Wave overtopping

Definition of Wave overtopping: Wave overtopping is the passage of wave-driven water over the crest of a coastal protection structure, such as a breakwater, seawall or dike, whose crest is above the still-water level (SWL). It is commonly quantified by the mean overtopping discharge q, expressed as volume per unit time and per meter crest length. This is the common definition for Wave overtopping, other definitions can be discussed in the article

Fig. 1a. Wave-overtopping of a breakwater in a flume test.	Fig. 1b. Wave-overtopping of a breakwater in nature.

Sloping structures

Pictures of wave overtopping in the laboratory and in the field are shown in Fig. 1; a schematic representation with definitions is displayed in Fig. 2. During overtopping, waves run up the seaward slope and, when the run-up exceeds the crest freeboard, a part of the uprush passes over the crest. This can contribute to wave transmission and water-level setup behind low-crested or permeable structures, see the articles Wave run-up and Wave set-up and wave transmission by low-crested breakwaters. The process of wave overtopping is highly random in time, space and volume; there is no permanent discharge over the crest of a structure. The highest waves may wash a large volume of water over the crest in a short period of time (less than a wave period), whereas lower waves may not produce any overtopping.

The overtopping formulas presented in this article are empirical or semi-empirical relations, mainly calibrated with laboratory flume experiments and, in some cases, validated or extended with numerical simulations and field data. These empirical formulas often have large scatter and should be used with safety factors, sensitivity tests or probabilistic methods for design.

Wave overtopping is governed primarily by the relation between the wave run-up level and the crest freeboard [math]R_c[/math]. If the freeboard is large compared with the run-up level, only very few waves pass over the crest. If the freeboard is small, overtopping increases rapidly and may occur in large intermittent pulses. Therefore small changes in still-water level, wave height, wave period, foreshore level or crest height can produce large changes in overtopping discharge. The mean discharge [math]q[/math] is useful for drainage and flood-volume estimates, but damage to dike crests, landward slopes, roads and people behind the structure often depends on individual overtopping volumes, flow depths and flow velocities.

An empirical formula for the average water discharge [math]q[/math] per linear meter by non-breaking waves over a rubble-mound protection structure is given in the EurOtop manual

[math]\Large\frac{q}{\sqrt{gH_{m0}^3}}\normalsize \equiv q^* = 0.09 \exp(-\large\frac{1.5 R_c}{\gamma_f H_{m0}}\normalsize) \, , \qquad (1)[/math]

for perpendicular wave incidence. Since overtopping rates of more than a factor 10 higher were observed in several experiments, a revised formula was developed by Eldrup et al. (2022). The general form of the wave overtopping discharge is

[math]q^* = a \, \exp\Big[ -\big(b \large\frac{R_c}{H_{m0}}\normalsize \big)^c \Big] \, , \qquad (2)[/math]

Fig. 2. Waves overtopping a breakwater; definition of symbols.

where the coefficients [math]a, b, c[/math] depend on (see Fig. 2):

• The spectral wave height at the toe of the structure [math]H \equiv H_{m0}[/math] (approximately equal to the significant wave height [math]H_s[/math], see Statistical description of wave parameters)
• The spectral wave energy period [math]T \equiv T_{m-1,0}[/math] at the toe of the structure
• The wave steepness [math]s = H_{m0}/L[/math], where [math]L=g T^2 / (2 \pi)[/math] is the deep-water offshore wavelength
• The wave incidence angle [math]\theta[/math]
• The water depth [math]h_{toe}[/math] at the toe of the structure
• The breakwater wave run-up [math]R_2[/math] exceeded by only 2 % of the waves; [math]R_2/ H_{m0}[/math] depends on the roughness (and permeability) reduction factor [math]\gamma_f[/math], the surf similarity parameter [math]\xi[/math] and the wave obliqueness,
• The freeboard [math]R_c[/math] (the structure crest level relative to the still water level SWL);
• The seabed slope [math]m[/math]
• The front slope of the structure [math]\tan \alpha[/math]
• The crest width of the structure [math]G_c[/math]
• The surf similarity parameter (Iribarren number) [math]\xi = \tan \alpha / \sqrt{s}[/math]
• The roughness and permeability reduction factor [math]\gamma_f[/math] accounts for energy dissipation and percolation on the seaward slope. For smooth impermeable slopes [math]\gamma_f =1[/math]. For rough and permeable rubble-mound armour, [math]\gamma_f[/math] is smaller, often of order [math]\gamma_f \approx 1 – 0.7 \, (D_{n50}/ H_{m0})^{0.1} \sim 0.4 – 0.6[/math], but the appropriate value depends on armour type, permeability, wave steepness, slope angle and the calibration range of the chosen formula. Additional influence factors may be needed for crest width, berms, protruding crest walls, oblique wave attack, swell components and wind. Structures with a protruding crest wall can produce more overtopping than rubble-mound structures with the same crest elevation.

