Runup and Reverse Shoaling

This Javascript app calculates swash runup elevation on the foreshore. Input parameters include wave height H, wave period T, water depth h, beach slope βf, and γ, the ratio of Hb/hb (wave height to water depth at the break point).

Runup height estimates are based on beach slope in the foreshore region (landward of the break point) and deepwater parameters height Ho and wavelength Lo. Deepwater wavelength depends only on wave period as Lo = gT2/(2π) where g is gravitational acceleration (9.81 m/s). Runup height is a combination of setup <η> (elevation of the sea surface by shoaling) and variance of the swash associated with incident waves Sinc and low-frequency infragravity waves SIG (sometimes called surf beat). Total runup R2 (based on the highest 2%) is calculated as R2 = 1.1 (<η>+S/2) where S = sqrt(Sinc2 + SIG2; Stockdon et al., 2006, eqns. 9 and 7). Sinc and SIG are calculated from Stockdon et al. (2006), eqns. 11 and 12. For very dissipative conditions (low-angle beaches with Iribarren number ξ = βf sqrt(Lo/Ho)< 0.3), R2 = 0.043 sqrt(Ho Lo) (Stockdon et al., 2006, eqn. 18).

Runup distance x = R2/sin(arctanβf) is the distance that swash extends across the sloping shoreface from the mean waterline.

There are three different ways to use these calculations, depending on where the waves are measured. The radio-button selection allows you to specify this:

  1. Wave heights are measured at the break point hb – In this case, the associated water depth is the depth at the break point hb = Hb, and the offshore conditions are determined from reverse shoaling (see below).
  2. Wave heights are measured offshore in deep water – In this case, Ho is the input wave height H, and the break height is computed from an empirical formula: Hb = γ/g1/5 (Cgo Ho2)2/5 (Plant et al., 1999) where Cgo = 1/2 gT/(2π) is group velocity for deepwater waves and break depth is computed as hb = Hb/γ. Wavenumber k and shoaling coefficient Ks are computed for the input depth h.
  3. Wave heights measured at an intermediate depth h – In this case, deepwater wave height Ho is determined from reverse shoaling, and break height and break depth are computed as in (2) above.

Reverse shoaling involves estimating deepwater conditions from measurements at shallow or intermediate depths, using linear wave theory as follows. The wavenumber k at measurement depth h (or hb) is calculated from the dispersion relationship, shoaling coefficient Ks is calculated according to Neilsen (2009, eqn. 1.7.5), and deepwater wave height Ho is determined as Ho = Hb/Ks. Because wave period T remains constant as waves shoal, deepwater wavelength can be calculated as above.


height H
period T
Water depth h
slope βf
ratio γ

At break point hb
In deepwater ho
Intermediate depth h


height Ho
height Hb
depth hb
(total) R2
  Setup <η>
distance x
coefficient Ks
number ξ


Komar, P. D. (1976) Beach Processes and Sedimentation. Prentice-Hall, Inc., 429 pp.

Nielsen, P. (2009) Coastal and Estuarine Processes. Vol. 29, Advanced Series on Ocean Engineering, World Scientific, 343 pp.

Plant, N. G., R. A. Holman, M. H. Freilich, and W. A. Birkemeier (1999) A simple model for interannual sandbar behavior. Journal of Geophysical Research, 104(C7): 15,755-15,776.

Stockdon, H. F., R. A. Holman, P. A. Howd, and A. H. Sallenger Jr. (2006) Empirical parameterization of setup, swash, and runup. Coastal Engineering, 53: 573-588.


Although this program has been used by the USGS, no warranty, expressed or implied, is made by the USGS or the United States Government as to the accuracy and functioning of the program and related program material nor shall the fact of distribution constitute any such warranty, and no responsibility is assumed by the USGS in connection therewith.
Last revised: 28 April 2015

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