Relative Magnitude of Infragravity Waves at Coastal Dikes with Shallow Foreshores: A Prediction Tool

Acknowledgments

Notation

A

envelope of the sea and swell waves (m);

a

fitted coefficient in the prediction of ${\tilde{H}}_{\mathrm{IG}}$ (m⁻¹);

a_h

vegetation height (m);

b_v

vegetation stem diameter (m);

C_D

drag coefficient (−);

C_ηη

wave energy density (m²/Hz);

c

wave speed (m/s);

c_f

bed friction factor (−);

c_g

wave-group velocity (m/s);

c_in

incoming wave speed (m/s);

c_out

outgoing wave speed (m/s);

F_D

drag force (N/m);

F_v

depth-averaged vegetation force (N);

f_low

mean frequency of the IG wave at breakpoint (Hz);

f_node

nodal frequency of standing wave (Hz);

f_p

peak frequency (Hz);

${\tilde{H}}_{\mathrm{IG}}$

relative magnitude of the infragravity waves at the dike toe (−);

H_m0

local significant wave height (m);

H_m0,deep

significant wave height offshore in deep water (m);

H_m0,IG

significant wave height in the infragravity frequency band (m);

H_m0,IG,toe

significant wave height in the infragravity frequency band, at the dike toe (m);

H_m0,SS

significant wave height in the sea and swell frequency band (m);

H_m0,SS,toe

significant wave height in the sea and swell frequency band, at the dike toe (m);

h

local water depth (m);

h_b

water depth at breakpoint (m);

h_toe

initial water depth at the dike toe (m);

k

wave number (rad/m);

L

local wavelength (m);

L₀

wavelength in deep water (m);

m

mode of the dike–foreshore system (standing wave) (−);

N_v

vegetation stem density (stems/m²);

n

manning roughness coefficient (s/m^1/3);

$\overline{q}$

depth-averaged dynamic (nonhydrostatic) pressure normalized by the density (−);

Rel.bias

Relative Bias (−);

R_ηA

cross-correlation coefficient (−);

R²

coefficient of determination (−);

s

user-defined directional spreading factor (XBeach) (−);

s₀

deep-water wave steepness (−);

SCI

scatter Index (−);

T_m−1,0

spectral wave period (s);

T_p

peak wave period in deep water (s);

u

depth-averaged cross-shore velocity (m/s);

v_h

horizontal viscosity (m²/s);

W_veg

width of vegetated cover (m);

x

cross-shore location (m);

x_b

cross-shore breakpoint location (m);

x_dike

cross-shore dike location (m);

α_dike

dike slope angle (°);

α_fore

foreshore slope angle (°);

β_b

normalized bed slope (−);

$\overline{\gamma }$

Influence factor (−);

Δx

cross-shore grid spacing (m);

Δy

alongshore grid spacing (m);

η

surface elevation (m);

η^IG

low-pass filtered surface elevation (m);

$\overline{\eta }$

mean water level relative to the dike toe (m);

${\eta }_{\mathrm{in}}^{\mathrm{IG}}$

incoming low-pass filtered surface elevation (m);

${\eta }_{\mathrm{out}}^{\mathrm{IG}}$

outgoing low-pass filtered surface elevation (m);

η^SS

high-pass filtered surface elevation (m);

ξ₀

breaker index (Iribarren number) (−);

ρ

density of water (kg/m³);

σ

directional spreading (°);

σ_A

standard deviation of the wave envelope time series (m);

σ_η

standard deviation of the low-pass filtered surface elevation time series (m);

ω

angular frequency (rad/s); and

Ω

ratio of breaking waves to water depth (−).

References