Mathematical model for wave problems (Part I)
=============================================

Governing equations
~~~~~~~~~~~~~~~~~~~

Flow equation and boundary conditions
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Assuming incompressible flow :math:`(\nabla \cdot \boldsymbol{v})` and irrotational flow :math:`(\nabla \times \boldsymbol{v}_\phi = 0 \Rightarrow \boldsymbol{v}_\phi = \nabla \phi)`, then the flow governing equations are given by:

.. math::
   \begin{align}
        & \Delta \varphi = 0 & \text{in } \Omega
   \end{align}
   :label: GoverningEquations1

.. math::   
   \begin{align}
        & \frac{\partial \xi }{\partial t} + \frac{\partial \varphi}{\partial x} \frac{\partial \xi }{\partial x} + \frac{\partial \varphi}{\partial y} \frac{\partial \xi }{\partial y} - \frac{\partial \varphi}{\partial z} = 0 & \text{on } z = \zeta \\
   \end{align}
   :label: GoverningEquations2

.. math::
   \begin{align}       
        & \frac{\partial \varphi}{\partial t} + \frac{1}{2} \nabla \varphi \cdot \nabla \varphi + \frac{P_{fs}}{\rho } + g\xi  = 0 & \text{on } z = \zeta \\
   \end{align}  
   :label: GoverningEquations3

.. math::
   \begin{align}
        & \boldsymbol{v}_p \cdot \boldsymbol{n}_p + \boldsymbol{v}_\phi \cdot \boldsymbol{n}_p = 0 & \text{on } S_B \\
   \end{align}
   :label: GoverningEquations4

.. math::
   \begin{align}
        & P_p= -\rho \frac{\partial \varphi}{\partial t} -\frac{1}{2} \rho \nabla \varphi \cdot \nabla \varphi - \rho gz_p & \text{in } \Omega
   \end{align}
   :label: GoverningEquations5  

Solution approach
^^^^^^^^^^^^^^^^^

**Taylor series expansion**

Free surface and body boundary condition (BC) will be applied on :math:`z=0`. Taylor series expansion are carried out to both free surface BCs around :math:`z=0` to approximate the BC on :math:`z=\zeta`. Body boundary condition will be applied on :math:`S_B^0`. Taylor series expansion are carried out around :math:`S_B^0` to approximate the BC on :math:`S_B`. 

**Perturbed solution**

A perturbed solution based on stokes waves approximation is used, where the velocity potential and free surface elevation are perturbed as:

.. math::      
   \begin{aligned}
      & \varphi = \epsilon^1\varphi^1 + \epsilon^2\varphi^2 + \epsilon^3\varphi^3 + ...
   \end{aligned}
   :label: VelPotentialPerturbation

.. math::
   \begin{aligned}
      & \xi = \epsilon^1\xi^1 + \epsilon^2\xi^2 + \epsilon^3\xi^3 + ...
   \end{aligned}
   :label: FreeSurfaceElevationPerturbation

Body movement solution is also assumed to be a perturbed solution:

.. math::      
   \begin{aligned}
      & \boldsymbol{X} = \epsilon^1\boldsymbol{X} ^1 + \epsilon^2\boldsymbol{X}^2 + \epsilon^3\boldsymbol{X}^3 + ...
   \end{aligned}
   :label: BodyMovementPerturbed

.. math::      
   \begin{aligned}
      & \boldsymbol{V} = \epsilon^1\boldsymbol{V}^1 + \epsilon^2\boldsymbol{V}^2 + \epsilon^3\boldsymbol{V}^3 + ...
   \end{aligned}
   :label: BodyVelocityPerturbed

Then the translational vector of any point P on the body surface can be perturbed as:

.. math::
   \begin{aligned}
      & \boldsymbol{r}_p = \epsilon^1\boldsymbol{r}_p^1 + \epsilon^2\boldsymbol{r}_p^2 + \epsilon^3\boldsymbol{r}_p^3 + ...
   \end{aligned}
   :label: TranslationalVectorPerturbation

where

.. math::
   \begin{aligned}
      & \boldsymbol{X}^i = (\delta_x^i, \delta_y^i, \delta_z^i, \theta_x^i, \theta_y^i, \theta_z^i) = (\boldsymbol{\delta}^i, \boldsymbol{\theta}^i)
   \end{aligned}
   :label: BodyMovement

.. math::
   \begin{aligned}
      \boldsymbol{r}_p^1 = \boldsymbol{\delta}^1 + \boldsymbol{\theta}^1 \times \overrightarrow{\boldsymbol{R^0P^0}}
   \end{aligned}
   :label: FirstOrderTranslationVector

.. math::
   \begin{aligned}
      \boldsymbol{r}_p^{1+2} = \boldsymbol{\delta}^{1+2} + \boldsymbol{\theta}^{1+2} \times \overrightarrow{\boldsymbol{R^0P^0}} + \boldsymbol{\overline{\overline{H}}} \overrightarrow{\boldsymbol{R^0P^0}}
   \end{aligned}
   :label: SecondOrderTranslationVector

.. math::
   \begin{aligned}
      \boldsymbol{\overline{\overline{H}}} = \frac{1}{2}
      \begin{pmatrix}
         -(\theta_y^2 + \theta_z^2) & 0 & 0 \\
         2\theta_x\theta_y & -(\theta_x^2 + \theta_z^2) & 0 \\
         2\theta_x\theta_z & 2\theta_y\theta_z & -(\theta_x^2 + \theta_y^2)
      \end{pmatrix}
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      \boldsymbol{V}^i = (v_x^i, v_y^i, v_z^i, \omega_x^i, \omega_y^i, \omega_z^i) = (\boldsymbol{v}^i, \boldsymbol{\omega}^i)
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      \boldsymbol{v}_p^1 = \boldsymbol{v}^1 + \boldsymbol{\omega}^1 \times \overrightarrow{\boldsymbol{R^0P^0}}
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      \boldsymbol{v}_p^{1+2} = \boldsymbol{v}^{1+2} + \boldsymbol{\omega}^{1+2} \times \overrightarrow{\boldsymbol{R^0P^0}} + \dot{\boldsymbol{\overline{\overline{H}}}} \overrightarrow{\boldsymbol{R^0P^0}}
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      \boldsymbol{n}_p^1 = \boldsymbol{n}_p^0 + \boldsymbol{\theta}^1 \times \boldsymbol{n}_p^0
   \end{aligned}
   :label:

.. figure:: ../figures/seafem/fig1.png
   :align: center

   First and second order rigid body movements

  

First order approach
~~~~~~~~~~~~~~~~~~~~

