3.1. Equilibrium meniscus profile

The equilibrium meniscus profile $\eta = \eta _0(x)$ under the effect of both surface tension and gravity is determined by the Young–Laplace balance between hydrostatics and curvature,

(3.2) \begin{equation} \rho g \eta _0 = \sigma \kappa _0, \end{equation}

which can be rewritten as an equation for the meniscus curvature, i.e.

(3.3) \begin{equation} \kappa _0 = \frac {\eta _0''}{\left ( 1 + (\eta _0')^2 \right )^{3/2}} = \frac {\eta _0}{a^2}, \end{equation}

where $a$ defines a capillary length,

(3.4) \begin{equation} a = \sqrt {\sigma /\rho g}. \end{equation}

Here $\eta _0$ depends only on $x$ , so derivatives with respect to $x$ are strictly total derivatives: $\partial _x \eta _0 = \eta _0'$ and $\partial _x^2 \eta _0 = \eta _0''$ .

Figure 2.

Schematic for two differently oriented menisci: (a) a left-going meniscus, extending from the contact point to $x \to -\infty$ ; (b) a right-going meniscus, extending from the contact point to $x \to +\infty$ .

Integrating (3.3) once yields an implicit form of the meniscus slope $\eta _0'$ :

(3.5) \begin{equation} \frac {1}{ \sqrt { 1 + ( \eta _0' )^2 } } = 1 - \frac { \eta ^2_0 }{ 2 a^2 }. \end{equation}

This matches the flat boundary condition at the far end of the profile ( $\eta _0 =0$ and $\eta _0'=0$ as $|x|\rightarrow \infty$ ), and at the other end, the capillary rise at the contact points (denoted by $ \eta _0 = h$ ) leads to an equilibrium contact angle $ \theta$ (measured from the barrier’s vertical face to the tangent of the interface; see figure 1) given by

(3.6) \begin{equation} \sin \theta = 1 - \frac { h^2 }{ 2 a^2 }. \end{equation}

The capillary rise height $h$ can be either positive or negative to form a positive or negative meniscus, respectively.

Equation (3.5) can be reorganised as an explicit form of the meniscus slope, i.e.

(3.7) \begin{equation} - \eta _0' = \pm \frac {\eta _0}{a} \frac {\sqrt {1 - \eta _0^2/4a^2}}{1 - \eta _0^2/2a^2} , \end{equation}

where ‘ $\pm$ ’ distinguishes between the two kinds of meniscus: ‘ $-$ ’ for the left-going meniscus (figure 2 a), ‘ $+$ ’ for the right-going meniscus (figure 2 b).

Integrating (3.7) subsequently gives the implicit form for the equilibrium meniscus profile $\eta _0(x)$ (see Appendix A), i.e.

(3.8) \begin{equation} F(\eta _0) - F(h) = \pm (x-x_c), \quad F(\eta _0) \equiv a \cosh ^{-1} \left ( \frac { 2 a }{ \eta _0 } \right ) - \sqrt {4 a^2 - \eta ^2_0 }, \end{equation}

which matches the capillary rise height $\eta _0 = h$ at the contact points $x = x_c$ , and the asymptotic limit $\eta _0 \rightarrow 0$ at the limit of $x \rightarrow \pm \infty$ . For the barrier in figure 1, $x_c = \pm w/2$ . This expression agrees with the classical profile equation given in standard references (Landau & Lifshitz Reference Landau and Lifshitz1987; Gennes, Brochard-Wyart & Quéré Reference Gennes, Brochard-Wyart and Quéré2004), where the positive branch is typically presented.

3.2. Linearised free-surface boundary conditions on the meniscus

Recall the full nonlinear forms of the kinematic and dynamic boundary conditions at the free surface, (2.5) and (2.12):

(3.9a,b)

They serve as the fundamental boundary conditions to derive the free-surface boundary conditions for the linear waves on the meniscus.

Substituting (3.1) into the exact free-surface boundary conditions (3.9) leads to

(3.10a,b)

Here the surface normal $\hat{\boldsymbol{n}}$ and curvature $\kappa$ are now evaluated at $y = \eta _0 + \epsilon \eta _1$ with

(3.10c) \begin{equation} \kappa = -\boldsymbol{\nabla }\boldsymbol{\cdot }\hat{\boldsymbol{n}} \quad \text{and} \quad \hat{\boldsymbol{n}} = \frac {-\partial _x (\eta _0 + \epsilon \eta _1) \hat{\boldsymbol{e}}_x + \hat{\boldsymbol{e}}_y}{\sqrt {1+(\partial _x(\eta _0 + \epsilon \eta _1))^2}}. \end{equation}

Since $\eta _0$ is independent of $t$ and $\epsilon \eta _1$ is small, we linearise (3.10) about $y = \eta _0$ . We take a corresponding expansion of the curvature and normal, i.e.

(3.11) \begin{equation} \kappa = \kappa _0 + \epsilon \kappa _1 + O(\epsilon ^2), \quad \hat{\boldsymbol{n}} = \hat{\boldsymbol{n}}_0 + \epsilon \hat{\boldsymbol{n}}_1 + O(\epsilon ^2), \end{equation}

where $\kappa _1$ and $\hat{\boldsymbol{n}}_1$ are the first-order corrections from the equilibrium $\kappa _0$ and $\hat{\boldsymbol{n}}_0$ , and

(3.12) \begin{equation} \kappa _0 = -\boldsymbol{\nabla }\boldsymbol{\cdot }\hat{\boldsymbol{n}}_0, \quad \kappa _1 = -\boldsymbol{\nabla }\boldsymbol{\cdot }\hat{\boldsymbol{n}}_1, \end{equation}

where $\kappa _0$ is given in (3.3).

Considering the first-order terms $O(\epsilon )$ , (3.10) becomes

(3.13a,b)

where, following from (3.10c ), it has

(3.13c) \begin{align} & \hat{\boldsymbol{n}}_0 = \frac {-\eta _0' \hat{\boldsymbol{e}}_x + \hat{\boldsymbol{e}}_y}{\sqrt {1+(\eta _0')^2}}, \quad \hat{\boldsymbol{n}}_1 = -\frac {\partial _x\eta _1(\hat{\boldsymbol{e}}_x+\eta _0'\hat{\boldsymbol{e}}_y)}{\left (1+(\eta _0')^2\right )^{3/2}}, \\[-12pt]\nonumber \end{align}

(3.13d) \begin{align} & \kappa _1 = -\boldsymbol{\nabla }\boldsymbol{\cdot }\hat{\boldsymbol{n}}_1 = \frac {\partial }{\partial x}\left (\frac {\partial _x \eta _1}{\left (1+(\eta _0')^2\right )^{3/2}}\right )\!. \end{align}

Equations (3.13) constitute the linearised kinematic and dynamic boundary conditions for small-amplitude capillary–gravity waves, explicitly assuming that the perturbation $\eta _1(x,t)$ is a vertical displacement from the equilibrium meniscus profile $\eta _0(x)$ . In (3.13), the $\eta _0(x)$ and its derivatives $\eta _0'(x)$ and $\eta _0''(x)$ terms arise from the local profile, slope and curvature of the meniscus and are $x$ dependent.

We now implement this formulation of the linearised kinematic and dynamic conditions (3.13) on the equilibrium meniscus $y=\eta _0(x)$ specified by (3.8). By carrying out the differentiation in (3.13c ) and replacing the meniscus curvature and slope terms with expressions in terms of the meniscus profile $\eta _0$ itself (see Appendix B), we reduce the boundary conditions to

(3.14a,b)

where $a = \sqrt {\sigma /\rho g}$ is the capillary length defined in (3.4).

Equations (3.14) constitute the linearised kinematic and dynamic boundary conditions we have derived for capillary–gravity waves $\eta _1(x,t)$ propagating on a single-valued equilibrium meniscus surface $\eta _0(x)$ given by (3.8). On the right-hand side of (3.14b), the first two terms are from the surface tension force and the last term is from the gravity force. Recall that ‘ $\pm$ ’ distinguishes between the two kinds of meniscus (see figure 2): ‘ $+$ ’ for the right-going meniscus, ‘ $-$ ’ for the left-going meniscus. The conditions are expressed in the form (3.14) for ease of implementation in finite-element simulations.

3.3. Far-field form

At the far field $x\rightarrow \pm \infty$ , where the unperturbed surface is flat with $\eta _0 = 0$ , (3.14) reduce to the classical linearised boundary conditions for capillary–gravity waves on a flat surface (Fetter & Walecka Reference Fetter and Walecka2003):

(3.15a) \begin{align} \partial _y \phi _1 &= \partial _t \eta _1, \\[-12pt]\nonumber \end{align}

(3.15b) \begin{align} \rho \partial _t \phi _1 &= \sigma \partial _x^2\eta _1 - \rho g \eta _1. \end{align}

At the free surface of the fluid of depth $H$ , the incident surface wave of angular frequency $\omega$ and wavenumber $k$ has the form of (Fetter & Walecka Reference Fetter and Walecka2003)

(3.16a) \begin{align} \phi _1(x,y,t) &= \phi _A \frac {\cosh (k (y + H))}{\sinh (k H)} \mathrm{e}^{\mathrm{i} (k x -\omega t)} + \text{c.c.}, \\[-12pt]\nonumber \end{align}

(3.16b) \begin{align} \eta _1(x,t) &= \eta _A \mathrm{e}^{\,\mathrm{i} (k x -\omega t)} + \text{c.c.}, \end{align}

where, following from (3.15a ), amplitudes $\phi _A$ and $\eta _A$ have the relation

(3.17) \begin{equation} \phi _A = -\mathrm{i} \frac {\omega }{k} \eta _A, \end{equation}

and, following from (3.15b ), $\omega$ and $k$ satisfy the dispersion relation

(3.18) \begin{equation} \omega ^2 = \left(gk + \frac {\sigma }{\rho }k^3\right)\tanh (kH), \end{equation}

or equivalently,

(3.19) \begin{equation} \omega ^2 = \frac {\sigma }{\rho }k^3\,(1 + B)\tanh (kH), \end{equation}

where

(3.20) \begin{equation} B = \frac {\rho g}{\sigma k^2} \end{equation}

is the Bond number, characterising the relative effect between gravity and surface tension.

4.1. Steady fields

Specifically, we consider time-harmonic fields with

(4.1a) \begin{align} \phi _1(x,y,t) = \psi (x,y) \mathrm{e}^{-\mathrm{i}\omega t} + \text{c.c.}, \\[-12pt]\nonumber \end{align}

(4.1b) \begin{align} \eta _1(x,t) = \zeta (x) \mathrm{e}^{-\mathrm{i}\omega t} + \text{c.c.}, \end{align}

where $\psi (x,y)$ and $\zeta (x)$ define the steady problem of the velocity potential and the surface elevation. The steady far-field surface elevation takes the form

(4.2a) \begin{align} \zeta &= \eta _A (\mathrm{e}^{\mathrm{i} k x} + R \mathrm{e}^{-\mathrm{i} kx}) \quad\text{at } x \rightarrow -\infty , \\[-12pt]\nonumber \end{align}

(4.2b) \begin{align} \zeta &= \eta _A T \mathrm{e}^{\mathrm{i} k x}\quad\text{at } x \rightarrow +\infty , \end{align}

and the steady velocity potential takes the form

(4.3a) \begin{align} \psi &= \phi _A \frac {\cosh (k (y + H))}{\sinh (k H)} ( \mathrm{e}^{\mathrm{i} k x } + R \mathrm{e}^{-\mathrm{i} kx})\quad\text{at } x \rightarrow -\infty , \\[-12pt]\nonumber \end{align}

(4.3b) \begin{align} \psi &= \phi _A \frac {\cosh (k (y + H))}{\sinh (k H)} T \mathrm{e}^{\mathrm{i} k x }\quad\text{at } x \rightarrow +\infty , \end{align}

where the amplitude and the dispersion relation are given by (3.17)–(3.20).

For the steady fields, the free-surface boundary conditions (3.14) reduce to

(4.4a,b)

where the equilibrium meniscus profile $\eta _0$ is given by (3.8), $a=\sqrt {\sigma /\rho g}$ is the capillary length and the ‘ $\pm$ ’ distinguishes between the two kinds of meniscus: ‘ $-$ ’ is for the left-going meniscus (figure 2 a), ‘ $+$ ’ is for the right-going meniscus (figure 2 b).

4.2. Coupled boundary-value problems

Following the approach in Zhang & Thiessen (Reference Zhang and Thiessen2013) and Liu & Zhang (Reference Liu and Zhang2025), we now form the problem as solving two coupled boundary-value problems, the steady potential $\psi (x,y)$ and the steady surface elevation $\zeta (x)$ , which are coupled through the free-surface boundary conditions, where one acts as the source of the other.

Specifically, for the $\psi (x,y)$ problem, we solve the Laplace equation

(4.5a) \begin{align} & \boldsymbol{\nabla} ^2 \psi = 0 \quad \text{in the fluid domain} \ -H \leqslant y \leqslant \eta _0(x) \end{align}

together with boundary conditions at solid surfaces, far fields and the free-surface kinematic boundary condition where $\zeta$ acts as a source, i.e.

(4.5b) \begin{align} & \hat{\boldsymbol{n}}_0 \boldsymbol{\cdot }\boldsymbol{\nabla }\psi = -\mathrm{i} \omega \left ( 1 - \frac {\eta ^2_0}{2 a^2} \right ) \zeta \quad \text{at } y=\eta _0, \\[-12pt]\nonumber \end{align}

(4.5c) \begin{align} & \hat{\boldsymbol{n}}_{\mathrm{b}} \boldsymbol{\cdot }\boldsymbol{\nabla }\psi = 0 \quad \text{at solid surfaces with } \hat{\boldsymbol{n}}_{\mathrm{b}} {\text{the outward normal to the solid boundary,}} \\[-12pt]\nonumber \end{align}

(4.5d) \begin{align} & \psi = \phi _A \frac {\cosh (k (y + H))}{\sinh (k H)} ( \mathrm{e}^{\mathrm{i} k x } + R \mathrm{e}^{-\mathrm{i} kx}) \quad \text{at } x \rightarrow -\infty , \\[-12pt]\nonumber \end{align}

(4.5e) \begin{align} & \psi = \phi _A \frac {\cosh (k (y + H))}{\sinh (k H)} T \mathrm{e}^{\mathrm{i} k x } \quad \text{at } x \rightarrow +\infty . \end{align}

For the $\zeta (x)$ problem, defined outside the barrier span, $x \in (-\infty ,-w/2]\cup [w/2,\infty )$ , we solve the dynamic boundary equation where $\psi (x,\eta _0(x))$ acts as a source, i.e.

(4.6a) \begin{align} & \left ( 1 - \frac {\eta ^2_0}{2 a^2} \right ) ^3 \frac {d^2\zeta }{\text{d}x^2} \pm \frac {3 \eta _0^2}{a^3} \left ( 1 - \frac {\eta ^2_0}{2 a^2} \right ) \sqrt {1 - \frac {\eta _0^2}{4a^2}}\; \frac{\text{d}\zeta }{\text{d}x} - a^{-2} \zeta = -\mathrm{i} \omega \frac {\rho }{\sigma } \psi (x,\eta _0(x)), \end{align}

together with boundary conditions at the contact points and far fields:

(4.6b) \begin{align} & \zeta = 0 \quad \text{at contact points } x = \pm w/2, \\[-12pt]\nonumber \end{align}

(4.6c) \begin{align} &\zeta = \eta _A (\mathrm{e}^{\mathrm{i} k x} + R \mathrm{e}^{-\mathrm{i} kx}) \quad \text{at } x \rightarrow -\infty , \\[-12pt]\nonumber \end{align}

(4.6d) \begin{align} &\zeta = \eta _A T \mathrm{e}^{\mathrm{i} k x} \quad \text{at } x \rightarrow +\infty . \end{align}

Here the coefficients in (4.6a ) are functions of the meniscus profile $\eta _0(x)$ given by (3.8) and depend on the spatial variable $x$ .

4.3. Truncated domain

For numerical implementation using finite-element methods, we truncate the $x$ direction to $x\in [-L, L]$ , where $L$ is large compared with the wavelength so that radiation boundary conditions at $x=\pm L$ approximate the far-field behaviour and allow outgoing waves to exit.

Specifically, for $\psi (x,y)$ , we impose (Zhang & Thiessen Reference Zhang and Thiessen2013)

(4.7a) \begin{align} &\partial _x \psi + \mathrm{i} k \psi = 2 \mathrm{i} k \phi _A \frac {\cosh (k(y + H))}{\sinh (kH)} \mathrm{e}^{ - \mathrm{i} k L } \quad \text{at } x=-L, \\[-12pt]\nonumber \end{align}

(4.7b) \begin{align} &\partial _x \psi - \mathrm{i} k \psi = 0 \quad \text{at } x=L, \end{align}

which replace the far-field conditions (4.5d ) and (4.5e ), and for $\zeta (x)$ , we impose (Zhang & Thiessen Reference Zhang and Thiessen2013)

(4.8a) \begin{align} &\partial _x \zeta + \mathrm{i} k \zeta = 2 \mathrm{i} k \eta _A \mathrm{e}^{ - \mathrm{i} k L } \quad \text{at } x=-L, \\[-12pt]\nonumber \end{align}

(4.8b) \begin{align} &\partial _x \zeta - \mathrm{i} k \zeta = 0 \quad \text{at } x=L, \end{align}

which replace (4.6c ) and (4.6d ).

We then extract $R$ and $T$ by comparing the computed $\zeta$ at $x=\pm L$ with the known incident wave amplitude $\eta _A$ :

(4.9) \begin{align} R &= \left [ \frac {\zeta (x)\mathrm{e}^{\mathrm{i} k x}}{\eta _A} - \mathrm{e}^{\mathrm{i} 2 k x}\right ]_{x=-L}, \\[-12pt]\nonumber \end{align}

(4.10) \begin{align} T &= \left [ \frac {\zeta (x)\mathrm{e}^{- \mathrm{i} k x}}{\eta _A} \right ]_{x=L}. \end{align}

4.4. Numerical set-up and convergence tests

We numerically solve the steady fields using finite-element methods by implementing the two sets of coupled boundary-value equations, (4.5) and (4.6), into commercial software COMSOL $\textrm {Multiphysics}^{\circledR }$ . Instead of using its built-in models, we use its ‘Laplace Equation’ and ‘Coefficient Form Boundary PDE’ solvers within a ‘Blank Model’, where the coupled equations are implemented manually.

We take water with density $\rho = 997\,\mathrm{kg\,m}^{-3}$ , surface tension $\sigma = 71.99\,\mathrm{mN\,m}^{-1}$ , gravitational acceleration $g=9.8$ ms $^{-2}$ and fluid depth $H = 9.2\,\mathrm{cm}$ . The barrier thickness is $w=3.18\,\mathrm{mm}$ . A wave of frequency $f=15\,\mathrm{Hz}$ implies an angular frequency $\omega = 2\pi f = 94.2\,\mathrm{rad\,s}^{-1}$ . The corresponding wavenumber is $k = 408.71\,\mathrm{rad\,m}^{-1}$ , giving a Bond number $B \approx 0.813$ and wavelength $\lambda \approx 15.4\,\mathrm{mm}$ . The capillary length is $a=2.7\,\mathrm{mm}$ . The amplitude of the incident wave is set to $\eta _A = 0.01 a$ , though this value serves only as a scaling parameter in the linearised problem for reflection/transmission calculated from (4.9). These parameters match the experiment by Wang et al. (Reference Wang, Liu and Zhang2025). We vary $h$ from $-\sqrt {2}a$ to $\sqrt {2}a$ , corresponding to equilibrium contact angles spanning nearly $180^\circ$ to nearly $0^\circ$ (figure 5), as follows from (3.6).

We take the domain size as $L = 10 \lambda$ . Convergence tests of $L$ were performed by doubling $L$ for typical cases, which revealed a change of transmission $|T|$ less than 0.3 %. We set up the mesh by specifying a maximum mesh size $\lambda /25$ and a minimum mesh size $\lambda /10\,000$ , which automatically generates finer elements near the bottom of the barrier (figure 3). The output mesh size is about 0.7 mm away from the meniscus and about 0.1 mm near the bottom of the barrier in both $x$ and $y$ directions. Convergence tests with finer and coarser meshes (doubled and halved mesh densities) indicated that changes in the computed transmission coefficient $|T|$ remained below 1 %, demonstrating sufficient convergence.

Figure 3.

Schematic of the mesh condition near the barrier used in the numerical modelling calculations in the finite-element solver. The coordinate axes are in the space of $kx$ and $ky$ , where $k=2\pi /\lambda$ . The truncated window is zoomed in with a range from $-1.5\lambda$ to $1.5\lambda$ for visualisation of the near-field finer meshes.

5.1. Validation against theoretical formulas in limiting cases by Liu & Zhang (Reference Liu and Zhang2025)

We start by examining the limiting case of transmission through an infinitesimal barrier. The results serve to validate our model by comparison with an analytical formula recently reported in Liu & Zhang (Reference Liu and Zhang2025) for a flat surface without a meniscus effect under the deep-water assumption (namely, capillary rise $h=0$ , equilibrium contact angle $\theta = 90^\circ$ and fluid depth $H\gg \lambda$ ). The infinitesimal barrier corresponds to our modelling in the limit of zero barrier width $w=0$ . Numerically, we approximate such an infinitesimal barrier by imposing a zero-velocity condition at a single mesh point located at $(x,y)=(0,h)$ , where both the horizontal and vertical components of the fluid velocity are set to zero, resembling a pinned contact point at a fixed infinitesimal barrier (see top panels in figure 4). Mesh refinement tests confirm that the computed transmission converges as the local mesh is refined near this point. We run our numerical calculations for wave frequency $f=15$ Hz, corresponding to Bond number $B=0.813$ in (3.20), and the capillary rise $h$ ranges from $-\sqrt {2}a$ to nearly $\sqrt {2}a$ (negative to positive meniscus).

The computed results (figure 4) show that the transmission as a function of capillary rise $h$ varies by approximately 0.095 per 1 mm increase in $h$ . The results are shown for contact angles down to $4.89^\circ$ , approaching the limit of a vanishingly thin fluid layer beneath the barrier, which would occur at $0^\circ$ . At the flat-surface limit corresponding to a contact angle of $90^\circ$ (i.e. capillary rise height $h=0$ ), the results agree with the theoretical value, providing validation of the model against the closed-form formulas in Liu & Zhang (Reference Liu and Zhang2025). These formulas follow from (7.9) therein for the case of pinned contact points, where the reflection and transmission coefficients, $T$ and $R$ , are given as a function of the Bond number $B$ :

(5.1) \begin{align} T = \frac {X}{X+\mathrm{i}}, \,\, R = -\frac {\mathrm{i}}{X+\mathrm{i}}, \,\, \text{and} \,\, X = \frac {1}{\pi }\left [ \frac {3+2B}{\sqrt {3+4B}}\tan ^{-1}\big (\sqrt {3+4B}\big ) + \frac {1}{2}\ln (1+B) \right ]\!. \end{align}

Figure 4.

Transmission coefficient $|T|$ as a function of height $h$ for an infinitesimally thin barrier in the presence of a meniscus. The top panels illustrate the free-surface profiles and typical barrier heights corresponding to equilibrium contact angles $\theta = 180,\,90,\,0^\circ$ , where $a=2.7$ mm is the capillary length given in (3.4). The bottom plot compares numerical results from a zero-width barrier with the analytical solution in the limiting case according to (5.1). The results are for a 15 Hz wave; see table 1 for the associated data.

Figure 5.

Transmission coefficient $|T|$ versus barrier height $h$ for a 15 Hz wave interacting with a 3.18 mm-wide, surface-piercing plate barrier, comparing model results and experiments using superhydrophobic (yellow), hydrophobic (blue) and hydrophilic (orange) coated barriers by Wang et al. (Reference Wang, Liu and Zhang2025). The four insets at the top illustrate the meniscus profiles and barrier heights corresponding to equilibrium contact angles $\theta = 180,\,90,\,0\,\textrm {and}\,{-}90^\circ$ .

5.2. Numerical results and comparisons with measurements by Wang et al. (Reference Wang, Liu and Zhang2025)

In the experiment by Wang et al. (Reference Wang, Liu and Zhang2025), wave transmission $|T|$ is measured as a function of barrier height $h$ (see figure 5). The barrier is hydrophobically coated and initially positioned at $h = -1.1$ mm so that the pinned contact point forms an equilibrium contact angle $\theta = 113.6^\circ$ . From this configuration, raising the barrier forces the meniscus to rise along with the edge of the barrier, reducing $\theta$ below $113.6^\circ$ . As the meniscus is pulled higher, it may overturn (corresponding to $\theta \lt 0^\circ$ ) and eventually break once its curvature becomes too large to sustain (at around $h=4.8$ mm). The incident wave in the experiment is a time-harmonic surface disturbance generated by a paddle wavemaker located at a distance of $10\lambda$ from the barrier and driven by horizontal oscillation. The oscillation frequency and amplitude are adjustable to control the wave parameters. The surface-elevation amplitude was set to be of the order of $0.01\lambda$ , where the waves remained well within the linear regime.

Figure 5 compares experimentally measured transmission coefficients $|T|$ (coloured solid lines) by Wang et al. (Reference Wang, Liu and Zhang2025) with our numerical predictions (black solid line) for $0^\circ \leqslant \theta \leqslant 180^\circ$ , corresponding to the single-valued equilibrium meniscus regime. Both experimental and numerical results consider a 15 Hz capillary–gravity wave scattered by a 3.18 mm-wide surface-piercing plate barrier in a 9.2 cm depth tank. Initially, the barrier is positioned so that the equilibrium contact angle is $\theta = 180^\circ$ , and raising the barrier reduces $\theta$ from hydrophobic ( $\gt 90^\circ$ ) to $90^\circ$ (flat meniscus) and then toward $0^\circ$ (positively curved meniscus). Across this range, $|T|$ increases steadily as the meniscus transitions from hydrophobic to positively curved, and the model captures this trend accurately, showing how moderate meniscus curvature enhances transmission beyond what a flat interface would predict.

As $\theta$ approaches $0^\circ$ , the experimental $|T|$ begins to saturate. For $\theta \lt 0^\circ$ , the meniscus becomes overturned (multi-valued) and $|T|$ subsequently decreases. Our linearised model relies on a single-valued representation $y = \eta (x)$ and, therefore, cannot represent the overturned geometry, making it inapplicable for predicting transmission in this regime. Consequently, comparison between model and experiment is restricted to the single-valued regime $0^\circ \leqslant \theta \leqslant 180^\circ$ . In comparison with the infinitesimal-barrier results, the transmission exhibits a similar rate of variation, increasing by about 0.08 for every 1 mm increase in capillary height, although the overall transmission is symmetrically reduced by approximately 0.2 due to the finite barrier width.

Different coatings on the barrier can produce different degrees of effective slip along the bottom surface of the barrier (Truesdell et al. Reference Truesdell, Mammoli, Vorobieff, van Swol and Brinker2006; Rothstein Reference Rothstein2010), which may contribute to the quantitative differences among the experimental results in figure 5. The discrepancy between the numerical predictions and experimental measurements is of the same order as the variation among the experimental results themselves for different coatings. In our model, the potential-flow assumption implies a free-slip condition at the solid bottom surface of the barrier. In contrast, hydrophobic or hydrophilic coated barriers may exhibit partial-slip behaviour, which may partly explain the quantitative differences between model and measurements. Overall, these results demonstrate the accuracy of our linearised model for gentle, single-valued menisci by comparison with experimental results.

6.1. Flow fields

While the experiments in Wang et al. (Reference Wang, Liu and Zhang2025) only measured the surface elevation acoustically, our numerical modelling provides the entire flow field. Figure 6 shows snapshots of the numerical velocity potential field and the surface elevation for a negative ( $\theta = 180^\circ$ ), flat ( $\theta = 90^\circ$ ) and positive ( $\theta = 0^\circ$ ) meniscus. The results highlight how the meniscus geometry and equilibrium contact angle $\theta$ affect both the wave profile and the underlying flow in the model.

Figure 6.

Snapshots of the normalised velocity potential $\phi /2\phi _A$ at $t=0$ for three characteristic equilibrium contact angles: $\theta = 180^\circ , 90^\circ , 0^\circ$ . The dashed black curves denote the unperturbed meniscus, whereas the perturbed (wave-bearing) surface $\eta$ is shown by the solid black curve. The barrier is drawn in the centre and the colour scale denotes the normalised velocity potential field $\phi /2\phi _A$ at $t=0$ , where red/blue represent positive/negative values. The amplitude of the surface elevation $\eta _A$ has been set to $0.15 a$ for visualisation purposes only. See also supplementary movie 1 for the corresponding time-dependent evolution.

From figure 6, showing the normalised snapshots $\phi _1/2\phi _A$ at $t=0$ , we note the following key observations, aligned with the interpretations in Wang et al. (Reference Wang, Liu and Zhang2025).

1. (i) For $\theta = 180^\circ$ : the disturbance of the downstream free surface is slight, indicating that the barrier size effect in blocking the transmission of the capillary–gravity wave is significant when the barrier is below the water level.

2. (ii) Transition from $\theta = 180^\circ$ to $\theta = 90^\circ$ : as the meniscus evolves from a negative shape to a nearly flat interface, the wave amplitude on the upstream side grows and the overall transmission rises. Reducing the meniscus deformation increases the effective coupling between the upstream and downstream flows.

3. (iii) Transition from $\theta = 90^\circ$ to $\theta = 0^\circ$ : transmission continues to increase as a positively curved meniscus develops. The formation of the water column beneath the barrier enhances the coupling of the flow from the incident side to the transmitted side, leading to an increase in transmission.

Snapshots of the velocity potential field may also be interpreted as snapshots of the pressure field perturbed by the waves. Writing the fluid pressure as the sum of the hydrostatic pressure $p_0 = p_{\textit{atm}} - \rho g y$ (in the absence of waves) and the dynamic pressure perturbation $p_1(x,y,t)$ ,

(6.1) \begin{equation} p = p_0 + \epsilon p_1, \end{equation}

and linearising Bernoulli’s equation (2.9) about the equilibrium state while neglecting the quadratic term, the $O(\epsilon )$ equation gives

(6.2) \begin{equation} p_1 = -\rho \partial _t \phi _1. \end{equation}

For purely time-harmonic motion, using the complex potential notation (4.1a ),

(6.3) \begin{equation} \phi _1(x,y,t) = \psi (x,y)e^{-\mathrm{i}\omega t} + \text{c.c.}, \end{equation}

which implies that

(6.4) \begin{equation} p_1(x,y,t) = \mathrm{i}\omega \rho \psi (x,y)e^{-\mathrm{i}\omega t}+\text{c.c.} = \rho \omega \,\phi _1(x,y,t-T/4). \end{equation}

Thus, a snapshot of the velocity potential field corresponds to a snapshot of the dynamic pressure field with the amplitude scaled by $\rho \omega$ and with a phase shift of $90^\circ$ (equivalently a time shift of one quarter of the period $T$ ).

Videos of the dynamic pressure field together with particle motion in the fluid are provided in supplementary movies 2–4 (corresponding to $\theta =180^\circ , 90^\circ , 0^\circ$ , respectively) available at https://doi.org/10.1017/jfm.2026.11584, where the particle trajectories are computed as

(6.5) \begin{equation} \Delta \boldsymbol{x}(x,y,t) = (-\mathrm{i}\omega )^{-1} \boldsymbol{\nabla }\psi (x,y) \, e^{- \mathrm{i} \omega t} + \text{c.c.}, \end{equation}

with $\Delta \boldsymbol{x}(x,y,t)$ denoting the instantaneous displacement of a particle from its equilibrium position $(x,y)$ .

6.2. Interpretations using energy-flux fields

We now calculate the energy-flux field using the velocity potential solution to gain insight into the influence of the meniscus near the barrier. The energy conservation equation defines the energy flux, whose average over a wave cycle can be written in terms of the velocity potential as (see Appendix C)

(6.6) \begin{align} \langle \boldsymbol{j} \rangle = - \rho \, \langle \partial _t \phi _1 \boldsymbol{\nabla }\phi _1 \rangle . \end{align}

For a time-harmonic field,

(6.7) \begin{align} \phi _1(x,y,t) = \psi (x,y)\mathrm{e}^{-\mathrm{i}\omega t} + \text{c.c.}, \end{align}

where substitution into (6.6) yields

(6.8) \begin{align} \langle \boldsymbol{j} \rangle = 2\rho \omega \,\text{Im}[\psi ^* \boldsymbol{\nabla }\psi ], \end{align}

with $\psi ^*$ denoting the complex conjugate of the steady complex potential $\psi$ and $\text{Im}$ denoting the imaginary part. We use this expression to compute the flux fields from the steady complex potential and its spatial gradient.

Figure 7.

Meniscus-induced energy-flux fluctuations for three characteristic equilibrium contact angles (top: $\theta =180^\circ$ ; middle: $\theta =90^\circ$ ; bottom: $\theta =0^\circ$ ). Left panels: time-averaged energy flux (arrows) and its vertical component $\langle j_y\rangle$ (colours), calculated from (6.8). Right panels: vertical profiles of the horizontal energy flux $\langle j_x\rangle$ along the centreline $x=0$ , where the horizontal dashed lines indicate the water level and vertical dashed lines denote the energy-flux value right beneath the bottom of the barrier. The energy-flux arrows are scaled by the transmitted surface energy flux, while the components $\langle j_x\rangle$ and $\langle j_y\rangle$ are normalised by the incident surface energy flux $\langle j_0\rangle$ in the absence of both the barrier and the meniscus.

Figure 7 presents the numerical results of the energy-flux fields for a 15 Hz capillary–gravity wave scattered by a 3.18 mm-wide surface-piercing plate barrier in a 9.2 cm depth tank. The left panels illustrate the time-averaged energy flux (arrows) and its vertical component (colours) computed by (6.8). A particular observation is that the energy bypasses underneath the barrier for both negative and flat menisci, generating a downward (blue) and then upward (red) behaviour of its vertical component. In contrast, a positive water column allows energy to pass through it almost horizontally at the water level, providing an additional up–down mode of vertical flux. The results support the interpretation that the water column enlarges the effective cross-section for energy passing through and enhances the coupling of the water surface on the two sides of the barrier.

From the right panels of figure 7, we observe that the horizontal component of the energy flux is significantly larger above the water level when a positive water column exists, consistent with enhanced transmission beneath the barrier. The positive meniscus-induced water column is therefore consistent with stronger free-surface coupling and increased energy transmission. Overall, these results reveal how the meniscus curvature impacts small-scale wave–structure interactions.

To further verify global energy conservation, we computed the vertically integrated horizontal energy flux,

(6.9) \begin{align} P(x)=\int _{-H}^{\eta _0(x)} \langle j_x \rangle \, \text{d}y, \end{align}

and confirmed that $P(x)$ evaluated at the left boundary, right boundary and along the central line beneath the barrier are equal within numerical tolerance. This demonstrates that the total wave power transported through the domain is conserved and is consistent with the scattering relation $1-|R|^2 = |T|^2$ obtained from the far-field amplitudes.

7.1. Dispersion analysis at leading order

To leading order, we apply the classical flat-surface boundary conditions (3.15), evaluated at $y=\eta _0(x)$ :

(7.6a,b)

Substituting (7.2) and (7.5) yields a $2\times 2$ homogeneous system for $(A,B)$ . Requiring a non-trivial solution gives the dispersion relation

(7.7) \begin{equation} \omega ^2 = \left(gk + \frac {\sigma }{\rho }k^3\right)\tanh\, (kd), \end{equation}

where

(7.8) \begin{equation} d \equiv H+\eta _0(x) \end{equation}

is the vertical distance from the bottom $y=-H$ to the local free surface $y=\eta _0(x)$ .

Thus, at leading order the meniscus modifies the dispersion only through the effective depth $H+\eta _0(x)$ . In the deep-water regime ( $kd\gg 1$ ), this correction is small since $\tanh (kd)\approx 1$ . Consequently, this leading-order approximation does not modify wave transmission to the extent predicted by the exact linearised boundary conditions (figure 8 a).

Figure 8.

Transmission coefficient $|T|$ as a function of barrier height $h$ . Blue squares in (a) denote results from the leading-order formulation (7.6) and orange diamonds in (b) from the small-slope approximation (7.12). Both are compared with black circles obtained from the exact linearised boundary conditions (figure 5). Results are shown for the $15\,\mathrm{Hz}$ wave and the barrier of width $3.18\,\mathrm{mm}$ .

7.2. Small-slope dispersion analysis

We now include first-order corrections in the meniscus slope. Let

(7.9) \begin{equation} \delta _0=\max |\eta _0'|\ll 1, \end{equation}

and since the meniscus varies over the capillary length scale $a$ , we estimate

(7.10) \begin{equation} \eta _0' \sim \frac {\eta _0}{a} \sim O(\delta _0). \end{equation}

Specifically, under the small-slope approximation, the Young–Laplace equation (3.3) reduces to $\eta _0'' = \eta _0/a^2$ , which yields the equilibrium profile,

(7.11) \begin{equation} \eta _0 = h \exp (-|x-x_c|/a), \end{equation}

where $x_c$ denotes contact-point locations and $|x|\geqslant |x_c|$ .

Expanding the boundary conditions (3.14) to $O(\delta _0)$ gives

(7.12a,b)

Substitution yields

(7.13) \begin{equation} \omega ^2 = \left(gk + \frac {\sigma }{\rho }k^3\right) \left [ \tanh (kd) - \mathrm{i}\,\eta _0' \right ]\!, \end{equation}

with the local effective depth $d = H + \eta _0(x)$ given in (7.8). In deep water,

(7.14) \begin{equation} \omega ^2 = \left(gk + \frac {\sigma }{\rho }k^3\right) \left ( 1 - \mathrm{i}\,\eta _0' \right )\!. \end{equation}

Thus, at this order, the slope $\eta _0'$ introduces an imaginary contribution to the dispersion relation. For real $\omega$ , this implies a complex spatial wavenumber $k$ , corresponding to spatial amplification or attenuation induced by the meniscus geometry.

A further calculation of the transmission using the small-slope $O(\delta _0)$ boundary conditions (7.12) shows close agreement with that obtained from the full set of linearised boundary conditions (3.13); see figure 8 b. The results further show that the small-slope approximation remains accurate beyond its formal asymptotic regime of validity.

7.3. Dispersion analysis beyond small-slope meniscus

Substituting (7.2) and (7.5) into the full set of linearised boundary conditions (3.13) yields a homogeneous linear system for $(A,B)$ , i.e.

(7.15) \begin{equation} \begin{pmatrix} k\big (\sinh (kd)-\mathrm{i}\,\eta _0'\cosh (kd)\big ) & -\,\mathrm{i}\omega \\[6pt] -\,\mathrm{i}\omega \cosh (kd) & g+\dfrac {\sigma }{\rho }\left ( \dfrac {k^2}{Q^{3/2}} +\mathrm{i} k\,\dfrac {3\eta _0'\eta _0''}{Q^{5/2}} \right ) \end{pmatrix} \begin{pmatrix} A\\[4pt] B \end{pmatrix} = \begin{pmatrix} 0\\[4pt] 0 \end{pmatrix}, \end{equation}

where $d = H+\eta _0(x)$ is the effective depth given in (7.8) and

(7.16) \begin{equation} Q \equiv 1 + (\eta _0')^2. \end{equation}

A non-trivial solution $(A,B)\neq (0,0)$ requires the determinant to vanish, yielding

(7.17) \begin{equation} \omega ^2 = \left [ kg+\frac {\sigma }{\rho }k^3\left ( \frac {1}{Q^{3/2}} +\mathrm{i} \,\frac {3\eta _0'\eta _0''}{k^2 Q^{5/2}} \right ) \right ] \big (\tanh (kd)-i\eta _0'\big ). \end{equation}

This dispersion relation explicitly incorporates the effects of the local elevation $\eta _0(x)$ through the effective depth $d=H+\eta _0(x)$ , the local slope $\eta _0'$ through the imaginary correction terms and the local curvature $\eta _0''$ through the higher-order geometric contribution in the surface tension term.

In the small-slope regime,

(7.18) \begin{equation} \frac {1}{Q^{3/2}} = 1 + O\big(\delta _0^2\big), \quad \frac {\eta _0'\eta _0''}{Q^{5/2}} = O \big(\delta _0^2\big), \end{equation}

so (7.17) reduces to the small-slope $O(\delta _0)$ dispersion (7.13), providing a consistency check.

Figure 9.

(a) Equilibrium meniscus profile with capillary rise height $h=a=2.7$ mm. (b) Real (black) and imaginary (grey) parts of the complex wavenumber $k(x)$ , which approach the flat-surface limit $k_0$ far from the meniscus (dots). (c) Imaginary part of $\exp \!(\mathrm{i} k(x)\,x)$ (black) compared with a sinusoidal wave of constant wavenumber $k_0$ (dots). (d) Dimensionless validity measure $\delta _k(x)$ (given in (7.4)). Results correspond to the $15\,\mathrm{Hz}$ wave. The locations $x_c$ denote the contact points.

7.4. Local eigenvalue calculations

In the present configuration, the base state is spatially varying due to the localised meniscus. Rather than performing a global eigenvalue analysis about a spatially uniform base state, we instead formulate a spatial eigenvalue problem by computing the complex wavenumber

(7.19) \begin{equation} k(x)=k_r(x)+\mathrm{i}\,k_i(x), \end{equation}

using (7.17) for a prescribed real forcing frequency of $15\,\mathrm{Hz}$ and a given meniscus with capillary rise height $h=a$ as an example (figure 9 a). In the far field (figure 9 b), $k\to k_0$ , the flat-surface wavenumber, consistent with radiation conditions.

Once $k(x)$ is obtained, the local wave profile takes the form

(7.20) \begin{equation} \eta _1 \sim \exp (\mathrm{i} kx -\mathrm{i} \omega t) = \exp (- k_i x)\exp (\mathrm{i} k_r x - \mathrm{i} \omega t). \end{equation}

The imaginary part $k_i(x)$ represents spatial attenuation ( $k_i x\gt 0$ ) or amplification ( $k_i x\lt 0$ ) induced by the meniscus geometry (figure 9 c). The validity of the local (WKB) approximation is monitored via the parameter $\delta _k(x)$ defined in (7.4). For the experimental configurations considered, $\delta _k \ll 1$ except in a narrow region near the contact points, indicating that the spatial eigenvalue description remains accurate over most of the meniscus region (figure 9 d).

This spatial eigenvalue formulation provides a stability interpretation of the transmission problem: the meniscus modifies the local spectrum and thereby controls phase modulation and attenuation of propagating waves. The present linearised problem could also be formulated as a global eigenvalue problem (Wei et al. Reference Wei, Rivero-Rodríguez, Zou and Scheid2021; Keeler & Blyth Reference Keeler and Blyth2024). Such a formulation would treat the frequency $\omega$ (or phase speed $c=\omega /k$ ) as an eigenvalue of a boundary-value problem posed on the full fluid domain with radiation conditions imposed in the far field. Such a global eigenvalue computation lies beyond the scope of the present study, which focuses instead on the experimentally relevant forced problem and on spatial modulation of propagating modes. Nevertheless, the local spatial eigenvalue analysis developed above may be viewed as a WKB reduction of this global formulation, valid when the meniscus geometry varies slowly relative to the wavelength. In this sense, the present approach captures the leading-order stability and transmission mechanisms induced by the meniscus.