====== Diagram Orbital Motion in a Wave ====== In this diagram, the water level \( \eta\), the horizontal and vertical orbital velocities u and v, and the orbital accelerations (\( \frac{\delta u}{\delta t}\) and \( \frac{\delta v}{\delta t}\)) can be displayed, which occur at the water level surface for an selectable wave in the water depth h. According to STOKES theory of 3rd order, the following equations are used to determine the free surface, orbital velocities and accelerations corresponding to EAK: \\ \\ \( \eta\ = a \cdot \cos \theta + \frac{k}{4} a^2 \cdot \frac{\cosh(kh) [2 + \cosh(2 kh)]}{\sinh^3(kh)} \cdot \cos(2\theta) + \frac{3}{64} k^2 a^3 \cdot \frac{1+8 \cosh^6(kh)}{\sinh^6(kh)} \cdot \cos(3\theta) \) \\ \\ \( u = c \cdot [k \cdot a \cdot \frac{\cosh[k(z + h)]}{\sinh(kh)} \cdot \cos \theta + \frac{3}{4} k^2 a^2 \cdot \frac{\cosh[2k(z + h)]}{\sinh^4(kh)} \cdot \cos(2\theta) + \frac{3}{64} k^3 a^3 \cdot \frac{11 - 2 \cosh(kh)}{\sinh^7(kh)} \cdot \cosh[3k (z+h)] \cdot \cos(3\theta)] \) \\ \( v = c \cdot [k \cdot a \cdot \frac{\sinh[k(z + h)]}{\sinh(kh)} \cdot \sin \theta + \frac{3}{4} k^2 a^2 \cdot \frac{\sinh[2k(z + h)]}{\sinh^4(kh)} \cdot \sin(2\theta) + \frac{3}{64} k^3 a^3 \cdot \frac{11 - 2 \cosh(kh)}{\sinh^7(kh)} \cdot \sinh[3k (z+h)] \cdot \sin(3\theta)] \) \\ \\ \(\frac{\delta u}{\delta t} = c \cdot [k \cdot \omega \cdot a \cdot \frac{\cosh[k(z + h)]}{\sinh(kh)} \cdot \sin \theta + \frac{3}{2} k^2 \omega a^2 \cdot \frac{\cosh[2k(z + h)]}{\sinh^4(kh)} \cdot \sin(2\theta) + \frac{9}{64} k^3 \omega a^3 \cdot \frac{11 - 2 \cosh(2kh)}{\sinh^7(kh)} \cdot \cosh[3k (z+h)] \cdot \sin(3\theta)] \) \\ \(\frac{\delta v}{\delta t} = c \cdot [-k \cdot \omega \cdot a \cdot \frac{\sinh[k(z + h)]}{\sinh(kh)} \cdot \cos \theta - \frac{3}{2} k^2 \omega a^2 \cdot \frac{\sinh[2k(z + h)]}{\sinh^4(kh)} \cdot \cos(2\theta) - \frac{9}{64} k^3 \omega a^3 \cdot \frac{11 - 2 \cosh(2kh)}{\sinh^7(kh)} \cdot \sinh[3k (z+h)] \cdot \cos(3\theta)] \) \\ \\ here are \( k = \frac{2\pi}{L}\), \( \omega = \frac{2\pi}{T}\) and \( \theta = kx - \omega t\). For simplification, the amplitude for calculation input will be determined from \(a=\frac{H}{2}\) here. The first component's amplitude \(a\) can also be determined from the implicit equation (EAK, p. 43): \(H=2a+\frac{3}{32}\cdot k^2 \cdot a^3 \cdot \lbrack \frac{1+8\cosh ^6(kh)}{\sinh ^6(kh)} \rbrack \). In addition, the wave theory application diagram shows which wave theory can be applied for the selected wave parameters and water depth. Lernplattform des Leichtweiß-Institut für Wasserbau
  •  ___ eta
  •  ___ u
  •  ___ v
  •  ___ u/t
  •  ___ v/t

Wellenlänge: 4 m

H/gT² = ...

h/gT² = ...

Wellenzahl k = ... 1/m

Kreisfrequenz ω = ... 1/s

Wellenschnelligkeit c = ... m/s

Wasserstandsanstieg Δh = ... m

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.