Theory: Second-order equations

Continuing the expansion of the governing equations to second order in wave steepness gives a boundary value problem for the second-order complex potential, $\phi\so$. The boundary value problem for $\phi\so$ is the same as the boundary value problem for $\phi\fo$, except that the forcing functions $q_B$ and $q_F$ on the right hand side now include nonlinear effects and are therefore more complicated.

With the boundary value problem taking the same general form, it is again valid to consider wave frequencies individually. At second order, this takes the form \begin{equation} \Phi\so(\vec{X},t) = \sum_{ij}\Re\left\{ A_i A_j \phi^+_{ij}(\vec{X}) \textrm{e}^{\textrm{i}(\omega_i+\omega_j) t} + A_i A_j^* \phi^-_{ij}(\vec{X}) \textrm{e}^{\textrm{i}(\omega_i-\omega_j) t} \right\} \end{equation} where, without loss of generality \begin{equation} \begin{aligned} & \omega_i \ge \omega_j > 0 \\ & \phi^+_{ij}(\vec{X}) = \phi^+_{ji}(\vec{X}) \\ & \phi^-_{ij}(\vec{X}) = \phi^{-*}_{ji}(\vec{X}) \end{aligned} \end{equation} Here, $\omega_i$ and $\omega_j$ are the frequencies of incident waves and $\omega_i \pm \omega_j$ (the sum and difference frequencies) are the frequencies at which second-order effects are excited.

The linearity of the problem allows a decomposition of $\phi^{\pm}$ as \begin{equation} \phi^{\pm} = \phi^{\pm}_{I} + \phi^{\pm}_{R} + \phi^{\pm}_{S} \label{eqPotentialDecomposition} \end{equation} where

$\phi^{\pm}_I$ is the second-order potential in the absence of the body.

$\phi^{\pm}_R$ is the potential due to the second-order body motion.

$\phi^{\pm}_S$ is the remainder of $\phi^{\pm}$.

Boundary value problems

The incident and scattered potentials in (\ref{eqPotentialDecomposition}) satisfy a boundary value problem of the same general form as the first-order problems \begin{equation} \begin{aligned}%implicit alignment is {rlrlrl...} \nabla^2\phi^{\pm} & = 0 & & \vec{X} \in \fV \\ \PD{\phi^{\pm}}{n} & = q_B^{\pm}(\vec{X}) & & \vec{X} \in \SB \\ g\PD{\phi^{\pm}}{Z} - (\omega_i \pm \omega_j)^2\phi^{\pm} & = q_F^{\pm}(\vec{X}) & & \vec{X} \in \SF \\ \PD{\phi^{\pm}}{Z} & = 0 & & Z\rightarrow - \infty \textrm{ (or on seabed)} \end{aligned} \label{eqGeneralSecondOrderBVP} \end{equation} where the forcing functions $q_B^{\pm}$ and $q_F^{\pm}$ now include a variety of second-order effects.

The second-order incident wave potential

The incident wave potential, $\phi^{\pm}_I$, is the solution to the BVP (\ref{eqGeneralSecondOrderBVP}) in the absence of any body ($q_B^{\pm} = 0)$ and with surface forcing (eqs 3.8-9 of Lee 1995) \begin{equation} \begin{aligned} q^+_F(\vec{X}) = & \frac{\textrm{i}}{4g}\omega_i\phi_i \left(-\omega_j^2\PD{\phi_j}{Z} + g\PDD{\phi_j}{Z}\right) + \frac{\textrm{i}}{4g}\omega_j\phi_j \left(-\omega_i^2\PD{\phi_i}{Z} + g\PDD{\phi_i}{Z}\right) - \frac{\textrm{i}}{2}(\omega_i + \omega_j)\nabla\phi_i \cdot \nabla\phi_j \\ q^-_F(\vec{X}) = & \frac{\textrm{i}}{4g}\omega_i\phi_i \left(-\omega_j^2\PD{\phi_j^*}{Z} + g\PDD{\phi_j^*}{Z}\right) - \frac{\textrm{i}}{4g}\omega_j\phi_j^* \left(-\omega_i^2\PD{\phi_i}{Z} + g\PDD{\phi_i}{Z}\right) - \frac{\textrm{i}}{2}(\omega_i - \omega_j)\nabla\phi_i \cdot \nabla\phi_j^* \end{aligned} \label{eqQFpmDefinition} \end{equation} where