First order governing equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

After carrying out the Taylor series expansions, using the perturbed solution, and retaining terms of order :math:`\epsilon`, the first order governing equations become:

.. math::
   \begin{align}
        & \Delta \varphi^1=0 & \text{in } \Omega
   \end{align}
   :label: FirstOrderGoverningEquations1

.. math::   
   \begin{align}
        & \frac{\partial \xi^1}{\partial t} - \frac{\partial \varphi^1}{\partial z} = 0 & \text{in } z = 0 \\
   \end{align}
   :label: FirstOrderGoverningEquations2

.. math::
   \begin{align}       
        & \frac{\partial \varphi^1}{\partial t} \frac{P_{fs}}{\rho } + g\xi^1 = 0 & \text{in } z = 0 \\
   \end{align}  
   :label: FirstOrderGoverningEquations3

.. math::
   \begin{align}
        & \boldsymbol{v}_p^1 \cdot \boldsymbol{n}_p^0 + \boldsymbol{v}_\phi^1 \cdot \boldsymbol{n}_p^0 = 0 & \text{in } S_B \\
   \end{align}
   :label: FirstOrderGoverningEquations4 

and the first order pressure at a point :math:`P_p^1` on the body surface is

.. math::
   \begin{aligned}
      & P_p^1 = P_H^0 + P_H^1 + P_D^1 & \text{in } \boldsymbol{S_B}
   \end{aligned}
   :label:

where :math:`P_D^1=-\rho (\partial \varphi^1  )/\partial t, P_H^0=-\rho gz_p`, and :math:`P_H^1=-\rho gr_{pz}^1`.

First order decomposition solution
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The total velocity potential can be decomposed as:

.. math::
   \begin{aligned}
      \varphi^1 = \psi^1 + \phi^1
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      \xi^1 = \zeta^1 + \eta^1
   \end{aligned}
   :label:

where :math:`\psi^1` is the incident wave potential, and :math:`\phi` is the diffraction-radiation wave potential.

