# Wave run-up

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 Definition of Wave run-up: Landward incursion of a wave. Wave run-up is usually expressed as the maximum onshore elevation reached by a wave, relative to the wave-averaged shoreline position. This is the common definition for Wave run-up, other definitions can be discussed in the article

Wave run-up is an important parameter for assessing the safety of sea dikes or coastal settlements. Wave run-up is the sum of wave set-up and swash uprush (see Swash zone dynamics) and must be added to the water level reached as a result of tides and storm setup.

By waves is meant: waves generated by wind (locally or on the ocean) or waves generated by incidental disturbances of the sea surface such as tsunamis, seiches or ship waves. Wave run-up is often indicated with the sympol $R$.

For waves collapsing on the beach, the wave run-up can be estimated in first approach with the formula of Hunt (1959) [1],

$R = H \xi ,$

where $H$ is the offshore wave height and $\xi$ is the wave similarity parameter,

$\xi = \Large\frac{S}{\sqrt{H/L}}\normalsize = S \, T \Large\sqrt{\frac{g}{4\pi H}}\normalsize ,$

where $L = g T^2/(2 \pi)$ is the offshore wave length, $S$ is the beach slope and $T$ is the wave period. The horizontal wave incursion is approximately given by $R / S$.

For more precise estimates of wave run-up see:

Swash zone dynamics
Tsunami

## References

1. Hunt, I.A. 1959. Design of seawalls and breakwaters. J. Waterw. Harbors Division ASCE 85: 123–152