Lernplattform des Leichtweiß-Institut für Wasserbau

====== 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 = 2. order amplitude η₂ = 3. order amplitude η₃ = 4. order amplitude η₄ = 5. order amplitude η₅ = Amplitude Stokes 5 wave ∑ηᵢ =

___ 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 **[[https://www.tu-braunschweig.de/en/lwi/hyku/teaching/master/basic-coastal-engineering|Basic Coastal Engineering]]** within the specialization Coastal and Ocean Engineering of the master programmes Civil and Environmental Engineering.