Wave overtopping experiments by van Gent et al. (2022) are best represented by formula (2) with

[math]a = \dfrac{0.0016}{s} \, , \quad b = \dfrac{2.4}{\gamma_f \, \gamma_{\theta} \, \gamma_{cw} } \, , \quad c=1 \, . \qquad (3)[/math].

Here is [math]s= H_{m0}/L[/math] the wave steepness, [math]\gamma_f[/math] the roughness and permeability factor, [math]\gamma_{cw}[/math] the influence factor for crest walls and [math]\gamma_{\theta}[/math] a reduction factor that accounts for oblique wave incidence. The factor [math]\gamma_{cw}[/math] may have a value larger than 1 for rubble mound structures. The value of [math]\gamma_{\theta}[/math] decreases from [math]\gamma_{\theta}=1[/math] for normal wave incidence ([math]\theta=0[/math]) to [math]\gamma_{\theta} \sim 0.85-0.95[/math] for [math]\theta=30°[/math] and [math]\gamma_{\theta} \sim 0.7-0.85[/math] for [math]\theta=60°[/math].

This empirical formula for overtopping discharges was derived from flume experiments that simulate typical design values in the range: [math] H_{m0} \sim[/math]2-5 m, [math]T \sim[/math]8-15 s, [math]h_{toe}/ H_{m0} \sim [/math]1-10, [math]R_2/ H_{m0} \sim [/math]2-6, [math]R_c/ H_{m0} \sim [/math]0.5-2.5, [math]G_c/ H_{m0} \sim [/math]1-5, [math]\tan \alpha \sim [/math]0.25-0.6, [math]m \sim[/math]0.005-0.05, [math]\xi \sim [/math]1.5-8.

The same typical design values were considered in the experiments by Etemad-Shahidi et al. (2022), who proposed a formula for the mean overtopping discharge of rubble mound breakwaters by surging (non-breaking) waves:

[math]q^* = (1.22 \pm 0.13) 10^{-4} \, \exp \big[ (3.5 \pm 0.13) \large\frac{R_2 -R_c}{ H_{m0}}\normalsize – (0.64 \pm 0.07) \large\frac{G_c}{ H_{m0}}\normalsize \big] . \qquad (4)[/math]

The standard deviation of observational data with respect to this formula is about a factor 3. This uncertainty is mainly related to estimating the wave run-up. The EurOtop manual proposes for approximately normal wave incidence:

[math]R_2 = 1.65 \, \gamma_f \, H_{m0} \, \xi \;[/math], with a maximum of [math]\gamma_{sf} \, H_{m0} \, (4 - 1.5 \, \xi^{-1/2})[/math]. The roughness factor [math]\gamma_{sf}[/math] for surging waves is estimated as [math]\gamma_{sf} = \gamma_f + 0.12 \, (1-\gamma_f)(\xi-1.8)[/math] with a minimum value equal to [math]\gamma_f[/math].

In shallow-water conditions, especially when strong breaking and low-frequency wave generation occur on the foreshore, standard overtopping formulas may perform poorly or overpredict the mean overtopping discharge. For this case, de Ridder et al. (2026) proposed several formulas with similar root-mean-square deviation from the overtopping discharges measured in their laboratory experiments. The simplest formula is

[math]q^* = 0.16 \exp \Big[ -6.93 \, s^{0.33}_{HF} \, \dfrac{R_c }{\gamma_f \gamma_{\theta} H_{m0}} \Big] \, , \quad \gamma_f=0.55, \; \gamma_{\theta}=0.35+0.65 \cos^2 \theta \, , \qquad (5)[/math]

where the subscript [math]HF[/math] refers to the contribution of the high-frequency waves in the shallow-water wave spectrum and [math]s_{HF}[/math] is the corresponding wave steepness. The experiments covered the following parameter ranges: [math]\; 1.41\lt h/ H_{m0}\lt 2.15 \, , \; 0.98\lt R_c/ H_{m0}\lt 1.94 \, , \; 0.012\lt s_{HF}\lt 0.049 \, , \; 0\lt \theta\lt 30° \, [/math]. The directional spreading of incident waves was varied over [math]\sigma_{\theta}=8-25°[/math].

