r-Z 



Later Profile yr 



Sequence of changes in elevation S = ] Z, , Zj , ••• !„ [ 



CVC = AX X ( Z, + Z2 + ■•• + Zn) 

 Ttiickness of volume spread uniformly Tti = CVC /AX • n 



Cumulative volume change 

 per unit width 



Distance Offshore 

 Figure 12. Schematic diagram of volume change calculations. 



Farther offshore the cumulative volume curve usually crosses the zero line 

 several times before finally smoothing out. A positive cumulative volume 

 curve indicates additional sediment was supplied from outside the profile 

 area. If this sediment came from offshore, the cumulative volume curve would 

 approach zero again if extended to the pinch-out depth. As it turned out, 

 changes in depth over the 4- to 6-year period were so small near the pinch-out 

 depth that the total volume change summed over the entire active beach was 

 relatively unaffected by deliberate extensions of the cumulative volume curve, 

 and therefore, even more insensitive to actual uncertainties encountered in 

 selecting the pinch-out depth (Fig. 11). Although the cumulative volume curve 

 approached a constant near the pinch-out depth, it usually was not zero. The 

 value of the cumulative volume curve at the pinch-out depth, representing the 

 net change summed over the active profile, is tabulated by station from north 

 to south in Table 1. The concept that equivalent volumes are eroded from the 

 upper beach and deposited offshore (sometimes called Bruun's rule) is clearly 

 invalid when applied to single profiles. In fact, with the given profile 

 spacing, there is no sediment balance even when volume changes are calculated 

 over reaches several kilometers in length. Rosen (1978) pointed out similar 

 local imbalances in the Chesapeake Bay. 



23 



