# Talk:Beach Cusps

The proportion of uprush that is diverted from the incipient horns to the incipient embayments is maximum when the swash diversion (uprush and downrush) equals half the embayment width, i.e. half the beach cusp spacing $L_y$, see Fig. 1. Half the maximum diversion occurs at time $T$ when the uprush has reached its highest point. In the simplest model of frictionless particle motion over a slope, the time $T$ depends on the ratio of distance to slope, for both the cross-shore and longshore directions. The maximum downrush in the embayment thus corresponds to the situation where the ratio longshore uprush diversion $L_y / 4$ to longshore slope $\beta_y$ equals the ratio cross-shore uprush $L_x$ to cross-shore slope $\beta_x$. The beach cusp spacing $L_y$ is four times the longshore uprush diversion, thus proportional to $\beta_y L_x / \beta_x$. The longshore slope $\beta_y$ is inversely proportional to the beach cusp spacing $L_y$ and proportional to the elevation difference $Z$ between horn and valley. This yields the formula $L_y \propto \beta_y L_x / \beta_x \propto Z L_x / (L_y \beta_x)$. The elevation difference $Z$ cannot be larger than the vertical uprush (wave run-up) $R = \beta_x L_x$ and will become a certain proportion of $R$ in the course of cusp development. Substitution gives $L_y \propto Z L_x / (L_y \beta_x) \propto R L_x / (L_y \beta_x) \propto L_x^2 / L_y$ which implies $L_y \propto L_x$. Maximum scouring of the embayment thus occurs when the cusp spacing $L_y$ is proportional to the wave uprush $L_x$.