Results: Sea state RAOs

These are reported on the sea state RAOs sheet of the results tables. Sea state RAOs give results at field points for the pressure in the fluid and the velocity of the fluid. The first-order complex pressure and velocity are given in terms of the total first-order potential, $\phi$, by \begin{equation} \begin{aligned} p(\vec X) & = -\textrm{i}\omega\rho\phi(\vec X) \\ \vec{v}(\vec X) & = \nabla\phi(\vec X) \end{aligned} \end{equation} The velocity results are given in the global coordinates $\GXYZ$.

The complex elevation of the free surface, $\eta(X,Y)$, can also be obtained by placing a field point at $(X,Y,0)$ and using the relation \begin{equation} \eta(X,Y)=\frac{p(X,Y,0)}{\rho g} \end{equation} The potential is obtained at a field point in terms of the incoming wave field, $\phi_I$, and the body potential, $\phi_B$, via $\phi = \phi_I + \phi_B$. The method for calculating the body potential depends on the solve type, as detailed below.

Modified formulae for the body potential hold in the presence of a damping lid or dipole panels.

Potential formulation

If only the potential formulation is solved, then the body potential is obtained from an application of Green's theorem \begin{equation} \begin{aligned} \phi_B(\vec X) & = \frac{1}{4\pi}\int_{\SB} \left\{\PD{\phi_B(\vec{\xi})}{n_{\xi}} G - \phi_B(\vec{\xi}) \PD{G}{n_{\xi}} \right\} \ud S_{\xi} \\ \nabla\phi_B(\vec X) & = \frac{1}{4\pi}\int_{\SB} \left\{\PD{\phi_B(\vec{\xi})}{n_{\xi}} \nabla_x G - \phi_B(\vec{\xi}) \nabla_x \PD{G}{n_{\xi}} \right\} \ud S_{\xi} \end{aligned} \end{equation}

Source formulation

If the source formulation is solved, then the body potential is obtained from the source functions for the scattered and radiation potentials \begin{equation} \begin{aligned} \phi_B(\vec X) & = \int_{\SB} \Big\{\sigma_R(\vec{\xi}) + \sigma_S(\vec{\xi}) \Big\} G \ud S_{\xi} \\ \nabla\phi_B(\vec X) & = \int_{\SB} \Big\{\sigma_R(\vec{\xi}) + \sigma_S(\vec{\xi}) \Big\} \nabla_x G \ud S_{\xi} \end{aligned} \end{equation}