**First-order incident wave solution**

The incident wave velocity potential :math:`\psi^1` fulfils the following equations:

.. math::
   \begin{aligned}
      & \nabla \psi^1 = 0 & \text{in } \Omega
   \end{aligned}
   :label: incident1

.. math::
   \begin{aligned}
      & \frac{\partial \zeta^1}{\partial t} - \frac{\partial \psi^1}{\partial z} = 0 & \text{in } z = 0
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \frac{\partial \psi^1}{\partial t} + g\zeta^1 = 0 & \text{in } z = 0
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \frac{\partial \psi^1}{\partial z} = 0 & \text{in } z = -H
   \end{aligned}
   :label: incident4

Eqs. :eq:`eq:incident1` - :eq:`eq:incident4`  have an analytical solution, known as the Airy wave solution:

.. math::
   \begin{aligned}
      & \psi^1=\sum_i \frac{A_i g}{\omega_i} \frac{cosh⁡(|\boldsymbol{k}_i |(H+z))}{cosh⁡(|\boldsymbol{k}_i |H)} sin⁡(\boldsymbol{k}_i \boldsymbol{x} - \omega_i t+\alpha_i) \\
      & \zeta^1=\sum_i A_i cos⁡(\boldsymbol{k}_i \boldsymbol{x} - \omega_i t+\alpha_i)
   \end{aligned}
   :label:

where :math:`A` is the wave amplitude, :math:`H` is a constant water depth, :math:`\boldsymbol{k}=2π/L (cos⁡(\Gamma ),sin⁡(\Gamma ) )`, :math:`L` is the wave length, :math:`\Gamma ` is the wave propagation direction, :math:`\omega=2π/T`, :math:`T` is the wave period, and :math:`\alpha` is the wave phase delay. The following dispersion relation holds:

.. math::
   \begin{aligned}
      \omega_i^2 = g|\boldsymbol{k}_i | tanh⁡(|\boldsymbol{k}_i |H)
   \end{aligned}
   :label:

and the fluid pressure induced by the Airy wave in a point :math:`P` is given by:

.. math::
   \begin{aligned}
      P_p\psi^1=\sum_i \rho gA_i cosh⁡(|\boldsymbol{k}_i |(H+z))/cosh⁡(|\boldsymbol{k}_i |H) cos (\boldsymbol{k}_i \boldsymbol{x} - \omega_i t+\alpha_i)
   \end{aligned}
   :label:
   
In the asymptotic case of infinite water depth :math:`(H→\infty )`, the factor :math:`cosh⁡(|\boldsymbol{k}|(H+z))/cosh⁡(|\boldsymbol{k}|H) →exp⁡(|\boldsymbol{k}|z)`.

**First-order diffraction-radiation wave problem**

The governing equations for the diffraction-radiation velocity potential :math:`\phi^1` is given by:

.. math::
   \begin{aligned}
      & \Delta \phi^1=0 & \text{in } \Omega
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \frac{\partial \eta^1}{\partial t} - \frac{\partial \phi^1}{\partial z} = 0	& \text{on } z=0
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \frac{\partial \phi^1}{\partial t} + \frac{P_{fs}}{\rho } + g\eta^1 = 0	& \text{on } z=0
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{v}_\phi^1 \cdot \boldsymbol{n}_p^0  = -\boldsymbol{v}_p^1 \cdot \boldsymbol{n}_p^0 - \boldsymbol{v}_\psi^1 \cdot \boldsymbol{n}_p^0	& \text{on } P \in S_B
   \end{aligned}
   :label:

and the fluid pressure at a point :math:`P` on the body surface is given by:

.. math::
   \begin{aligned}
      & P_p^1 = P_{ph}^0 + P_{ph}^1 + P_{p\psi}^1 + P_{p\phi}^1 &	\text{on } P \in S_B
   \end{aligned}
   :label:

where :math:`P_{p\phi}^1 = -\rho (\partial \phi^1)/\partial t`, :math:`P_{ph}^0 = -\rho gz`, and :math:`P_{ph}^1 = -\rho gr_{pz}^1`.

