The propagating medium is considered to consist of two homogeneous layers of fluid with very different properties. The upper layer is free surface water, infinitesimally thick as skin and characterised by capillary-gravity wave propagation. The lower layer is the inner water with finite or semi-infinite thickness characterised by acoustic wave propagation. Therefore, two different models must be introduced to account for these features: an internal fluid model and a surface model.

We assume that the flow is characterized by two variables modelling the mass density ${\rho }_{tot}$ and the velocity potential Φ that satisfy:

$\frac{\partial {\rho }_{tot}}{\partial t}+div\left({\rho }_{tot}\nabla \Phi \right)=0\text{in}\Omega \times \left]0,T\right[.$(6)

$\frac{\partial \Phi }{\partial t}+\frac{1}{2}{\Vert \nabla \Phi \Vert }^2+F\left({\rho }_{tot}\right)=0\text{in}\Omega \times \left]0,T\right[$(7)

where $F\left({\rho }_{tot}\right)={\displaystyle \underset{{\rho }_0}{\overset{{\rho }_{tot}}{\int }}\frac{1}{\rho }\cdot \frac{\partial p}{\partial \rho }\text{d}\rho }+F\left({\rho }_0\right)$ is the barotropic potential, $p$, the fluid pressure and T, the simulation time.

Applying Newton’s second law of motion to an infinitesimal small surface element of thickness $2\epsilon$ that vertically moves of a displacement ${\eta }_{tot}$, leads to the free surface equilibrium equation:

$2\epsilon {\rho }_{tot}\frac{D^2{\eta }_{tot}}{D{t}^2}=-{\rho }_{tot}\frac{\partial \Phi }{\partial t}-\frac{{\rho }_{tot}}{2}{\Vert {\nabla }_s\Phi \Vert }^2+\sigma {\Delta }_s{\eta }_{tot}-{\rho }_{tot}g{\eta }_{tot}\text{in}{\Gamma }_s\times \left]0,T\right[$(8)

where $D/Dt$ stands for material derivative. In comparison with the dynamic boundary condition Equation (3) that Equation (8) replaces, the inertia of the surface has been taken into account. The $\epsilon$ value is chosen in our case so that the characteristic velocity of Equation (8), namely $\sqrt{\sigma /2{\rho }_{tot}\epsilon }$, is equal to the velocity $c_r=\sqrt[4]{4g\sigma /{\rho }_{tot}}$ which corresponds to the special case of a non-dispersive capillary-gravity wave with $\lambda ={\lambda }_c$ in Equation (5) [20] . However, this value could be chosen arbitrarily to take account of the presence of particles on the surface.

The kinematic boundary condition Equation (2) becomes the continuity of normal velocity at the interface:

$\frac{\partial {\eta }_{tot}}{\partial t}+{\nabla }_s\Phi \cdot {\nabla }_s{\eta }_{tot}=\frac{\partial \Phi }{\partial \nu }\text{in}{\Gamma }_s\times \left]0,T\right[$(9)

taking account of the rotation of the normal to the surface.

The global nonlinear dynamical model obtained is linearized around a main steady state. Therefore, the global solution is split into a steady state obtained and a transient one. The lateral boundary conditions are defined separately according to the nature of the state. For the steady flow, the most realistic condition is to fix the normal velocity. For transient flow, non-reflecting boundary conditions have to be prescribed for the inlet and the outlet of Ω in order to avoid any spurious reflections of the waves reaching the boundaries of the domain.

We introduce ${\phi }_0$, the velocity potential corresponding to a steady quasi-uniform horizontal steady flow of an incompressible fluid that enters and leaves Ω at constant unit horizontal velocity with normal surface displacement variation along ${\text{Γ}}_s$ regarded as negligible and non penetrability condition on ${\text{Γ}}_0\cup {\text{Γ}}_b$ satisfied. This background flow is stationary with respect to the boat, which was chosen as the frame of reference. ${\phi }_0$ is the solution to the following problem:

$\{\begin{array}{l}-\Delta {\phi }_0=0\text{in}\Omega \text{and}{\displaystyle {\int }_{{\Gamma }_s}{\phi }_0}=0,\\ \frac{\partial {\phi }_0}{\partial \nu }=0\text{on}{\Gamma }_0\cup {\Gamma }_b\cup {\Gamma }_s,\\ \frac{\partial {\phi }_0}{\partial \nu }=\left(e_1\cdot \nu \right)\text{in}{\Gamma }_1\cup {\Gamma }_2.\end{array}$(10)

