# GenCade Basic Governing Equations

The governing equation of GenCade is primarily formed by conservation of sand volume. The assumption is that the beach profile translates seaward or shoreward along a section of coast without changing shape. During an interval of time (Δ*t*), a net amount of sand enters or leaves the section. The change in shoreline position is Δ*y*, the length of the shoreline segment is Δ*x*, and the profile moves within a vertical extent defined by the berm elevation *D _{B}* and the closure depth

*D*, both measured from the vertical datum (for example, MSL or MLLW).

_{C}The resulting change in volume along the four sides of the area is therefore calculated as:

A definition sketch is shown in Figure 1.

If there is a difference between the updrift and downdrift longshore sediment transport (*Q*, in cubic meters/second), then the volume either expands or contracts shifting the shoreline seaward or shoreward, respectively. The contribution from a line source or sink is included as the term *q*, where *q* is the sum of the addition or removal of sand per unit width of beach from either the shoreward or offshore sides. Thus:

As Δ*t* goes to zero, a differential equation is produced that is the governing equation for the rate of change of the shoreline position:

This equation is solved with the inputs of boundary conditions and values for *Q*, *q*, *D _{B}* and

*D*given.

_{C}