2. Preliminaries

Consider an inviscid, irrotational disturbance of a slightly compressible fluid of depth $h$ , with free surface $z=\eta (x,t)$ and rigid bottom $z=-h$ . The fluid is assumed to be barotropic, with constant sound speed $c=(\mathrm{d}p/\mathrm{d}\rho )^{1/2}=1.5\times 10^3\,\mathrm{m\,s^{-1}}$ , where $p$ is the pressure and thermodynamic relation $p(\rho) = P_a+c^2(\rho-\rho_a)$  (Zuccoli & Kadri Reference Zuccoli and Kadri2025). For standard water–air interface, the value of the atmospheric pressure is $P_{a}=101.3\,\mathrm{kPa}$ and that of the uniform density is $ \rho _{a}=1000\,\mathrm{Kg\,m^{-3}}$ . In addition, the fluid is subject to both gravitational acceleration $ g=9.81\,\mathrm{m\,s^{-2}}$ and surface tension $\gamma =0.072\,\mathrm{N\,m^{-1}}$ . For water at laboratory depths, $h\sim 1\,\mathrm{m}$ , two parameters control compressibility and capillarity effects, namely $\mu =gh/c^2 \sim \mathcal{O}(10^{-6})\ll 1$ and $\sigma =\gamma /(\rho _a g h^2) \sim \mathcal{O}(10^{-6}) \ll 1$ , respectively.

The governing equations are the compressible Euler equations, expressing conservation of momentum and conservation of mass

(2.1) \begin{align} \partial _{t}\boldsymbol{u} + (\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{u} + \frac {c^2}{\rho }\boldsymbol{\nabla }\rho + g\boldsymbol{\hat {z}} = 0, \quad \partial _{t}\rho + \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }\rho + \rho (\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}) = 0. \end{align}

We assume irrotationality of the fluid motion so that the fluid velocity can be written in terms of a potential $\varphi$ as $\boldsymbol{u}=\boldsymbol{\nabla }\varphi$ in two dimensions $(x,z)$ . The governing equations become

(2.2) \begin{align} \partial _{t}\varphi + \frac {1}{2}|\boldsymbol{\nabla }\varphi |^2 + c^2\log {\rho } + gz = 0, \quad (\partial _{t} + \boldsymbol{\nabla }\varphi \boldsymbol{\cdot }\boldsymbol{\nabla })\log {\rho } + \Delta \varphi = 0, \end{align}

where $\Delta = \partial ^{2}_x + \partial ^{2}_z$ is the Laplacian operator in two spatial coordinates. Applying the convective operator $\partial _t + \boldsymbol{\nabla }\varphi \boldsymbol{\cdot }\boldsymbol{\nabla}$ to the Bernoulli equation (2.2) (left), we obtain a single equation for the potential

(2.3) \begin{align} \partial ^{2}_{t}\varphi - c^2\Delta \varphi + g\partial _{z}\varphi = -\partial _{t}|\boldsymbol{\nabla }\varphi |^{2} - \frac {1}{2}\boldsymbol{\nabla }\varphi \boldsymbol{\cdot }\boldsymbol{\nabla }|\boldsymbol{\nabla }\varphi |^2. \end{align}

Regarding the boundary conditions, the flow is unbounded in the $x$ -direction and must satisfy kinematic and dynamic boundary conditions on the free surface

(2.4a) \begin{align} & \partial _{t}\eta + \partial _x\varphi \partial _x\eta = \partial _{z}\varphi , \quad \textrm {on} \quad z = \eta ,\\[-7pt]\nonumber \end{align}

(2.4b) \begin{align} & \partial _{t}\varphi + \frac {1}{2}|\boldsymbol{\nabla }\varphi |^2 + gz - \frac {\gamma }{\rho _a} \big [1 + (\partial _x\eta )^2\big ]^{-3/2}\partial ^{2}_x\eta = 0, \quad \textrm {on} \quad z = \eta . \end{align}

At the bottom, $\partial _z\varphi =0$ on $z=-h$ .