To obtain the related displacement on the surface ${\eta }_0$ of this flow for our model, the stationary balance of forces on free surface Equation (8) along with homogeneous Neumann boundary conditions are applied. According to the frame of reference, it leads in stationary case to:

$\{\begin{array}{l}2\epsilon {\rho }_0{U}^2{\nabla }_s{\phi }_0\cdot {\nabla }_s\left({\nabla }_s{\phi }_0\cdot {\nabla }_s{\eta }_0\right)+\sigma {\Delta }_s{\eta }_0-{\rho }_{0}g{\eta }_0=-\frac{{\rho }_0}{2}{U}^2{\Vert {\nabla }_s{\phi }_0\Vert }^2\text{in}{\Gamma }_s,\hfill \\ \frac{\partial {\eta }_0}{\partial {\nu }_s}=0\text{on}\partial {\Gamma }_s\hfill \end{array}$(11)

with ${\rho }_0$, the fluid density and $g$, the gravity acceleration. The solution of Equation (10) corresponds to a steady quasi-uniform horizontal flow with a digging effect due to the term $-{\rho }_0{U}^2{\Vert {\nabla }_s{\phi }_0\Vert }^2/2$, as shown in Figure 2 . The order of magnitude of free surface strain is about 10⁻³ m and, therefore, negligible in comparison with initial computational domain Ω dimensions.

We study the evolution of a small disturbance around the steady state $\left({\rho }_0,{\phi }_0,{\eta }_0\right)$. The unsteady waves in the fluid are represented by the perturbation functions $\rho ,\phi ,\eta$ of variables $x\left(x_1,x_2\right)$ and t. The problem is formulated with them wherein ${\rho }_{tot}\left(x,t\right)={\rho }_0\left(x\right)+\rho \left(x,t\right)$, $\text{Φ}\left(x,t\right)=U{\phi }_0\left(x\right)+\phi \left(x,t\right)$, ${\eta }_{tot}\left(x,t\right)={\eta }_0\left(x\right)+\eta \left(x,t\right)$. The solution is split into a main background steady-state component and a transient one. The domain Ω is then cropped by ${\text{Γ}}_0$, ${\text{Γ}}_b$, ${\text{Γ}}_s={\eta }_0$, ${\text{Γ}}_{1equi}$ and ${\text{Γ}}_{2equi}$, as shown in Figure 3 . The new lateral boundaries ${\text{Γ}}_{1equi}$ and ${\text{Γ}}_{2equi}$ correspond to equipotential lines of ${\phi }_0$ passing respectively through the left upper domain corner and right upper corner of Ω. As a result, the main steady flow crosses perpendicularly ${\text{Γ}}_{1equi}$ and ${\text{Γ}}_{2equi}$ and no reflected flow remains in the new domain Ω. The artificial boundaries are chosen far enough from the rigid body to consider that steady-state flow is uniform in this area, so the corners of the new domain are right-angled. The disturbance is so small that it is then reasonable to neglect the nonlinear terms in the governing equations Equation (6)-Equation (9) to obtain Equation (12)-Equation (15). Convective derivatives with flow velocity $U\nabla {\phi }_0$ are used to derive linearized equations Equation (13) and Equation (14). Hence, $\left(\rho ,\phi ,\eta \right)$ are assured of satisfying the linearized enhanced Neumann-Kelvin’s model with capillarity:

$\frac{\partial \rho }{\partial t}+U\nabla \rho \cdot \nabla {\phi }_0+{\rho }_0\Delta \phi =0\text{in}\Omega \times \left]0,T\right[.$(12)

$\frac{{\partial }^2\phi }{\partial t^2}+2U\nabla {\phi }_0\cdot \nabla \left(\frac{\partial \phi }{\partial t}\right)+U^2\nabla {\phi }_0\cdot \nabla \left(\nabla {\phi }_0\cdot \nabla \phi \right)-c_f^2\Delta \phi =0\text{in}\Omega \times \left]0,T\right[$(13)

given that $\frac{1}{2}{\left|\nabla {\phi }_0\right|}^2+\nabla F\left({\rho }_0\right)=0$ and $\frac{\partial F\left({\rho }_0\right)}{\partial {\rho }_{tot}}=\frac{c_f^2}{{\rho }_0}$.

