# Mean wave drift forceΒΆ

The generalized mean wave drift force, given by the linear approximation, is:

\[\mathbf{f}_{WD} = 2 \int_0^{2\pi} \int_0^{\infty} S(\omega,\theta) \mathbf{C}(\omega_e,\alpha) d\omega d\theta\]

where

- \(S(\omega,\theta)\) is the wave spectrum amplitude for the circular frequency \(\omega\), and wave direction \(\theta\);
- \(\omega_e\) is the encounter wave frequency, which depends on \(\omega\), \(\theta\) and the constant speed of the vessel \(\mathbf{U}\);

\[\omega_e = \omega - k_{\omega} \mathbf{U} \cdot x_w\]

where \(k_{\omega}\) is the wave number and \(x_w\) is the wave direction of the corresponding wave component.

- \(\alpha\) is the relative angle between the wave direction and vessel heading, with respect to the equilibrium frame;

- \(\mathbf{C}(\omega_e,\alpha)\) are the polar wave drift coefficients, in \(N/m^2\) which depend on \(\omega_e\) and \(\alpha\).