Diagram Group Velocity

The displayed wave profiles 1 and 2 are calculated according to the linear wave theory of Airy/Laplace.
Therefore, the water surface elevation is represented by the equation \( \eta_i (x,t) = \frac{H}{2} \cdot \cos(k_i \cdot x - \omega_i \cdot t)\).
The remaining parameters of waves 1 and 2 are calculated as follows: Wave celerity \( c_i = \frac{L_i}{T_i} \), Wave number \( k_i = \frac{2 \cdot \pi}{L_i} \), Angular frequency \( \omega_i = \frac{2 \cdot \pi}{T_i} \).

The resulting oscillation shown is the sum of waves 1 and 2 and follows the equation \( \eta_s (x,t) = H \cdot \cos(k_m \cdot x - \omega_m \cdot t) \cdot \cos(k_g \cdot x - \omega_g \cdot t)\).
The parameters of the resulting oscillation of waves 1 and 2 are calculated as follows: Wave number \( k_m = \frac{k_1 + k_2}{2} \), Angular frequency \( \omega_m = \frac{\omega_1 + \omega_2}{2} \), Wave celerity \( c_m = \frac{\omega_m}{k_m} \).

The envelope function is represented by the equation \( \eta_g (x,t) = H \cdot \cos(k_g \cdot x - \omega_g \cdot t)\).
The corresponding parameters are calculated as follows: Wave number \( k_g = \frac{k_1 - k_2}{2} \), Angular frequency \( \omega_g = \frac{\omega_1 - \omega_2}{2} \), Group velocity \( c_g = \frac{\omega_g}{k_g} = \frac{\omega_1 - \omega_2}{k_1 - k_2} \).

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.