Wave damping by vegetation

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.

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.

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	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 by

[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 surface per unit seabed surface. Assumptions used in the formula (A1) are: (1) The shallow water approximation [math]kh \lt \lt 1[/math], and (2) the wave orbital velocity in the canopy is not much smaller than the surface wave orbital velocity.

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. 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) 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.

The expression (A3) assumes submerged vegetation with rigid stems. For vegetation with flexible stems and leaves (e.g. seagrass), the equations describing the energy dissipation are more complex, involving the elastic restoring force due to blade stiffness.

An approach to avoid these complications, proposed by Luhar and Nepf (2016), consists of the replacing the rigid stem length [math]l[/math] in the wave energy dissipation equation (A1) with a reduced length [math]l_e[/math]. They also proposed an empirical formula in which [math]l_e[/math] depends on the blade flexibility through the elastic Young modulus [math]E[/math]. While this approach worked well for individual seagrass plants, a different behavior was observed in seagrass meadows. In seagrass meadows, the blades are bending more strongly, thus opposing less resistance to the wave orbital flow. For strong steady or tidal currents, it was observed that bending of the blades strongly reduces the drag force exerted by the seagrass meadow. However, this is not the case in wave orbital flow, where, according to flume experiments, the drag coefficients for rigid and flexible vegetation are similar. Reis et al. (2024) suggested that this may be explained by the influence of the swaying motion of flexible blades on the acceleration of the wave orbital flow. See also 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

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 (2025): Wave damping by vegetation. Available from http://www.coastalwiki.org/wiki/Wave_damping_by_vegetation [accessed on 15-04-2025] 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