$\begin{array}{l}2\epsilon {\rho }_0\left(\frac{{\partial }^2\eta }{\partial t^2}+2U{\nabla }_s{\phi }_0\cdot {\nabla }_s\left(\frac{\partial \eta }{\partial t}\right)+U^2{\nabla }_s{\phi }_0\cdot {\nabla }_s\left({\nabla }_s{\phi }_0\cdot {\nabla }_s\eta \right)\right)\\ =\sigma {\Delta }_s\eta -{\rho }_{0}g\eta -{\rho }_0\frac{\partial \phi }{\partial t}-{\rho }_{0}U{\nabla }_s{\phi }_0\cdot {\nabla }_s\phi \text{in}{\Gamma }_s\times \left]0,T\right[\end{array}$(14)

where $\cdot$ denotes the scalar product. Since the domain of study was reshaped, ${\nabla }_s{\eta }_0$ and ${\text{Δ}}_s{\eta }_0$ are set to zero on ${\text{Γ}}_s$.

$\frac{\partial \phi }{\partial \nu }=\frac{\partial \eta }{\partial t}+U{\nabla }_s{\phi }_0\cdot {\nabla }_s\eta \text{in}{\Gamma }_s\times \left]0,T\right[.$(15)

$\frac{\partial \phi }{\partial \nu }=0\text{in}{\Gamma }_0\cup {\Gamma }_b\times \left]0,T\right[.$(16)

Non-reflecting boundary condition applied on inner fluid lateral edges is:

$a\frac{\partial \phi }{\partial t}+\left(U\frac{\partial {\phi }_0}{\partial \nu }+c_f\right)\frac{\partial \phi }{\partial \nu }=0\text{in}{\Gamma }_{1equi}\cup {\Gamma }_{2equi}\times \left]0,T\right[$(17)

where $a$ is a parameter related to the angle of incidence waves with respect to the normal of the boundary surface. This relation is consistent with Sommerfeld-like or zero-order non-reflecting boundary conditions for a wave that propagates at the phase velocity $c_f$ corrected by the normal to the boundary component of velocity of the main background steady flow $U\partial {\phi }_0/\partial \nu$. In the following, the parameter $a$ is set to 1, this implies that the angles of incidence of impinging disturbances are close to the normal of the boundary. For such an order of approximation of absorbing boundary conditions, it is not beneficial to set $a\ne 1$ [14] .

On the surface bounds $\partial {\text{Γ}}_s$, a Sommerfeld-like non-reflecting boundary condition is difficult to apply since the surface is a very dispersive propagating medium, as can be stated from the dispersion relation Equation (5). Therefore, a value of velocity that approaches the apparent wave velocity value at boundaries can hardly be set. The more different this value is, the more spurious reflections you get. Furthermore, in the case of multi-layer models, numerical instabilities due to singularities appear at these points [25] .

As the conditions are to be set for the points $\partial {\text{Γ}}_s$ that belong to the surface ${\text{Γ}}_s$ and to the lateral boundaries ${\text{Γ}}_{1equi}$ or ${\text{Γ}}_{2equi}$, the equations Equation (14), Equation (15) and Equation (17) are considered to devise the new boundary conditions. The guiding idea is to extend the non-reflective boundary condition applied on the lateral boundaries ${\text{Γ}}_{1equi}$ or ${\text{Γ}}_{2equi}$ to the surface ${\text{Γ}}_s$ intersecting points $\partial {\text{Γ}}_s$ by means of the normal velocity continuity condition. Using the simplifying assumptions ${\phi }_0=x_1$, $s=x_1$ and $\left({\nu }_s\cdot e_1\right)=\pm 1$ led by choosing ${\text{Γ}}_{1equi}$ and ${\text{Γ}}_{2equi}$ away from rigid body, the following simplified equations must be satisfied on $\partial {\text{Γ}}_s$:

$\frac{{\partial }^2\eta }{\partial t^2}+2U\frac{{\partial }^2\eta }{\partial x_1\partial t}+\left(U^2-c_r^2\right)\frac{{\partial }^2\eta }{\partial x_1^2}+\frac{1}{2\epsilon }\left(\frac{\partial \phi }{\partial t}+U\frac{\partial \phi }{\partial x_1}\right)+\frac{g}{2\epsilon }\eta =0,$(18)

$\frac{\partial \phi }{\partial x_2}=\frac{\partial \eta }{\partial t}+U\frac{\partial \eta }{\partial x_1},$(19)