$\phi_i$ and $\phi_j$ are the first-order incident wave potentials at frequencies $\omega_i$ and $\omega_j$

The right hand sides of (\ref{eqQFpmDefinition}) are evaluated on $Z=0$

The solution for $\phi^{\pm}_I$ is known analytically \begin{equation} \begin{aligned} \phi_I^{\pm}(\vec{X}) = & C^{\pm}_I f(\kappa^{\pm}_{ij} Z) \textrm{e}^{-\textrm{i}\left(\vec{k}_i \pm \vec{k}_j\right)\cdot\vec{X}} \\ \vec{k}_i = & \kappa_i(\cos\beta_i, \sin\beta_i, 0) \\ \kappa^{\pm}_{ij} = & \left| \vec{k}_i \pm \vec{k}_j \right| \\ C^{\pm}_I = & \frac{\mp\frac{\textrm{i}}{2}g^2 \left\{\frac{\kappa_j^2-\nu_j^2}{2\omega_j} \pm \frac{\kappa_i^2-\nu_i^2}{2\omega_i} \pm \frac{(\omega_i\pm \omega_j)(\vec{k}_i\cdot\vec{k}_j \mp \nu_i\nu_j)}{\omega_i\omega_j}\right\}} {-(\omega_i\pm\omega_j)^2 + g \kappa^{\pm}_{ij}\tanh \kappa^{\pm}_{ij}h} \end{aligned} \end{equation} where

$f(\kappa Z)$ is the function of depth that appears in the first-order incident potential

In water of infinite depth, $\kappa_i = \nu_i$ and $\kappa_j = \nu_j$ and therefore $\phi_I^+ = 0$ if $\beta_i = \beta_j$.

The second-order scattered potential

The scattered potential, $\phi_S^{\pm}$, is the solution to the BVP (\ref{eqGeneralSecondOrderBVP}) with body forcing (eqs 3.11-2 of Lee 1995) \begin{equation} \begin{aligned} q^+_B(\vec{x}) = & -\PD{\phi^+_I}{n} +\frac{\textrm{i}}{2}(\omega_i + \omega_j)\vec{n}\cdot H^+\vec{x} \\ & +\frac{1}{4}\Big\{ \left(\vec{\alpha}_i \times\vec{n}\right) \cdot \left(\textrm{i}\omega_j\vec{d}_j - \nabla\phi_j\right) + \left(\vec{\alpha}_j \times\vec{n}\right) \cdot \left(\textrm{i}\omega_i\vec{d}_i - \nabla\phi_i\right) \Big\} \\ & - \frac{1}{4}\vec{n}\cdot\Big\{ \vec{d}_i\cdot\nabla \nabla\phi_j + \vec{d}_j\cdot\nabla \nabla\phi_i \Big\} \\ q^-_B(\vec{x}) = & -\PD{\phi^-_I}{n} +\frac{\textrm{i}}{2}(\omega_i - \omega_j)\vec{n}\cdot H^-\vec{x} \\ & +\frac{1}{4}\Big\{ \left(\vec{\alpha}_i \times\vec{n}\right) \cdot \left(-\textrm{i}\omega_j\vec{d}_j^* - \nabla\phi_j^*\right) + \left(\vec{\alpha}_j^* \times\vec{n}\right) \cdot \left(\textrm{i}\omega_i\vec{d}_i - \nabla\phi_i\right) \Big\} \\ & - \frac{1}{4}\vec{n}\cdot\Big\{ \vec{d}_i\cdot\nabla \nabla\phi_j^* + \vec{d}_j^*\cdot\nabla \nabla\phi_i \Big\} \\ \end{aligned} \end{equation} where

$\vec{x}$ is the position in body coordinates $\Bxyz$

$\phi_i$ and $\phi_j$ are the total first-order potentials at frequencies $\omega_i$ and $\omega_j$

$\vec{\xi}_i$ and $\vec{\alpha}_i$ are the complex displacement RAOs for translation and rotation

