Wave runup
Definition of Wave runup:
Wave runup is the maximum onshore elevation reached by waves, relative to the shoreline position in the absence of waves.
This is the common definition for Wave runup, other definitions can be discussed in the article

Notes
Wave runup is the sum of wave setup and swash uprush (see Swash zone dynamics) and must be added to the water level reached as a result of tides and wind setup (Fig. 1). Wave runup on a beach is generally due to socalled swash bores: the uprush of waves after final collapse on the beach. Wave runup is an important parameter for assessing the safety of sea dikes or coastal settlements.
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 runup is often indicated with the symbol [math] R [/math].
For waves collapsing on the beach, a first orderofmagnitude estimate is given by the empirical formula of Hunt (1959) ^{[1]}^{[2]}^{[3]},
[math]R \sim \eta_u + H \xi , [/math]
where [math]\eta_u \sim 0.2 H[/math] is the wave setup, [math]H[/math] is the offshore significant wave height and [math]\xi[/math] is the surf similarity parameter,
[math]\xi = \Large\frac{\tan \beta}{\sqrt{H/L}}\normalsize = T \tan \beta \Large\sqrt{\frac{g}{2\pi H}}\normalsize , \qquad (2)[/math]
where [math]L = g T^2/(2 \pi)[/math] is the offshore wave length, [math]\beta[/math] is the beach slope and [math]T[/math] is the wave period. The horizontal wave incursion is approximately given by [math] R / \tan \beta[/math].
Many other empirical formulas have been proposed for the runup. A popular formula for the runup [math] R_2[/math] exceeded by only 2 % of the waves has been developed by Stockdon et al. (2006^{[4]}), based on a large dataset:
[math] R_2 = 1.1 \; (\eta_u + 0.5 \sqrt{S_w^2 + S_{ig}^2} \, ) , \qquad \xi \ge 0.3 , \qquad R_2= 0.43 \; \sqrt{HL} , \qquad \xi \lt 0.3 , \qquad (3) [/math]
where [math]\eta_u = 0.35 H \xi[/math] is the wave setup, [math]S_w=0.75 H \xi[/math] is the swash uprush related to incident waves and [math]S_{ig}=0.06 \sqrt{HL}[/math] is the additional uprush related to infragravity waves. The factor 1.1 takes into account the nonGaussian distribution of runup events.
From an inventory of runup formulas by Gomes da Silva et al. (2020^{[5]}), it appears that for steep beaches ([math]\tan \beta \gt 0.1[/math]) the runup increases with increasing beach slope (approximately linear dependance^{[6]}), while for gently sloping dissipative beaches ([math]\tan \beta \lt 0.1[/math]) the dependence on beach slope is weak or absent^{[6]}^{[4]}. In these latter cases, runup is dominated by infragravity waves, that yield a small runup that increases with increasing wave height (approximately linear dependence^{[7]}^{[8]}).
The general applicability of empirical formulas of runup based on simple parametric representations of beach and shoreface is limited due to the influence of the more detailed characteristics of the local shoreface bathymetry. This is similar to the limited applicability of empirical formulas for the wave setup, which is a substantial component of the runup. Accurate estimates of the wave runup require insitu observations or detailed numerical models.
Related articles
References
 ↑ Hunt, I.A. 1959. Design of seawalls and breakwaters. J. Waterw. Harbors Division ASCE 85: 123–152
 ↑ Holman, R.A. and Sallenger, A.H. 1985. Setup and swash on a natural beach. J. Geophys. Res. 90: 945–953
 ↑ Atkinson, A.L., Power, H.E., Moura, T., Hammond, T., Callaghan, D.P. and Baldock, T.E. 2017. Assessment of runup predictions by empirical models on nontruncated beaches on the southeast Australian coast. Coast. Eng. 119: 15–31
 ↑ ^{4.0} ^{4.1} Stockdon, H.F., Holman, R.A., Howd, P.A. and Sallenger, A.H. 2006. Empirical parameterization of setup, swash, and runup. Coast. Eng. 53: 573–588
 ↑ Gomes da Silva, P., Coco, G., Garnier, R. and Klein, A.H.F. 2020. On the prediction of runup, setup and swash on beaches. EarthScience Reviews 204, 103148
 ↑ ^{6.0} ^{6.1} Nielsen, P. and Hanslow, D.J. 1991. Wave runup distributions on natural beaches. J. Coast. Res. 7: 1139–1152
 ↑ Ruessink, B.G., Kleinhans, M.G. and Van Den Beukel, P.G.L. 1998. Observations of swash under highly dissipative conditions. J. Geophys. Res. 103: 3111–3118
 ↑ Ruggiero, P., Holman, R. A. and Beach, R. A. 2004. Wave runup on a highenergy dissipative beach. J. Geophys. Res. 109, C06025, doi:10.1029/2003JC002160