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 \( \frac{H_0}{L_0} = \frac{1}{7} \).

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.