Diagram Wave Profiles by Stokes Wave Theory

The displayed wave profiles are calculated according to Stokes' fifth order wave theory.
For convenience the formulae are given at the bottom of this page. These formulae are taken from Skjelbreia und Hendrickson (1961).

According to the dispersion relation the wave length is calculated as $L = \frac{g \cdot T^2}{2 \cdot \pi} \cdot \tanh( \frac{2 \cdot \pi}{L} \cdot d) = \frac{g \cdot T}{\omega} \cdot \tanh(k \cdot d)$ .
The remaining specified parameters are as follows: Wave number $k = \frac{2 \cdot \pi}{L}$ , angular frequency $\omega = \frac{2 \cdot \pi}{T}$ , Ursell parameter $U_R = \frac{H}{L} \cdot ( \frac{L}{d} )^3 = \frac{H \cdot L^2}{d^3}$ , acceleration due to gravity $g = 9.81 \frac{m}{s^2}$ .

The range of possible inputs is restricted to the left by the limit $U_R = 26$ and upwards by the maximum wave steepness in deep water $\frac{H_0}{L_0} = \frac{1}{7}$ .

Lernplattform des Leichtweiß-Institut für Wasserbau

Wave period T [s]:

Wave length L = 72.7 m
d/gT² = 0.04368
H/gT² = 0.0009007
Wave number k = 0.08642
Angular frequency ω = 0.8975
Ursell parameter U_R = 0.23894

1. order amplitude η₁ = 0.2165 m 2. order amplitude η₂ = 0.06846 m 3. order amplitude η₃ = 0.018 m 4. order amplitude η₄ = 0.0054 m 5. order amplitude η₅ = 0.001469 m Amplitude Stokes 5 wave ∑ηᵢ = 0.3099 m

___ 1. order share
___ 2. order share
___ 3. order share
___ 4. order share
___ 5. order share
___ Stokes 5 wave profile

*Illustration taken from: EAK 2002, Empfehlungen für Küstenschutzwerke, Korrigierte Ausgabe 2007
des Kuratoriums für Forschung im Küsteningenieurwesen (published in: Die Küste, Booklet 65, 2002)

Stokes' fifth order wave theory according to Skjelbreia und Hendrickson (1961):

$k \cdot \eta = k \cdot \eta_1 (x,t) + k \cdot \eta_2 (x,t) + k \cdot \eta_3 (x,t) + k \cdot \eta_4 (x,t) + k \cdot \eta_5 (x,t)$

$\qquad = k \cdot a \cdot \cos(k \cdot x - \omega \cdot t) \\ \qquad \quad \qquad + \big[ (k \cdot a)^2 \cdot B_{22} + (k \cdot a)^4 \cdot B_{24} \big] \cdot \cos \big[ 2 \cdot (k \cdot x - \omega \cdot t) \big] \\ \qquad \quad \qquad \qquad + \big[ (k \cdot a)^3 \cdot B_{33} + (k \cdot a)^5 \cdot B_{35} \big] \cdot \cos \big[ 3 \cdot (k \cdot x - \omega \cdot t) \big] \\ \qquad \quad \qquad \qquad \qquad + (k \cdot a)^4 \cdot B_{44} \cdot \cos \big[ 4 \cdot (k \cdot x - \omega \cdot t) \big] \\ \qquad \quad \qquad \qquad \qquad \qquad + (k \cdot a)^5 \cdot B_{55} \cdot \cos \big[ 5 \cdot (k \cdot x - \omega \cdot t) \big]$
with:
$\qquad B_{22} = \frac{c \cdot (2 \cdot c^2 + 1)}{4 \cdot s^3}$
$\qquad B_{24} = \frac{c \cdot (272 \cdot c^8 - 504 \cdot c^6 - 192 \cdot c^4 + 322 \cdot c^2 + 21)}{384 \cdot s^9}$
$\qquad B_{33} = \frac{3 \cdot (8 \cdot c^6 + 1)}{64 \cdot s^6}$
$\qquad B_{35} = \frac{88128 \cdot c^{14} - 208224 \cdot c^{12} + 70848 \cdot c^{10} + 54000 \cdot c^8 - 21816 \cdot c^6 + 6264 \cdot c^4 - 54 \cdot c^2 - 81}{12288 \cdot s^{12} \cdot (6 \cdot c^2 - 1)}$
$\qquad B_{44} = \frac{c \cdot (768 \cdot c^{10} - 448 \cdot c^8 - 48 \cdot c^6 + 48 \cdot c^4 + 106 \cdot c^2 - 21)}{384 \cdot s^9 \cdot (6 \cdot c^2 - 1)}$
$\qquad B_{55} = \frac{192000 \cdot c^{16} - 262720 \cdot c^{14} + 83680 \cdot c^{12} + 20160 \cdot c^{10} - 7280 \cdot c^8 + 7160 \cdot c^6 - 1800 \cdot c^4 - 1050 \cdot c^2 + 225}{12288 \cdot s^{10} \cdot (6 \cdot c^2 - 1) \cdot (8 \cdot c^4 - 11 \cdot c^2 + 3)}$
$\qquad s = \sinh(k \cdot d)$
$\qquad c = \cosh(k \cdot d)$
$\qquad a = \frac{H}{2}$

The field of Wave Theories is part of the module Basic Coastal Engineering within the specialization Coastal and Ocean Engineering of the master programmes Civil and Environmental Engineering.