107 



z = zi + Z2(x) (7.12) 



and Eq. 7.11 becomes 



X. , X 



;'Sd. = ;^gid..;2dz2^.„ ,,^„, 



x^ x^ x^ 



Since Z]^ >^ f(x) and in fact az/3t = dS/dt over the active profile length, 

 L, and by the chain rule, 



|^ = ^g (7.14) 



at 5x 3t 



and ax/3t is uniform over depth and equal to — dR/dt, Eq. 7.13 becomes 



zi(xi) 



and the active depth is h*+B, we finally obtain 



M = as L 



at at (h*+B) (7.16) 



which is the Bruun Rule. 



We now return to a somewhat more conceptual version of the cross -shore 

 sediment budget equation, expanding the source term. Considering the case 

 of shoreline stability for illustration purposes 



