Wave damping by vegetation

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Vegetation can make an important contribution to shore protection by damping incident waves, see Nature-based shore protection and Shore protection vegetation. Wave damping by vegetation is generally expressed by means of a friction drag coefficient [math]C_D[/math]. This coefficient depends on plant characteristics and wave parameters. Vegetation and water flow interact in coupled, nonlinear ways[1]. This interaction is dynamic since the structure of aquatic plant fields changes with time and is exposed to variable physical forcing of the water flow[2]. The damping effect of vegetation is stronger for incident waves with high frequencies than for incident waves with the low frequencies[3].


Parameters relevant for wave-plant interaction

Wave attenuation by vegetation depends on many parameters referring to hydrographic conditions (mainly the parameters wave height, wave period and wave incidence direction) and vegetation characteristics (plant geometry, buoyancy, density, stiffness, degrees of freedom and vegetation sensity and spatial configuration). Relevant parameters are represented by the symbols listed below[4].


Parameter Symbol Typical value, range Parameter Symbol Typical value, range Parameter Symbol Typical value, range
wave amplitude [[math]m[/math]] [math]a[/math] 0.2-1 water depth [[math]m[/math]] [math]h[/math] 0.5-3 wave orbital velocity amplitude [[math]m s^{-1}[/math]] [math]u_0 \approx a c / h[/math] 0.2-1
blade / stem width
[[math]m[/math]]		 [math]b[/math] 0.003-0.3 wave damping coefficient [[math]m^{-2}[/math]] [math]K_D[/math] 0.001-0.01 vegetation cross-shore length [[math]m[/math]] [math]x[/math] 10-1000
wave celerity [[math]m s^{-1}[/math]] [math]c \approx \sqrt{gh}[/math] 2-5 Keulegan-Carpenter number [math]KC = u_0 T/ b[/math] 5-2000 water density [[math]kg m^{-3}[/math]] [math]\rho[/math] 1025
friction drag coefficient [math]C_D[/math] 0.5-3 submerged blade / stem length [[math]m[/math]] [math]l[/math] (0.2-1) h wave radial frequency
[[math]s^{-1}[/math]]		 [math]\omega = 2 \pi / T[/math] 0.5-2
gravitational acceleration [[math]m s^{-2}[/math]] [math]g[/math] 9.8 plant spacing [m] [math]s[/math] 0.02 - 0.07


Wave energy dissipation equation

The wave energy dissipation [math]E_{dis}[/math] per unit seabed surface [[math] kg s^{-3}[/math]] due to the drag force of rigid vegetation, can be approximated for regular waves by[5]

[math]E_{dis} \approx ½ \rho C_D \Large\frac{b l}{s^2}\normalsize \lt |u_0 \sin \omega t|^3 \gt \approx \rho C_D \Large\frac{2}{3 \pi}\frac{bl}{s^2} (\frac{a c}{h})^3\normalsize \, .\qquad (A1)[/math].

The factor [math]\; bl/s^2 \;[/math] is the plant frontal surface per unit seabed surface. Assumptions used in the formula (A1) are: (1) non-broken waves, (2) the shallow water approximation [math]kh \lt \lt 1[/math], and (3) the wave orbital velocity in the canopy is not much smaller than the surface wave orbital velocity. For irregular waves an expression for the energy dissipation [math]E_{dis}[/math] similar to (A1) can be derived[2] with the replacements [math]a \rightarrow \frac{1}{2} H_{rms}[/math] and [math]\dfrac{2}{3 \pi} \rightarrow \dfrac{1}{2 \sqrt{\pi}}[/math]. It should be noted that this substitution does not take into account the dependence of the dissipation on the wave frequency.

For intermediate water ([math]kh \sim 1[/math]) a factor [math]\dfrac{4 k h^2}{3l}\dfrac{\sinh^3 kl + 3 \sinh kl}{(\sinh 2kh + 2kh)\sinh kh}[/math] must be included in the r.h.s. of Eq. (A1).

Wave dissipation causes a decrease of the incident wave energy flux [math]dF/dx \; [kg s^{-3}][/math] given by

[math]dF/dx \approx ½ \rho g c \Large\frac{d a^2}{dx}\normalsize \, ,\qquad (A2)[/math]

where [math]a[/math] is the wave amplitude. This formula assumes a horizontal seabed and shallow water where the wave group velocity [math]c_g[/math] can be approximated by [math]c \approx \sqrt{gh}[/math].

The decay of the wave amplitude [math]a(x)[/math] can be found by equating [math]dF/dx = E_{dis}[/math] with the result

[math]a(x) = \dfrac{a_0}{1 + K_D a_0 x} , \quad K_D = \dfrac{2}{3 \pi} C_D \dfrac{b l}{h^2 s^2} \, , \qquad (A3)[/math]

where [math]a_0[/math] is the amplitude of the wave entering the vegetated area.

Empirical expressions of the friction drag coefficient [math]C_D[/math]

Many empirical expressions for the coefficient [math]C_D[/math] can be found in the literature, based on flume experiments or field observations, with rigid or flexible stems and synthetic or natural plants[6]. Most formulas relate the friction drag coefficient to the Reynolds number or to the Keulegan-Carpenter number [math]KC[/math]. For example, Chen et al. (2018[7]) propose

[math]C_D \approx 1.2 + 13 \, KC^{-1.25} \qquad (A4)[/math]

For emerging vegetation (e.g. mangroves), the length [math]l[/math] has to be taken equal to the water depth [math]h[/math]. Typical values of [math]C_D[/math] for vegetation with rigid stems are in the range 0.5-3.