$\vec{d}_i(\vec{x}) = \vec{\xi}_i+\vec{\alpha}_i\times\vec{x}$

and the rotation matrices are given by \begin{equation} \begin{aligned} H^+ = & \mathcal{M}\left(\vec{\alpha}_i, \vec{\alpha}_j\right) \\ H^- = & \mathcal{M}\left(\vec{\alpha}_i, \vec{\alpha}_j^*\right) \\ \mathcal{M}\left(\vec{\alpha}_i, \vec{\alpha}_j\right) = & \frac{1}{2}\left( \begin{array}{ccc} -\alpha_{2i}\alpha_{2j}-\alpha_{3i}\alpha_{3j} & 0 & 0 \\ \phantom{-}\alpha_{1i}\alpha_{2j} + \alpha_{2i}\alpha_{1j} & -\alpha_{1i}\alpha_{1j}-\alpha_{3i}\alpha_{3j} & 0 \\ \phantom{-}\alpha_{1i}\alpha_{3j} + \alpha_{3i}\alpha_{1j} & \phantom{-}\alpha_{2i}\alpha_{3j}+\alpha_{3i}\alpha_{2j} & -\alpha_{1i}\alpha_{1j}-\alpha_{2i}\alpha_{2j} \end{array}\right) \end{aligned} \end{equation}

The surface forcing for $\phi_S^{\pm}$ takes the same functional form as (\ref{eqQFpmDefinition}) but with $\phi_i$ and $\phi_j$ being the total first-order potentials at frequencies $\omega_i$ and $\omega_j$, minus the surface forcing already accounted for in $\phi_I^{\pm}$.

Integral equations

The integral equations for $\phi_S^{\pm}$ take the same form as at first order.

Evaluating the surface integrals of forcing functions that appear on the right hand side of the integral equations is significantly more complex than in the first-order problem. Both forcing functions, $q_B^{\pm}$ and $q_F^{\pm}$, include second-order derivatives of first-order potentials that cannot be evaluated with reliable accuracy, so the integrals must be transformed into alternative forms involving only first-order derivatives. In addition, evaluating the integral of $q_F^{\pm}$ over $\SF$ is challenging because the integral extends to infinity over the entire free surface.

Body forcing

The forcing function $q_B^{\pm}$ must be integrated over the body surface $\SB$. The required integral is transformed using Stokes' Theorem into an equivalent form that involves only first-order derivatives \begin{equation} \begin{aligned} \int_{\SB}q^{\pm}_B(\vec{\xi})G\ud S_{\xi} = & \int_{\SB} \vec{n} \cdot \vec{U}^{\pm}_B(\vec{\xi}) G \ud S_{\xi} + \int_{\SB} \left\{\vec{n}\times\vec{V}^{\pm}_B(\vec{\xi})\right\} \cdot \nabla_{\xi} G \ud S_{\xi} + \oint_{\CWL} G \vec{V}^{\pm}_B(\vec{\xi}) \cdot \ud \vec{l}_{\xi} \\ \vec{U}^+_B(\vec{x}) = & -\nabla \phi^+_I +\frac{\textrm{i}}{2}(\omega_i+\omega_j) H^+\vec{x} +\frac{\textrm{i}}{4} \Big(\omega_j\vec{d}_j\times\vec{\alpha}_i + \omega_i\vec{d}_i\times \vec{\alpha}_j\Big) \\ \vec{V}^+_B(\vec{x}) = & -\frac{1}{4}\Big(\nabla\phi_j\times\vec{d}_i + \nabla\phi_i\times\vec{d}_j\Big) \\ \vec{U}^-_B(\vec{x}) = & -\nabla \phi^-_I +\frac{\textrm{i}}{2}(\omega_i-\omega_j) H^-\vec{x} +\frac{\textrm{i}}{4} \Big(-\omega_j\vec{d}_j^*\times\vec{\alpha}_i + \omega_i\vec{d}_i\times \vec{\alpha}_j^*\Big) \\ \vec{V}^-_B(\vec{x}) = & -\frac{1}{4}\Big(\nabla\phi_j^*\times\vec{d}_i + \nabla\phi_i\times\vec{d}_j^*\Big) \end{aligned} \end{equation}

Surface forcing

The forcing function $q_F^{\pm}$ must be integrated over the entire free surface $\SF$, which extends to infinity. The required integral is evaluated by dividing $\SF$ into three zones \begin{equation} \int_{\SF} q_F^{\pm}(\vec{\xi}) G(\vec{X},\vec{\xi}) \ud S_{\xi} = \int_{PZ+QZ+AZ} q_F^{\pm}(\vec{\xi}) G(\vec{X},\vec{\xi}) \ud S_{\xi} \end{equation} where

• The panelled zone ($PZ$) is close to the body (or bodies). All bodies must be entirely within the panelled zone. The outer boundary of the panelled zone is a circle, centred on the global origin, called the inner circle ($IC$).
• The quadrature zone ($QZ$) is an annular zone between the inner circle and outer circle ($OC$).
• The asymptotic zone ($AZ$) is the remainder of $\SF$ outside the outer circle and extending to infinity.

