# Wave excitation¶

$\mathbf{f}_E(t) = \mathbf{f}_D(t) + \mathbf{f}_{FK}(t)$

In the case of a linear wave excitation model, the diffraction and Froude-Krylov loads are computed together:

$\mathbf{f}_E(t) = \sum_m \sum_n \Im\left(A_{mn} \mathbf{f}_e(\omega_m,\theta_n) e^{j(k_m\bar{x}_n - \omega_m^e t + \Phi_{mn})}\right)$

with

• $$\omega_m$$ the wave frequency;
• $$\theta_n$$ the wave direction;
• $$\bar{x}_n = x \cos(\theta_n) + y \sin(\theta_n)$$;
• $$\omega_m^e = \omega_m - k_mU$$ is the encounter circular frequency, calculated using the steady forward speed;
• $$A_{mn}$$ the wave amplitude for the wave frequency $$\omega_m$$ and the wave direction $$\theta_n$$;
• $$\mathbf{f}_e(\omega_m,\theta_n)$$ is the frequency-domain excitation force component (with both the diffraction and Froude-Krylov components) for the wave frequency $$\omega$$ and the wave direction $$\theta_n$$.

The frequency-domain excitation forces are obtained from a linear potential flow-based solver such as Nemoh or WAMIT.

## Froude-Krylov force¶

The Froude-Krylov loads are due to the integration of the pressure of the incident wave field over the wetted body surface. Several approximations can be defined, as for the hydrostatic loads:

• a linear model;
• a weakly nonlinear model;
• a fully nonlinear model.

### Linear model¶

The Froude-Krylov force, given by the linear approximation, is:

$\mathbf{f}_{FK}(t) = \sum_m \sum_n \Im\left(A_{mn} \mathbf{f}_{fk}(\omega_m,\theta_n) e^{j(k_m\bar{x}_n - \omega_m^e t + \Phi_{mn})}\right)$

with

• $$\omega_m$$ the wave frequency;
• $$\theta_n$$ the wave direction;
• $$\bar{x}_n = x \cos(\theta_n) + y \sin(\theta_n)$$, with $$x$$ and $$y$$ the coordinates of the equilibrium frame
• $$\omega_m^e = \omega_m - k_mU$$ is the encounter circular frequency, calculated using the steady forward speed;
• $$A_{mn}$$ the wave amplitude for the wave frequency $$\omega_m$$ and the wave direction $$\theta_n$$;
• $$\mathbf{f}_{fk}(\omega_m,\theta_n)$$ is the frequency-domain Froude-Krylov force component for the wave frequency $$\omega$$ and the wave direction $$\theta_n$$.

Note

The frequency-domain Froude-Krylov forces are obtained from a linear potential flow-based solver such as Nemoh or WAMIT.

### Weakly nonlinear model¶

Following the same approximations made for the weakly nonlinear hydrostatic model, the weakly nonlinear Froude-Krylov load is computed by integrating the incident pressure over the wetted body surface, $$S_0$$, defined by the exact position of the body mesh, clipped by the plane $$z = 0$$.

$\mathbf{f}_{FK}(t)= -\iint_{S_0} P_I \mathbf{n} dS$

where

• $$P_I$$ is pressure of the incident wave field;
• $$\mathbf{n}$$ is normal vector, pointing outward the body surface;

The Froude-Krylov torque at the center of gravity of the body is expressed by:

$\mathbf{\Gamma}_{FK}(t)= -\iint_{S_0} P_I (\mathbf{OM}-\mathbf{OG})\times\mathbf{n} dS$

with

• $$\mathbf{OG}$$ the position of the center of gravity of the body;
• $$\mathbf{OM}$$ the position of the centroid of the panel.

### Fully nonlinear model¶

Following the same approximation made for the nonlinear hydrostatic model, the nonlinear Froude-Krylov load is computed by integrating the incident pressure over the exact wetted body surface, $$S_I$$, defined by the exact position of the body mesh, clipped by the undisturbed free surface : $$z = \eta_I$$.

$\mathbf{f}_{FK}(t)= -\iint_{S_I} P_I \mathbf{n} dS$
$\mathbf{\Gamma}_{FK}(t)= -\iint_{S_I} P_I (\mathbf{OM}-\mathbf{OG})\times\mathbf{n} dS$

## Diffraction force¶

### Linear Model¶

The diffraction force, given by the linear approximation, is:

$\mathbf{f}_D(t) = \sum_m \sum_n \Im\left(A_{mn} \mathbf{f}_d(\omega_m,\theta_n) e^{j(k_m\bar{x}_n - \omega_m^e t + \Phi_{mn})}\right)$

with

• $$\omega_m$$ the wave frequency;
• $$\theta_n$$ the wave direction;
• $$\bar{x}_n = x \cos(\theta_n) + y \sin(\theta_n)$$, with $$x$$ and $$y$$ the coordinates of the equilibrium frame
• $$\omega_m^e = \omega_m - k_mU$$ is the encounter circular frequency, calculated using the steady forward speed;
• $$A_{mn}$$ the wave amplitude for the wave frequency $$\omega_m$$ and the wave direction $$\theta_n$$;
• $$\mathbf{f}_D(\omega_m,\theta_n)$$ is the frequency-domain diffraction force component for the wave frequency $$\omega$$ and the wave direction $$\theta_n$$.

Note

The frequency-domain diffraction forces are obtained from a linear potential flow-based solver such as Nemoh or WAMIT.