Second order approach
~~~~~~~~~~~~~~~~~~~~~

Second-order governing equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

After carrying out the Taylor series expansions, using the perturbed solution, and retaining terms up to order :math:`\epsilon^2`, and considering that :math:`\varphi^{1+2}=\varphi^1+\varphi^2`, :math:`\xi^{1+2}=\xi^1+\xi^2` , :math:`P_p^{1+2}=P_p^1+P_p^2`, :math:`X_B^{1+2}=X_B^1+X_B^2`, :math:`V_B^{1+2}=V_B^1+V_B^2`, :math:`v_p^{1+2}=v_p^1+v_p^2`, :math:`v_\phi^{1+2}=v_\phi^1+v_\phi^2, r_p^{1+2}=r_p^1+r_p^2`, the governing equations become:

.. math::
   \begin{aligned}
      & \Delta \varphi^{1+2}=0 &	\text{in } \omega
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \frac{\partial \xi^{1+2}}{\partial t} - \frac{\partial \varphi^{1+2}}{\partial z} = \xi^1 \frac{\partial}{\partial z} \left(\frac{\partial \varphi^1}{\partial z}\right) - \frac{\partial \xi^1}{\partial x} \frac{\partial \varphi^1}{\partial x} - \frac{\partial \xi^1}{\partial y} \frac{\partial \varphi^1}{\partial y} &	\text{in } z=0
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \frac{\partial \varphi^{1+2}}{\partial t} + \frac{P_{fs} }{\rho } + g\xi^{1+2} = -\xi^1 \frac{\partial}{\partial z} \left(\frac{\partial \varphi^1}{\partial t}\right) -\frac{1}{2} \nabla \varphi^1 \cdot \nabla \varphi^1	& \text{in } z=0
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & (\boldsymbol{v}_p^1 + \boldsymbol{v}_\phi^1 ) \cdot \boldsymbol{n}_p^1 + (\boldsymbol{v}_p^2 + \boldsymbol{v}_\phi^2)\cdot \boldsymbol{n}_p^0 = -(\boldsymbol{r}_p^1 \cdot \nabla \boldsymbol{v}_\phi^1) \cdot \boldsymbol{n}_p^0 & \text{in } S_B^0
   \end{aligned}
   :label:

and the pressure at a point :math:`P` on the body surface is:

.. math::
   \begin{aligned}
      & P_p^{1+2} = P_H^0+P_H^{1+2}+P_D^{1+2} &	in S_B^0
   \end{aligned}
   :label:

where  :math:`P_H^0=-\rho gz_p`, and :math:`P_H^{1+2} = -\rho gr_{pz}^{1+2}`, and :math:`P_D^{1+2} = -\rho (\partial \varphi^{1+2})/\partial t-\rho r_p^1 \cdot \nabla ((\partial \varphi^1)/\partial t)-\rho 1/2 \nabla \varphi^1 \cdot \nabla \varphi^1`.

Second-order decomposition solution
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

**Second-order incident wave solution**

The second-order total velocity potential can be decomposed as:

.. math::
   \begin{aligned}
      \varphi^2=\psi^2+\phi^2
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      \xi^2=\zeta^2+\eta^2
   \end{aligned}
   :label:

where :math:`\psi^2` is the second-order incident wave potential, and :math:`\phi^2` is the second-order diffraction-radiation wave velocity potential. The up to second-order incident wave potential and free surface elevation fulfils the following equations:

.. math::
   \begin{aligned}
      & \Delta \psi^{1+2}=0 &	\text{in } \Omega
   \end{aligned}
   :label:2ndOrderGovEquations1

.. math::
   \begin{aligned}
      & \frac{\partial \zeta^{1+2}}{\partial t} - \frac{\partial \psi^{1+2}}{\partial z} = \zeta^1 \frac{\partial ^2 \psi^1}{\partial z^2} - \frac{\partial \zeta^1}{\partial x} \frac{\partial \psi^1}{\partial x} - \frac{\partial \zeta^1}{\partial y} \frac{\partial \psi^1}{\partial y} &	\text{on } z=0
   \end{aligned}
   :label:2ndOrderGovEquations2

