3.1. Coupled potential-flow and floating thin-plate model

Following potential-flow theory, the water velocity field is defined as the gradient of a scalar potential function, ${\rm\Phi}(x,z,t)$ . The water motion is assumed to be time-harmonic at the frequency of the incident wave, and thus ${\it\Phi}(x,z,t)=\text{Re}\{{\it\phi}(x,z)\text{e}^{-\text{i}{\it\omega}t}\}$ , where the (reduced) velocity potential ${\it\phi}$ is a complex-valued function.

Thin-plate theory allows the vertical motion of the plate to be defined in terms of the displacement of its lower surface, $z={\it\eta}_{p}(x,t)$ . The motion is also assumed to be time-harmonic, and the displacement function is thus expressed as ${\it\eta}_{p}(x,t)=\text{Re}\{w(x)\text{e}^{-\text{i}{\it\omega}t}\}$ . The model does not include lateral motions of the plate. The (reduced) displacement function, $w$ , is decomposed into the orthonormal natural modes of vertical vibration of an in vacuo plate, $w_{n}(x)$ for $n=0,1,\dots ,$ with, as yet unknown, weights ${\it\xi}_{n}$ . Thus,

(3.1) $$\begin{eqnarray}w(x)=\mathop{\sum }_{n=0}^{\infty }{\it\xi}_{n}w_{n}(x),\quad \text{where}~w_{0}=\frac{1}{\sqrt{2L}}\text{ and }w_{1}=\sqrt{\frac{3}{2L^{3}}}x,\end{eqnarray}$$

are the rigid-body modes, corresponding to heave and pitch, respectively, and

(3.2a,b ) $$\begin{eqnarray}\begin{array}{@{}l@{}}w_{2n}=\displaystyle \frac{1}{\sqrt{2L}}\left(\frac{\text{cos}({\it\mu}_{2n}x)}{\text{cos}({\it\mu}_{2n}L)}+\frac{\text{cosh}({\it\mu}_{2n}x)}{\text{cosh}({\it\mu}_{2n}L)}\right)\quad \text{and}\\ w_{2n+1}=\displaystyle \frac{1}{\sqrt{2L}}\left(\frac{\text{sin}({\it\mu}_{2n+1}x)}{\text{sin}({\it\mu}_{2n+1}L)}+\frac{\text{sinh}({\it\mu}_{2n+1}x)}{\text{sinh}({\it\mu}_{2n+1}L)}\right)\end{array}\end{eqnarray}$$

for $n=1,2,\dots$ are the even and odd flexural modes, respectively, with real eigenvalues $0<{\it\mu}_{n}<{\it\mu}_{n+1}$ defined by

(3.3a,b ) $$\begin{eqnarray}\text{tan}({\it\mu}_{2n}L)+\text{tanh}({\it\mu}_{2n}L)=0\quad \text{and}\quad \text{tan}({\it\mu}_{2n+1}L)-\text{tanh}({\it\mu}_{2n+1}L)=0\end{eqnarray}$$

for $n=2,3,\dots ;{\it\mu}_{0}={\it\mu}_{1}=0$ are defined for convenience.

Linear theory is employed. Thus, the problem is posed on the equilibrium domain. The velocity potential satisfies Laplace’s equation in the equilibrium water domain, i.e.

(3.4a ) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}^{2}{\it\phi}=0\quad \text{for }-H<z<0,\text{where }\boldsymbol{{\rm\nabla}}=(\partial _{x},\partial _{z}).\end{eqnarray}$$

It also satisfies the impermeable-bed and linearised free-surface conditions, respectively,

(3.4b ) $$\begin{eqnarray}\partial _{z}{\it\phi}=\left\{\begin{array}{@{}ll@{}}0\quad & \text{on }z=-H\quad \text{and}\quad \\ \displaystyle \frac{{\it\omega}^{2}}{g}{\it\phi}\quad & \text{on }z=0\text{ for }x\notin (-L,L).\end{array}\right.\end{eqnarray}$$

In the far field the velocity potential satisfies the radiation conditions

(3.4c ) $$\begin{eqnarray}{\it\phi}\sim \frac{g}{\text{i}{\it\omega}}(A\text{e}^{\text{i}kx}+R\text{e}^{-\text{i}kx})\frac{\text{cosh}k(z+h)}{\text{cosh}(kh)}\quad \text{as}~x\rightarrow -\infty\end{eqnarray}$$

and

(3.4d ) $$\begin{eqnarray}{\it\phi}\sim \frac{gT\text{e}^{\text{i}kx}\;\text{cosh}k(z+h)}{\text{i}{\it\omega}\;\text{cosh}(kh)}\quad \text{as}~x\rightarrow \infty ,\end{eqnarray}$$

where $R$ and $T$ are, as yet unknown, complex-valued reflection and transmission amplitudes, respectively.

The velocity potential and displacement function are coupled at their equilibrium interface via kinematic and dynamic conditions. The kinematic condition equates the vertical velocity of the water particles at the water surface to the vertical velocity of the corresponding points of the plate. The dynamic condition uses Kirchoff–Love thin-plate theory, and considers the vertical motions of the plate to be forced by the pressure differential between its upper and lower surfaces. The conditions assume that the lower surface of the plate remains in contact with the water at all points and at all times during its motion. They are, respectively,

(3.4e,f ) $$\begin{eqnarray}\partial _{z}{\it\phi}=-\text{i}{\it\omega}\mathop{\sum }_{n=0}^{\infty }{\it\xi}_{n}w_{n}\quad \text{and}\quad \text{i}{\it\omega}{\it\phi}=\mathop{\sum }_{n=0}^{\infty }\{1+{\it\beta}{\it\mu}_{n}^{4}-{\it\omega}^{2}{\it\gamma}\}{\it\xi}_{n}w_{n}\end{eqnarray}$$

on $z=0$ for $x\in (-L,L)$ . Here, ${\it\beta}=Eh^{3}/\{12(1-{\it\nu}^{2}){\it\rho}g\}$ and ${\it\gamma}={\it\rho}_{p}h/{\it\rho}g$ are the scaled flexural rigidity and mass of the plate, respectively, where ${\it\nu}=0.4$ and ${\it\nu}=0.3$ are representative values of Poisson’s ratio for PP and PVC, respectively. The coupling conditions (3.4e,f ) use the zero-draught approximation (e.g. Meylan & Squire Reference Meylan and Squire1994). The Archimedean draught is used in § 3.3 to couple the surrounding water to the overwash.

The potential is expressed as the sum of scattering and radiation components, i.e.

(3.5) $$\begin{eqnarray}{\it\phi}={\it\phi}^{S}-\text{i}{\it\omega}\mathop{\sum }_{n=0}^{\infty }{\it\xi}_{n}{\it\phi}_{n}^{R}.\end{eqnarray}$$

The scattering potential ${\it\phi}^{S}$ satisfies the problem in which the plate is held in place, i.e. (3.4a ) and (3.4b ) and (3.4e ) with ${\it\xi}_{n}=0$ set for all $n$ . The radiation potential ${\it\phi}_{j}^{R}$ satisfies the problem in which the plate is forced to oscillate with unit amplitude in mode $j$ in the absence of incident forcing, i.e. (3.4a ) and (3.4b ) and (3.4e ) with ${\it\xi}_{n}={\it\delta}_{nj}$ and $A=0$ set.

The component potentials are calculated using the free-surface Green’s function, $G(x-x_{0},z)$ . It is the solution of the governing equations, forced at the point $x=x_{0}$ on the free surface in the absence of plate cover and incident wave forcing, i.e.

(3.6a,b ) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}^{2}G=0\quad \text{for}~-H<z<0,\quad \partial _{z}G=0\quad \text{on}~z=-H,\end{eqnarray}$$

(3.6c ) $$\begin{eqnarray}\partial _{z}G-{\it\omega}^{2}G/g={\it\delta}(x-x_{0})\quad \text{on}~z=0,\end{eqnarray}$$

and the radiation condition that $G$ represents outgoing waves in the far field. Meylan & Squire (Reference Meylan and Squire1994), for example, provide an explicit expression for a more general version of the Green’s function. Green’s theorem generates the integral equations

(3.7a ) $$\begin{eqnarray}{\it\phi}^{S}(x,0)=\frac{gA\text{e}^{\text{i}kx}}{\text{i}{\it\omega}}-\frac{{\it\omega}^{2}}{g}\int _{-L}^{L}G(x-x_{0},0){\it\phi}^{S}(x_{0},0)\,\text{d}x_{0}\end{eqnarray}$$

and

(3.7b ) $$\begin{eqnarray}{\it\phi}_{n}^{R}(x,0)=-\int _{-L}^{L}G(x-x_{0},0)\left(\frac{{\it\omega}^{2}}{g}{\it\phi}_{n}^{R}(x_{0})-w_{n}(x_{0})\right)\,\text{d}x_{0}\quad \text{for}~n=0,1,\dots\end{eqnarray}$$

for $x\in (-L,L)$ . The potentials are calculated numerically on the wetted surface of the plate, as the solutions of (3.7a ) and (3.7b ). The radiation potentials, ${\it\phi}_{n}^{R}$ , are calculated for $n=0,\dots ,N$ , where $N$ is finite and sufficiently large that the truncated version of (3.5) accurately approximates the full-linear solution.

Applying (3.4f ) to the truncated potential and taking inner products with respect to $w_{j}$ for $j=0,\dots ,N$ results in the system

(3.8) $$\begin{eqnarray}(\unicode[STIX]{x1D646}+\unicode[STIX]{x1D63E}-{\it\omega}^{2}\unicode[STIX]{x1D648}-{\it\omega}^{2}\unicode[STIX]{x1D63C}({\it\omega})-\text{i}{\it\omega}\unicode[STIX]{x1D63D}({\it\omega})){\bf\xi}=\boldsymbol{f}({\it\omega}).\end{eqnarray}$$

Here, $\unicode[STIX]{x1D646}$ , $\unicode[STIX]{x1D63E}$ and $\unicode[STIX]{x1D648}$ are diagonal stiffness, hydrostatic-restoring and mass matrices, respectively, $\unicode[STIX]{x1D63C}$ and $\unicode[STIX]{x1D63D}$ are added mass and damping matrices, respectively, $\boldsymbol{f}$ is the forcing vector and ${\bf\xi}$ is a vector containing the modal weights. Their entries are defined as

(3.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D612}_{jj}={\it\beta}{\it\mu}_{j-1}^{4},\quad \unicode[STIX]{x1D60A}_{jj}=1\quad \text{and}\quad \unicode[STIX]{x1D614}_{jj}={\it\gamma}\quad \text{for}~j=1,\dots ,N+1, & \displaystyle\end{eqnarray}$$

(3.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\omega}^{2}\unicode[STIX]{x1D608}_{ij}+\text{i}{\it\omega}\unicode[STIX]{x1D609}_{ij}=\frac{{\it\omega}^{2}}{g}\int _{-L}^{L}{\it\phi}_{j-1}^{R}(x,0)w_{i-1}(x)\,\text{d}x\quad \text{for}~i,j=1,\dots ,N+1, & \displaystyle\end{eqnarray}$$

(3.9c,d ) $$\begin{eqnarray}\boldsymbol{f}_{j}=\frac{\text{i}{\it\omega}}{g}\int _{-L}^{L}{\it\phi}^{S}(x,0)w_{j-1}(x)\,\text{d}x\quad \text{and}\quad {\bf\xi}_{j}={\it\xi}_{j-1}\quad \text{for}~j=1,\dots ,N+1.\end{eqnarray}$$

System (3.5) is solved for the vector of modal weights, ${\bf\xi}$ . The reflection and transmission amplitudes are subsequently calculated as $R=AI_{+}$ and $T=A(1+I_{-})$ , where

(3.10) $$\begin{eqnarray}I_{\pm }=\frac{-\text{i}\;\text{cosh}^{2}(kH)}{\text{cosh}(kH)\text{sinh}(kH)+kH}\int _{-L}^{L}\text{e}^{\pm \text{i}kx}\left(\frac{{\it\omega}^{2}}{g}{\it\phi}(x,0)-\partial _{z}{\it\phi}(x,0)\right)\,\text{d}x.\end{eqnarray}$$

3.2. Nonlinear shallow-water equations

It is assumed that (i) the vertical length scale of the overwash is much smaller than its horizontal length scale, (ii) the water particles’ vertical accelerations are much smaller than gravitational acceleration and, thus, the water pressure is approximately hydrostatic and (iii) the plate pitches at a small angle only. These assumptions are consistent with experimental observations. Assumption (ii) implies that the force of the plate’s motion on the overwash is negligible in comparison to gravitational force. Assumption (iii) implies that the gravitational force acts approximately perpendicularly to the plate’s upper surface. Further, turbulence and wave breaking are neglected as a simplifying assumption. Therefore, the nonlinear shallow-water equations model the overwash, i.e. the hyperbolic partial differential system

(3.11a ) $$\begin{eqnarray}\partial _{t}\boldsymbol{q}+\partial _{x}f(\boldsymbol{q})=0\quad \text{for}~x\in (-L,L),\end{eqnarray}$$

(3.11b,c ) $$\begin{eqnarray}\text{where}~\boldsymbol{q}=[q,qu]^{\text{T}}\quad \text{and}\quad f(\boldsymbol{q})=\left[qu,qu^{2}+{\textstyle \frac{1}{2}}gq^{2}\right]^{\text{T}}.\end{eqnarray}$$

Here, $u(x,t)$ is its depth averaged horizontal velocity and $q(x,t)$ is the overwash depth, which is defined as the height of the surface elevation above the plate’s upper surface.

The numerical scheme outlined by Kurganov & Tadmor (Reference Kurganov and Tadmor2000) is used to approximate spatial derivatives in system (3.11). It is a second-order-accurate finite-volume method. It is able to capture bores, which manifest as discontinuities in the solution. In this numerical scheme, horizontal space is discretised into $M+1$ uniformly spaced cells centred at the points $\{x_{0}=-L,x_{1}=-L+{\rm\Delta}x,\dots ,x_{M}=L\}$ , and the notation $\boldsymbol{q}_{j}(t)\equiv \boldsymbol{q}(x_{j},t)$ is employed. The spatial derivative in (3.11) is approximated as

(3.12) $$\begin{eqnarray}\partial _{x}f(\boldsymbol{q}_{j}(t))\approx -\mathscr{L}(\boldsymbol{q}_{j}(t))=\frac{\boldsymbol{B}_{j+1/2}(t)-\boldsymbol{B}_{j-1/2}(t)}{{\rm\Delta}x},\end{eqnarray}$$

where $\boldsymbol{B}_{j\pm 1/2}$ are fluxes on the $j$ th cell’s left ( $-$ ) and right ( $+$ ) boundaries, defined via

(3.13) $$\begin{eqnarray}\boldsymbol{B}_{j+1/2}(t)=\frac{a_{j+1/2}^{+}\,f(\boldsymbol{q}_{j+1/2}^{-})-a_{j+1/2}^{-}\,f(\boldsymbol{q}_{j+1/2}^{+})+a_{j+1/2}^{+}a_{j+1/2}^{-}(\boldsymbol{q}_{j+1/2}^{+}-\boldsymbol{q}_{j+1/2}^{+})}{a_{j+1/2}^{+}-a_{j+1/2}^{-}},\end{eqnarray}$$

where

(3.14a,b ) $$\begin{eqnarray}a_{j+1/2}^{\pm }=\pm \max \{\pm {\it\kappa}_{\pm }(\boldsymbol{q}_{j+1/2}^{-}),\pm {\it\kappa}_{\pm }(\boldsymbol{q}_{j+1/2}^{+}),0\}\quad \text{and}\quad {\it\kappa}_{\pm }(\boldsymbol{q})=u\pm \sqrt{gq},\end{eqnarray}$$

i.e. ${\it\kappa}_{\pm }(\boldsymbol{q})$ are the largest ( $+$ ) and smallest ( $-$ ) eigenvalues of the Jacobian of $f(\boldsymbol{q})$ .

The $\text{minmod}$ limiter presented in Kurganov & Tadmor (Reference Kurganov and Tadmor2000) is used to prevent unnatural oscillations from occurring either side of a shock. It is defined as

(3.15) $$\begin{eqnarray}\text{minmod}\{\bullet _{1},\bullet _{2},\dots \}=\left\{\begin{array}{@{}ll@{}}\text{min}_{j}\{\bullet _{j}\}\quad & \text{if}~\bullet _{j}>0\text{ for all }j,\\ \text{max}_{j}\{\bullet _{j}\}\quad & \text{if}~\bullet _{j}<0\text{ for all }j,\end{array}\right.\end{eqnarray}$$

and zero otherwise. It determines

(3.16) $$\begin{eqnarray}\boldsymbol{q}_{x,j+1/2}=\text{minmod}\left\{{\it\theta}\frac{\boldsymbol{q}_{j+1}-\boldsymbol{q}_{j}}{{\rm\Delta}x},\frac{\boldsymbol{q}_{j+1}-\boldsymbol{q}_{j-1}}{2{\rm\Delta}x},{\it\theta}\frac{\boldsymbol{q}_{j}-\boldsymbol{q}_{j-1}}{{\rm\Delta}x}\right\},\end{eqnarray}$$

where ${\it\theta}=1.5$ , which, in turn, determines the half-space values

(3.17a,b ) $$\begin{eqnarray}\boldsymbol{q}_{j+1/2}^{+}=\boldsymbol{q}_{j+1}-{\textstyle \frac{1}{2}}\boldsymbol{q}_{x,j+1/2}\quad \text{and}\quad \boldsymbol{q}_{j+1/2}^{-}=\boldsymbol{q}_{j}+{\textstyle \frac{1}{2}}\boldsymbol{q}_{x,j+1/2}\end{eqnarray}$$

used in (3.13) and (3.14).

The total variation diminishing Runge–Kutta 2 scheme is used to time step. It suppresses unnatural extrema as the solution evolves. The scheme is expressed as

(3.18) $$\begin{eqnarray}2\boldsymbol{q}_{j}(t+{\rm\Delta}t)=2\boldsymbol{q}_{j}(t)+{\rm\Delta}t\mathscr{L}(\boldsymbol{q}_{j}(t))+{\rm\Delta}t\mathscr{L}(\boldsymbol{q}_{j}(t)+{\rm\Delta}t\mathscr{L}(\boldsymbol{q}_{j}(t))),\end{eqnarray}$$

where ${\rm\Delta}t>0$ is a time step satisfying the Courant–Friedrichs–Lewy (CFL) condition (Gottlieb & Shu Reference Gottlieb and Shu1998).

3.3. Forcing the shallow-water equations

The coupled potential-flow/thin-plate model provides steady forcing for the shallow-water equations. It defines two conditions at each end of the interval occupied by the shallow water. This is a sufficient number of boundary conditions for the shallow-water equations. It allows for bores to be generated at the ends of the plate and also permits water to enter and exit the interval.

The overwash depth at the interface with the surrounding water is equated to the height of the free-surface elevation, ${\it\eta}(x,t)={\it\rho}\partial _{t}{\it\Phi}(x,0,t)/g$ , above the upper surface of the plate. It is set to zero if the height is negative. Thus,

(3.19) $$\begin{eqnarray}q(\pm L,t)=\max \{{\it\eta}(\pm L,t)-{\it\eta}_{p}(\pm L,t)+C,0\},\quad \text{where}~C=h(1-{\it\rho}_{p}/{\it\rho})\end{eqnarray}$$

is the plate’s equilibrium freeboard, which is calculated via Archimedes’ principle. Numerical tests showed evanescent wave motions have minor impact on the results presented in § 4. Thus, the free-surface elevations are approximated as

(3.20a,b ) $$\begin{eqnarray}{\it\eta}(-L,t)\approx \text{Re}\{(A\text{e}^{-\text{i}kL}+R\text{e}^{\text{i}kL})\text{e}^{-\text{i}{\it\omega}t}\}\quad \text{and}\quad {\it\eta}(L,t)\approx \text{Re}\{T\text{e}^{\text{i}kL-\text{i}{\it\omega}t}\}.\end{eqnarray}$$

Similarly, the horizontal velocity of the overwash at the interface with the surrounding water is equated to the horizontal velocity of the free-surface water particles adjacent to the plate edges, $U(x,t)=\partial _{x}{\it\Phi}(x,0,t)$ . It is set to zero if the free surface is not above the plate’s ends. Thus,

(3.21) $$\begin{eqnarray}u(\pm L,t)=\left\{\begin{array}{@{}ll@{}}U(\pm L,t)\quad & \text{if}~{\it\eta}(\pm L,t)\geqslant -{\it\eta}_{p}(\pm L,t)+C,\\ 0\quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$

Initially, the shallow water is set to a uniform depth of $q=1~{\rm\mu}\text{m}$ and to be stationary, i.e. $u=0$ . For a prescribed incident wave and plate, the shallow-water equations are run to a quasisteady state, i.e. until the overwash depth profile is approximately time-harmonic. For the results shown in § 4, this occurred after approximately 40 wave periods.