# Mean wave drift force¶

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;

Fig. 9 Representation of the wave directions and vessel orientation used for wave drift force computation.

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