.. math::
   \begin{aligned}
      & \frac{\partial \psi^{1+2}}{\partial t} + g\zeta^{1+2} = -\zeta^1 \frac{\partial }{\partial z} \left(\frac{\partial \psi^1}{\partial t}\right) - \frac{1}{2} \nabla \psi^1 \cdot \nabla \psi^1 &	\text{on } z=0
   \end{aligned}
   :label:2ndOrderGovEquations3

.. math::
   \begin{aligned}
      & \frac{\partial \psi^{1+2}}{\partial z}=0	& \text{on } z=-H
   \end{aligned}
   :label:2ndOrderGovEquations4

The solution to :eq:`eq:2ndOrderGovEquations1` - :eq:`eq:2ndOrderGovEquations4` is as follows:

.. math::
   \begin{aligned}
      \psi^{1+2} = \psi^1 + \sum_i B_{ij}^0 sin (2 (\boldsymbol{k_i} \boldsymbol{x} - \omega_i t + \alpha_i )) \\
      + \sum_{j>i} \sum_i B_{ij}^+ cosh⁡ (|\boldsymbol{k}_i + \boldsymbol{k}_j| (H+z)) sin ((\boldsymbol{k}_i \boldsymbol{x} - \omega_i t + \delta_i) + (\boldsymbol{k}_j \boldsymbol{x} - \omega_j t + \alpha_j )) \\
      + \sum_{j>i} \sum_i B_{ij}^- cosh⁡ (|\boldsymbol{k}_i - \boldsymbol{k}_j |(H+z)) sin ((\boldsymbol{k}_i \boldsymbol{x} - \omega_i t + \delta_i ) - (\boldsymbol{k}_j \boldsymbol{x} - \omega_j t + \alpha_j ))
   \end{aligned}
   :label:


where the coefficients are given in table 1. In the asymptotic case of infinite depth :math:`(H→\infty )` the coefficients :math:`B_{ij}^0→0` , :math:`D_{ij}^+→\infty `, :math:`D_{ij3}^-→\infty `, :math:`B_{ij}^+→0`, :math:`B_{ij}^-→0`. Then, the second order velocity potential becomes null. The wave elevation up to second order is obtained from:

.. math::
   \begin{aligned}
      \zeta^{1+2} = -\frac{1}{g} \left(\frac{\partial\psi^{1+2}}{\partial t} + \frac{P}{\rho } + \zeta^1 \frac{\partial}{\partial z} \left(\frac{\partial \psi^1}{\partial t}\right) + \frac{1}{2} \nabla \psi^1 \cdot \nabla \psi^1 - C^1\right)
   \end{aligned}
   :label:

and the fluid pressure induced by the second order wave potential at a point :math:`P` is:

.. math::
   \begin{aligned}
      P_{p\psi}^{1+2} = - \rho \left(\frac{\partial \psi^{1+2}}{\partial t} + \frac{1}{2} \nabla \psi^1 \cdot \nabla \psi^1\right) - \rho \left(\boldsymbol{r}_p^1 \cdot \nabla \left(\frac{\partial \psi^1}{\partial t}\right)\right)
   \end{aligned}
   :label:

In the asymptotic case of infinite depth :math:`(H→\infty )` the coefficients :math:`B_{ij}^0→0` , :math:`D_{ij}^+→\infty `, :math:`D_{ij3}^-→\infty `, :math:`B_{ij}^+→0`, :math:`B_{ij}^-→0`. Then, the second order velocity potential becomes null.

.. math::
   \begin{aligned}
      & B_{ij}^0 = \frac{3A_i^2 g|\boldsymbol{k}_i |}{8 \omega_i} \frac{cosh⁡(2|\boldsymbol{k}_i |(H+z))}{sinh^3⁡(|\boldsymbol{k}_i |H) cosh⁡(|\boldsymbol{k}_i |H)} \\
      & B_{ij}^+  =   \frac{\sum_{k=1}^3 C_{ijk}^+}{D_{ijk}^+} \\
      & B_{ij}^-  =   \frac{\sum_(k=1)^3 C_{ijk}^-}{D_{ijk}^-} \\
      & C_{ij1}^+ =   \frac{A_i A_j}{2g} \omega_i^2 (\omega_i + \omega_j) \\
      & C_{ij1}^- =   \frac{A_i A_j}{2g} \omega_i^2 (\omega_i - \omega_j) \\
      & C_{ij2}^+ =   \frac{A_i A_j}{4g} (\omega_i \omega_j - g^2 \frac{\boldsymbol{k}_i \cdot \boldsymbol{k}_j}{\omega_i \omega_j})(\omega_i+\omega_j) \\
      & C_{ij2}^- = - \frac{A_i A_j}{4g} (\omega_i \omega_j + g^2 \frac{\boldsymbol{k}_i \cdot \boldsymbol{k}_j}{\omega_i \omega_j})(\omega_i-\omega_j) \\