$\frac{\partial \phi }{\partial t}+\left(U+c_f\right)\frac{\partial \phi }{\partial x_1}=0\text{on}{\Gamma }_s\cap {\Gamma }_{2equi},$(20)

$\frac{\partial \phi }{\partial t}+\left(U-c_f\right)\frac{\partial \phi }{\partial x_1}=0\text{on}{\Gamma }_s\cap {\Gamma }_{1equi}.$(21)

For the right side (resp. the left side), the solution method consists in derivating first Equation (20) (resp. Equation (21)) with respect to $x_2$ and Equation (19) with respect to $x_1$ and $t$, in order to eliminate partial derivatives of $\phi$. Introducing the resulting expression into Equation (18) leads, after integrating with respect to time, to the following new boundary condition in cartesian coordinates for each edge,

$Z^{\pm }\frac{\partial \eta }{\partial t}+A^{\pm }\frac{\partial \eta }{\partial x_1}+B^{\pm }{\displaystyle {\int }_0^t\eta \text{d}s}+C^{\pm }\phi =0\text{on}\partial {\Gamma }_s^{\pm }$(22)

with $A^{\pm },B^{\pm },C^{\pm },Z^{\pm }$ depending on $U,c_f,c_r,ϵ,g$. The symbol ${}^{-}$ denotes that condition is on the left boundary of ${\text{Γ}}_s$ and ${}^{+}$ on the right one. The boundary conditions obtained are said non-local in time as they depend not only on the time $t$ but also on the entire history of $\eta$ on $\partial {\text{Γ}}_s$.

Multiplying Equation (13) by $\psi \in H^1\left(\text{Ω}\right)$ and Equation (14) by $v\in H^1\left({\text{Γ}}_s\right)$, respectively, together with Green’s formula application leads to the following coupled variational formulation: find functions $\left(\phi ,\eta \right)\in H^1\left(\Omega \right)\times H^1\left({\Gamma }_s\right)\times L^2\left(\left]0,T\right[\right)$ such that $\forall \left(\psi ,v\right)\in H^1\left(\Omega \right)\times H^1\left({\Gamma }_s\right)$

$\begin{array}{l}{\displaystyle {\int }_{\Omega }\ddot{\phi }\psi \text{d}\tau }+U{\displaystyle {\int }_{\Omega }\nabla {\phi }_0\cdot \left(\nabla \stackrel{\dot{}}{\phi }\psi -\stackrel{\dot{}}{\phi }\nabla \psi \right)\text{d}\tau }+c_f{\displaystyle {\int }_{\Gamma }\stackrel{\dot{}}{\phi }\psi \text{d}\tau }+c_f^2{\displaystyle {\int }_{\Omega }\nabla \phi \cdot \nabla \psi \text{d}\tau }\\ -U^2{c}_f^2{\displaystyle {\int }_{\Omega }\left(\nabla {\phi }_0\cdot \nabla \phi \right)\left(\nabla {\phi }_0\cdot \nabla \psi \right)\text{d}\tau }-c_f^2{\displaystyle {\int }_{{\Gamma }_s}\stackrel{\dot{}}{\eta }\psi \text{d}\tau }-U{c}_f^2{\displaystyle {\int }_{{\Gamma }_s}{\nabla }_s{\phi }_0\cdot {\nabla }_s\eta \psi \text{d}\tau }=0\end{array}$(23)

and

$\begin{array}{l}2\epsilon c_f^2{\displaystyle {\int }_{{\Gamma }_s}\left(\ddot{\eta }v+U{\nabla }_s{\phi }_0\left({\nabla }_s\stackrel{\dot{}}{\eta }v-\stackrel{\dot{}}{\eta }{\nabla }_{s}v\right)-U^2\left({\nabla }_s{\phi }_0\cdot {\nabla }_s\eta \right)\left({\nabla }_s{\phi }_0\cdot {\nabla }_{s}v\right)\right)\text{d}\sigma }\\ -2\epsilon c_f^{2}U{\displaystyle {\int }_{{\Gamma }_s}\left(Uv{\Delta }_s{\phi }_0\left({\nabla }_s{\phi }_0\cdot {\nabla }_s\eta \right)+v{\Delta }_s{\phi }_0\stackrel{\dot{}}{\eta }\right)\text{d}\sigma }+\frac{c_f^2}{{\rho }_0}{\displaystyle {\int }_{{\Gamma }_s}\sigma {\nabla }_s\eta \cdot {\nabla }_{s}v\text{d}\sigma }\\ +\frac{c_f^2}{{\rho }_0}{\displaystyle {\int }_{{\Gamma }_s}{\rho }_0\eta gv\text{d}\sigma }+c_f^2{\displaystyle {\int }_{{\Gamma }_s}\stackrel{\dot{}}{\phi }v\text{d}\sigma }-c_f^{2}U{\displaystyle {\int }_{{\Gamma }_s}{\nabla }_s{\phi }_0\cdot {\nabla }_{s}v\phi +v{\Delta }_s{\phi }_0\phi \text{d}\sigma }\\ +{\left[\left(E^{\pm }\stackrel{\dot{}}{\eta }+F^{\pm }{\displaystyle {\int }_0^t\eta \text{d}s}+G^{\pm }\phi \right)v\right]}_{\partial {\Gamma }_s}=0.\end{array}$(24)