Equation (A3) assumes submerged vegetation with rigid stems. For vegetation with flexible stems and leaves, such as seagrass, wave-induced energy dissipation is more complex because blade motion is governed by the balance between hydrodynamic forcing and elastic restoring forces associated with blade stiffness[8]. Reis et al. (2024[9]) showed that the swaying motion of flexible blades modifies the acceleration of the wave orbital flow, contributing to differences in wave damping between rigid and flexible vegetation.

To account for blade flexibility without explicitly resolving these dynamics, Luhar and Nepf (2016[8]) proposed replacing the rigid stem length [math]l[/math] in the wave energy dissipation formulation (A1) with a reduced effective blade length [math]l_e[/math]. They introduced an empirical relationship in which [math]l_e[/math] depends on blade flexibility, parameterized through the elastic Young’s modulus [math]E[/math]. Flume experiments indicate that, when expressed in terms of [math]l_e[/math], the drag coefficients of rigid and flexible vegetation are comparable[10]. Field observations of wave dissipation in seagrass meadows are consistent with this effective-length framework and further demonstrate that [math]l_e[/math] decreases with increasing incident wave height. Consequently, the ability of seagrass meadows to attenuate wave energy is substantially reduced under storm conditions.[11] Other field studies further show that bending of the blades by strong steady or tidal currents considerably reduces the drag force exerted by the seagrass meadow[12].

Wave forces on rigid stems are dealt with in Shallow-water wave theory#Vertical Piles.


Related articles

Nature-based shore protection
Seagrass meadows
Salt marshes
Mangroves
Shore protection vegetation
Climate adaptation measures for the coastal zone


References

  1. ↑ Koch, E.W., Sanford, L.P., Chen, S-N., Shafer, D.J., Mckee Smith, J., 2006. Waves in seagrass systems: Review and Technical recommendations. US Army Corps of Engineers®. Technical Report, ERDC TR-06-15
  2. ↑ 2.0 2.1 Mendez, F.J. and Losada, I.J. 2004. An empirical model to estimate the propagation of random breaking and nonbreaking waves over vegetation fields. Coastal Engineering 51: 103-118 Cite error: Invalid <ref> tag; name "ML" defined multiple times with different content
  3. ↑ Dermentzoglou, D., Tissier, M., Muller, J.R.M., Hofland, B., Lakerveld, S., Borsje, B.W. and Antonini, A. 2026. Transformation of 𝐻𝑚0 and 𝑇𝑚−1,0 over a model salt marsh. Coastal Engineering 204, 104900
  4. ↑ Vettori, D., Pezzutto, P., Bouma, T.J., Shahmohammadi, A. and Manes, C. 2024. On the wave attenuation properties of seagrass meadows. Coastal Engineering 189, 104472
  5. ↑ Dalrymple, R.A., Kirby, J.T. and Hwang, P.A. 1984. Wave diffraction due to areas of energy dissipation. J. Waterw. Port Coast. Ocean Eng. 110: 67–79
  6. ↑ Yin, K., Xu, S. Huang, W., Xu, H., Lu, Y. and Ma, M. 2024. A study on the drag coefficient of emergent flexible vegetation under regular waves. Ocean Modelling 191, 102422
  7. ↑ Chen, H., Ni, Y., Li, Y., Liu, F., Ou, S., Su, M., Peng, Y., Hu, Z., Uijttewaal, W. and Suzuki, T. 2018. Deriving vegetation drag coefficients in combined wave-current flows by calibration and direct measurement methods. Adv. Water Resour. 122: 217–227
  8. ↑ 8.0 8.1 Luhar, M. and Nepf, H.M. 2016. Wave-induced dynamics of flexible blades. J. Fluids Struct. 61: 20–41
  9. ↑ Reis, R.A., Fortes, C.J.E.M., Rodrigues, J.A., Hu, Z. and Suzuki, T. 2024. Experimental study on drag coefficient of flexible vegetation under non-breaking waves. Ocean Engineering 296, 117002
  10. ↑ Lei, J. and Nepf, H. 2019. Wave damping by flexible vegetation: Connecting individual blade dynamics to the meadow scale. Coast. Eng. 147: 138–148
  11. ↑ Contti Neto, N., Lowe, R. J., Ghisalberti, M., Pomeroy, A., Reidenbach, M., Conde‐Frias, M. and da Silva, R. F. 2025. Spectral wave energy dissipation by a seagrass meadow. Journal of Geophysical Research: Oceans 130, e2024JC020938
  12. ↑ Monismith, S. G., Hirsh, H., Batista, N., Francis, H., Egan, G. and Dunbar, R. B. 2019. Flow and drag in a seagrass bed. Journal of Geophysical Research: Oceans 124: 2153–2163


The main author of this article is Job Dronkers
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Citation: Job Dronkers (2026): Wave damping by vegetation. Available from http://www.coastalwiki.org/wiki/Wave_damping_by_vegetation [accessed on 21-02-2026]
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