline posi- 

 tion, and x denotes distance alongshore. It is assumed that the beach 

 profile translates seaward or shoreward along a section of coast without 

 changing shape when a net amount of sand enters or leaves the section during a 

 time interval At . The change in shoreline position is Ay , the length of 

 the shoreline segment is Ax , and the profile moves within a vertical extent 

 defined by the berm elevation D B and the closure depth D c , both measured 

 from the vertical datum (for example, MSL or MLLW) . 



108. The change in volume of the section is AV = AxAy(D B + D c ) and is 

 determined by the net amount of sand that entered or exited the section from 

 its four sides. One contribution to the volume change results if there is a 

 difference AQ in the longshore sand transport rate Q at the lateral sides 

 of the cells. This net volume change is AQAt = (3Q/3x)AxAt . Another 

 contribution can arise from a line source or sink of sand q = q s + q , which 

 adds or removes a volume of sand per unit width of beach from either the 

 shoreward side at the rate of q s or the offshore side at the rate of q . 

 These produce a volume change of qAxAt . Addition of the contributions and 

 equating them to the volume change gives AV = AxAy(D B + D c ) = (3Q/3x)AxAt 



+ qAxAt . Rearrangement of terms and taking the limit as At — > yields 

 the governing equation for the rate of change of shoreline position: 



£ * (dttdt, ( i -«.) " ° <» 



109. In order to solve Equation 1, the initial shoreline position over 

 the full reach to be modeled, boundary conditions on each end of the beach, 

 and values for Q , q , D B , and D c must be given. 



For convenience, symbols and abbreviation are listed in the Notation 

 (Appendix E) . 



50 