with $E^{\pm },F^{\pm },G^{\pm }$ depending on $U,c_f,c_r,ϵ,g$.

For physical field approximation, a finite dimension subspace $V_h\subset H^1\left(\bar{\Omega }\right)$ made of piecewise linear functions on a fixed mesh, characterized by element length $h$, is considered. $V_h$ is spanned by $\left({\phi }_1,\cdots ,{\phi }_{N1},{\eta }_1,\cdots ,{\eta }_{N2}\right)$ with ${\phi }_{i1\le i\le N1}$ and ${\eta }_{i1\le i\le N2}$ finite element shape functions on Ω and on ${\text{Γ}}_s$ respectively. $X$ is the coordinate vector of $\mathcal{X}\left(\phi ,\eta \right)$ relative to this basis. For the time domain approximation, a centered finite difference scheme for derivatives and a trapezoidal rule for the integral over time term are applied. The problem becomes finding $X^{\left(n\right)}\in R^N$, $N=N1+N2$, $n>1$, such that

$A_1{X}^{\left(n+1\right)}=A_2{X}^{\left(n\right)}+A_3{X}^{\left(n-1\right)}+F$(25)

where $A_1$, $A_2$ et $A_3$ are sparse matrices; $F$ a vector depending on $F^{\pm }$ and $\text{Δ}t$ accounting for non-local condition term in time Equation (22) with non-zero component corresponding to the surface edges $\partial {\text{Γ}}_s$. $X^{\left(0\right)},X^{\left(1\right)}$ are prescribed by initial conditions.

The computing process is fully automated. All the geometry operations and meshes are generated and updated automatically according to intermediate results by a batch program using Numpy and Scipy Python routines and GMSH. Due to the complexity of weak formulation terms, low-level generic assembly procedures of GETFEM++ is employed to make the assembly of the involved sparse matrices. To compute the solution of the large sparse system Equation (25), a parallel sparse direct solver (MUMPS) is used. For the post-processing handling, Matplotlib Python libraries, PARAVIEW and GMSH are utilized. A mesh convergence study is performed by reducing the characteristic size of elements, $h$, from $h={10}^{-2}$ to $h=6.25\times {10}^{-4}$. As shown in Figure 4 and Figure 5 , results converge upon the same solution as the mesh density increases. A satisfactory compromise between the accuracy of results and computing time can be achieved by choosing the value of $h=1.25\times {10}^{-3}$. This result is consistent with the order of magnitude of the wavelength of gravity-capillary waves of interest ( $\lambda \approx {10}^{-2}$ m). Indeed, the solution is curvy or even oscillates over the wavelength; this means using sufficient fine meshes or high-order piecewise polynomials to get a reliable approximation by the finite element method. A classic rule to reduce interpolation errors is to set the resolution of the wave $n_{res}=\lambda /h\approx 10$ (here $h/\lambda =8$) for a linear piecewise polynomial, but at a small wavelength, it appears to be insufficient due to numerical pollution identified in Helmholtz problems [26] . For numerical computations, values of parameters of Table 1 are used. The value of $H$ is set to ensure that surface waves progress over deep water. The value of $U$ is chosen to be less than $c_r\approx 0.23$ m·s⁻¹ so that the flow becomes a uniform stream with constant velocity $U$ at infinity [21] . The value of the half-thickness $\epsilon$ of the water surface, calculated accordingly $c_r$, is equal to 7 × 10⁻⁴ m.