      & C_{ij3}^+ = - \frac{A_i A_j}{2} \frac{g}{\omega_i} (|\boldsymbol{k}_i|^2 + \boldsymbol{k}_i \cdot \boldsymbol{k}_j) \\
      & C_{ij3}^- =   \frac{A_i A_j}{2} \frac{g}{\omega_i} (|\boldsymbol{k}_i|^2 - \boldsymbol{k}_i \cdot \boldsymbol{k}_j) \\
      & D_{ij}^+  =  |\boldsymbol{k}_i + \boldsymbol{k}_j| sinh(|\boldsymbol{k}_i + \boldsymbol{k}_j| H) -\frac{1}{g} (\omega_i + \omega_j)^2 cosh⁡(|\boldsymbol{k}_i + \boldsymbol{k}_j| H) \\
      & D_{ij3}^- =  |\boldsymbol{k}_i - \boldsymbol{k}_j| sinh(|\boldsymbol{k}_i - \boldsymbol{k}_j| H) -\frac{1}{g} (\omega_i - \omega_j)^2 cosh⁡(|\boldsymbol{k}_i + \boldsymbol{k}_j| H)
   \end{aligned}
   :label:

Table 1 Stoke´s second-order wave potential coefficients

**Second-order diffraction-radiation wave problem**

The governing equations for the diffraction-radiation velocity potential up to second-order :math:`\phi^{1+2}` is given by:

.. math::
   \begin{aligned}
      & \Delta \phi^{1+2}=0 & \text{in } \Omega
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \frac{\partial\eta^{1+2}}{\partial t} - \frac{\partial\phi^{1+2}}{\partial z} = -\frac{\partial\phi^1}{\partial x} \frac{\partial\eta^1}{\partial x} - \frac{\partial\phi^1}{\partial y} \frac{\partial\eta^1}{\partial y} - \frac{\partial\phi^1}{\partial x} \frac{\partial\zeta^1}{\partial x} - \\
      & -\frac{\partial\phi^1}{\partial y} \frac{\partial\zeta^1}{\partial y} - \frac{\partial\psi ^1}{\partial x} \frac{\partial\eta^1}{\partial x} - \frac{\partial\psi ^1}{\partial y} \frac{\partial\eta^1}{\partial y} & \text{in } z = 0
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \frac{\partial\phi^{1+2}}{\partial t} + \frac{P_{fs}}{\rho } + g\eta^{1+2} = -\eta^1 \frac{\partial}{\partial z} \left(\frac{\partial\phi^1}{\partial t}\right) - \zeta^1 \frac{\partial}{\partial z} \left(\frac{\partial\phi^1}{\partial t}\right) - \\
      & -\eta^1 \frac{\partial}{\partial z} \left(\frac{\partial\psi ^1}{\partial t}\right)
      - \frac{1}{2} \nabla \phi^1 \cdot \nabla \phi^1 - \nabla \psi ^1 \cdot \nabla \phi^1 & \text{in } z = 0
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      \boldsymbol{v}_\phi^2 \cdot \boldsymbol{n}_p^0 + \boldsymbol{v}_\phi^1 \cdot \boldsymbol{n}_p^1 = \\
      & -(\boldsymbol{v}_p^1 + \boldsymbol{v}_\psi ^1) \cdot \boldsymbol{n}_p^1 - (\boldsymbol{v}_p^2 + \boldsymbol{v}_\psi ^2 + \boldsymbol{r}_p^1 \cdot (\nabla \boldsymbol{v}_\phi^1 + \nabla \boldsymbol{v}_\psi ^1)) \cdot \boldsymbol{n}_p^0 & \text{in } \boldsymbol{P} \in \boldsymbol{S}_B
   \end{aligned}
   :label:

and the fluid pressure at a point :math:`P` on the body surface is given by:

.. math::
   \begin{aligned}
      & P_p^{1+2} = P_{ph}^0 + P_{ph}^{1+2} + P_{p\psi }^{1+2} + P_{p\phi}^{1+2} & \text{in } \boldsymbol{P} \in \boldsymbol{S}_B
   \end{aligned}
   :label:

where :math:`P_{ph}^0 = -\rho gz`, :math:`P_{ph}^{1+2} = -\rho gr_{pz}^{1+2}` and:

.. math::
   \begin{aligned}
      & P_{p\phi}^{1+2} = -\rho \frac{\partial\phi^{1+2}}{\partial t} - \rho \frac{1}{2} \nabla \phi^1 \cdot \nabla \phi^1 - \rho \nabla \psi ^1 \cdot \nabla \phi^1 - \rho r_p^1 \cdot \nabla \left(\frac{\partial\phi^1}{\partial t}\right) & \text{in } \boldsymbol{P} \in \boldsymbol{S}_B
   \end{aligned}
   :label: