# Wave excitation¶

The wave excitation loads represent the combination of both the diffraction loads and the Froude-Krylov loads.

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

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:

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 \(z = 0\).

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:

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 : \(z = \eta_I\).

## Diffraction force¶

### Linear Model¶

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

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*.