OrcaWave evaluates the integral over the asymptotic zone using far-field asymptotic approximations as described in the appendix of (Lee 1995). In brief, the Green's function, $G$, and the first-order potentials, $\phi_i$ and $\phi_j$, are all expressed as Fourier-Bessel asymptotic series, which can be multiplied together and integrated term-by-term. The approximations by asymptotic series and the resulting integration are valid provided the radius of the outer circle is large enough.

The integral over the panelled and quadrature zones is performed numerically. First, the integral is transformed using the Divergence Theorem into an equivalent form that involves only first-order derivatives \begin{equation} \begin{aligned} \int_{PZ+QZ} q^{\pm}_F(\vec{\xi}) G \ud S_{\xi} = & \int_{PZ+QZ} U^{\pm}_F(\vec{\xi}) G \ud S_{\xi} + \int_{PZ+QZ} \vec{V}^{\pm}_F(\vec{\xi})\cdot \nabla^h_{\xi} G \ud S_{\xi} \\ & - \oint_{\CWL + OC} \vec{n}^h \cdot \vec{V}^{\pm}_F(\vec{\xi}) G \ud l_{\xi} \\ U^+_F(X,Y) = & -\frac{\textrm{i}}{4}\left\{ \frac{\omega_i\omega_j}{g}\left(\omega_j\phi_i\PD{\phi_j}{Z} + \omega_i\phi_j\PD{\phi_i}{Z} \right) + (\omega_i + \omega_j)\left(\nabla\phi_i\cdot\nabla\phi_j + \PD{\phi_i}{Z}\PD{\phi_j}{Z} \right) \right\} \\ \vec{V}^+_F(X,Y) = & \frac{\textrm{i}}{4}\Big(\omega_i\phi_i \nabla^h\phi_j + \omega_j\phi_j \nabla^h\phi_i \Big) \\ U^-_F(X,Y) = & -\frac{\textrm{i}}{4}\left\{ \frac{\omega_i\omega_j}{g}\left(\omega_j\phi_i\PD{\phi^*_j}{Z} - \omega_i\phi^*_j\PD{\phi_i}{Z} \right) + (\omega_i - \omega_j)\left(\nabla\phi_i\cdot\nabla\phi^*_j + \PD{\phi_i}{Z}\PD{\phi^*_j}{Z} \right) \right\} \\ \vec{V}^-_F(X,Y) = & \frac{\textrm{i}}{4}\Big(\omega_i\phi_i \nabla^h\phi^*_j - \omega_j\phi^*_j \nabla^h\phi_i \Big) \end{aligned} \end{equation}

The integration over the panelled zone is performed in the usual way by evaluating the integrands at the panel centroids and multiplying by the panel area. The integration in the quadrature zone is performed using Gaussian quadrature. In the line integrals, $\vec{n}^h$ is a two dimensional normal to the line pointing out of $PZ+QZ$.

Second-order loads on a body

The total load on a body is given by the integration of fluid pressure over the exact instantaneous wet portion of the body's surface. Ogilvie (1983) and Lee (1995) show how to expand this complicated expression to first and second order in wave steepness to obtain a perturbation expansion for the load \begin{equation} \begin{aligned} \vec{F}(t) &= \vec{F}\zo(t) + \vec{F}\fo(t) + \vec{F}\so(t) + \ldots \end{aligned} \label{eqForcePerturbationExpansion} \end{equation} OrcaWave follows the same convention as Ogilvie (1983) and Lee (1995) for the second-order load $\vec{F}\so$, by expressing it in an inertial frame of reference: the body coordinates for the body in its mean position.

The second-order load can be written as the sum of three distinct contributions \begin{equation} \vec{F}\so = \vec{F}_p + \vec{F}_q + \vec{F}_r \end{equation} where

$\vec{F}_p$ is known as the potential load. It is the load arising from the second-order potential, $\phi\so$, analogous to the load RAO at first order. OrcaWave will evaluate the potential load if the solve type is full QTF calculation.

$\vec{F}_q$ is known as the quadratic load. It is the load that arises from products of two first-order quantities in $\vec{F}\so$. OrcaWave will evaluate the quadratic load using the selected calculation method(s).

$\vec{F}_r $ is the load that arises from second-order body motion. OrcaWave does not evaluate $\vec{F}_r $.