The mean overtopping discharge is the average over many individual waves. Some waves can produce overtopping volumes much larger than the average. The largest overtopping volumes of rubble-mound structures in shallow water occur for short waves riding on the crest of low-frequency (infragravity) waves. Flume experiments show that the overtopping volumes of individual waves follow approximately a log-normal distribution. Empirical formulas for the 2% largest overtopping volumes [math]V_2[/math] and related overtopping heights [math]h_2[/math] and overtopping velocities [math]u_2[/math] are

[math]V_2= 5.56 \, H_{m0}^2 \exp \Big( -9.94 \dfrac{R_c \, s^{0.57}_{HF}}{\gamma_f H_{m0}} \Big) \, , \quad h_2= 0.63 \, H_{m0} \exp \Big( -4.88 \dfrac{R_c \, s^{0.5}_{HF}}{\gamma_f H_{m0}} \Big) \, , \quad u_2= 1.26 \sqrt{gh} \Big( \dfrac{R_c \, s^{-0.24}_{HF}}{\gamma_f H_{m0}} \Big)^{-1.05} \, , \qquad (6)[/math]

where [math]h[/math] is the still water level at the structure's toe.

Astorga-Moar and Baldock (2023) conducted a laboratory investigation of wave overtopping of a beach with constant slope [math]\tan \alpha[/math] and berm height [math]R_c[/math] above still water, fronted by a low reef or shore platform. They found the following relationship of the mean wave overtopping discharge [math]q^*[/math] with the 2% wave run-up [math]R_2[/math], the spectral wave height [math] H_{m0}[/math] at the toe of the beach and the peak wave period [math]T_p[/math],

[math]q^* = 0.015 \, \xi_p \, \Big(\large\frac{R_2-R_c}{R_2}\normalsize \Big)^2 \, , \quad \xi_p = T_p \, \tan \alpha \, \sqrt{\large\frac{g}{2 \pi H_{m0}}} \, .\qquad (7)[/math]

Vertical walls

From numerical simulations validated with flume experiments, Tuozzo et al. (2024) found that the overtopping discharge of a vertical seawall in shallow water depends crucially on the upper tail [math]\zeta_{1/4}[/math] of the wave elevation distribution [math]f(\zeta)[/math] at the toe of the seawall:

[math]\zeta_{1/4} = 4 \int_{\zeta_{75}}^{\infty} \zeta f(\zeta) d \zeta ,[/math]

where [math]\zeta[/math] is the wave elevation at the toe of the seawall and [math]\zeta_{75}[/math] is the 75th percentile of the wave-elevation distribution. For linear (symmetric) waves this can be approximated by [math]\zeta_{1/4} = \mu + 0.32 \, H_{m0} [/math], where [math]\mu[/math] is the mean elevation and [math]H_{m0}[/math] the significant wave height at the toe.

The proposed formula for the overtopping discharge is

[math]q = g \, h_{toe} \, T_{\infty} \, m^{p_1} \, q^*(z) \, , \quad q^*(z) = min\big[ 0.0107 \, z^{-1.49}\, ; \, 0.068 \, z^{-3.04} \big] \, , \quad z=\Large\frac{p_2 R_c}{\zeta_{1/4}}\normalsize \, , \qquad (8)[/math]

where [math]m=\tan \beta_f[/math] is the foreshore slope in front of the vertical wall, [math]p_1=\big[ 2+100 \, \exp(- 10 \gamma_{\infty}) \big]^{-1}\, , \quad p_2 = max \Big[1\, ; \, \Large\frac{1}{\gamma_{\infty}}\normalsize \tanh\big(\large\frac{10 \pi s_{\infty}}{\gamma_{\infty}}\normalsize \big) \Big] \, , \quad \gamma_{\infty} = \Large\frac{H_{\infty}}{h_{toe}}\normalsize . [/math]

The subscript [math]\infty[/math] refers to deep water conditions and [math]s_{\infty}[/math] is the deep water wave steepness.

Onshore wind increases the overtopping discharge. Vegetation (e.g. mangroves, seagrass) or other nature-based elements in front of a seawall can reduce overtopping by attenuating waves and modifying water levels, but the effect depends on vegetation type, density, width, water depth and storm conditions, see Nature-based shore protection.

Related articles

Stability of rubble mound breakwaters and shore revetments

Wave set-up and wave transmission by low-crested breakwaters

Wave run-up

Overtopping resistant dikes

Wave collision on a vertical wall

Detached breakwaters

Modelling coastal hydrodynamics

References

The main author of this article is Job Dronkers Please note that others may also have edited the contents of this article. Citation: Job Dronkers (2026): Wave overtopping. Available from http://www.coastalwiki.org/wiki/Wave_overtopping [accessed on 18-06-2026] For other articles by this author see Category:Articles by Job Dronkers For an overview of contributions by this author see Special:Contributions/Dronkers J