We non-dimensionalise equations (2.3)–(2.4) based on length scale $h$ , time scale $\sqrt {h/g}$ and characteristic wave amplitude $a$ . This leads to the following complete dimensionless problem:

(2.5) \begin{align} & \partial ^2_{t}\varphi - \frac {1}{\mu }\Delta \varphi + \partial _z\varphi = -\epsilon \partial _{t}|\boldsymbol{\nabla }\varphi |^{2} - \frac {\epsilon ^2}{2}\boldsymbol{\nabla }\varphi \boldsymbol{\cdot }\boldsymbol{\nabla }|\boldsymbol{\nabla }\varphi |^2, \quad -1 \lt z \lt \epsilon \eta , \nonumber\\[3pt] & \partial _{t}\varphi + \frac {\epsilon }{2}|\boldsymbol{\nabla }\varphi |^2 + \eta - \sigma \big[1 + \epsilon ^2(\partial _x\eta )^2\big]^{-3/2}\partial ^{2}_x\eta = 0, \quad \mathrm{on} \quad z = \epsilon \eta , \nonumber\\[3pt] & \partial _{z}\varphi = \partial _t\eta + \epsilon \partial _x\varphi \partial _x\eta , \quad \mathrm{on} \quad z = \epsilon \eta , \nonumber\\[3pt] & \partial _z\varphi = 0, \quad \mathrm{on} \quad z = -1, \end{align}

where $\mu = {gh}/{c^2}$ measures the effects of compressibility, $\epsilon = {a}/{h}$ measures the effects of nonlinearity and $\sigma = {\gamma }/(\rho _agh^2)$ measures the effects of capillarity. Since the nonlinear terms are modulated by $\epsilon$ , the multiple-scale expansion is performed in $\epsilon$ only, while $\mu$ and $\sigma$ are treated as fixed parameters. The subsequent closed-form formulas therefore use a small- $\mu$ approximation only for the linear resonance problem, after which its accuracy is checked a posteriori. Similar orderings are used for weakly nonlinear capillary and compressible free-surface waves by Johnson (Reference Johnson1997), Viola et al. (Reference Viola, Brun and Gallaire2018), Viola & Gallaire (Reference Viola and Gallaire2018) and Zuccoli & Kadri (Reference Zuccoli and Kadri2026).

3. The linear solution and existence of resonant triads

We expand the velocity potential and the free-surface elevation in powers of $\epsilon$ as $\varphi = \varphi _0 + \epsilon \varphi _1 + \mathcal{O}(\epsilon ^2)$ and $\eta = \eta _0 + \epsilon \eta _1 + \mathcal{O}(\epsilon ^2)$ . Similarly, we expand the free-surface boundary conditions in Taylor series around $z = 0$ and retain only terms up to linear order in $\epsilon$ . At the leading order $(\epsilon =0)$ , (2.5) are linear and read

(3.1) \begin{align} \begin{aligned} & \partial ^2_{t}\varphi _0 - \frac {1}{\mu }\Delta \varphi _0 + \partial _z\varphi _0 = 0, \quad -1 \lt z \lt 0, \\ & \partial _{t}\varphi _0 + \eta _0 - \sigma \partial ^{2}_x\eta _0 = 0, \quad \mathrm{on} \quad z = 0, \\ & \partial _{z}\varphi _0 = \partial _t\eta _0, \quad \mathrm{on} \quad z = 0, \\ & \partial _z\varphi _0 = 0, \quad \mathrm{on} \quad z = -1. \end{aligned} \end{align}

Re-arranging the boundary conditions in terms of $\varphi _0$ only and decomposing the solution into normal modes of the form

(3.2) \begin{align} \varphi _0(x,z,t)=\sum _{n=1}^{\infty }A_nf_n(z)e^{\mathrm{i} k_nx-\mathrm{i}\omega _nt}+\mathrm{c.c.}, \end{align}

where $\mathrm{c.c.}$ denotes complex conjugate, the linear system (3.1) returns a polynomial eigenvalue problem in the eigenpair $(\omega _n,f_n)$

(3.3) \begin{align} \begin{aligned} & f''_{n} - \mu f'_{n} + (\mu \omega ^{2}_{n} - k^{2}_{n})f_n = 0, \\[4pt] & f'_{n}(0) = \frac {\omega ^{2}_{n}}{1 + \sigma k^{2}_n}f_{n}(0) \quad \mathrm{and} \quad f'_{n}(-1) = 0, \end{aligned} \end{align}

where $k_n$ and $\omega_n$ denote the dimensionless horizontal wavenumber and frequency of the $n$ th mode, respectively, and the prime symbol denotes differentiation with respect to the single variable $z$ . Problem (3.3) can be solved exactly in closed form. The roots of the characteristic polynomial associated with the differential equation are given by

(3.4) \begin{align} \lambda ^{(\pm )}_{n}(k_n,\omega _n)=\frac {\mu }{2}\pm \sqrt {\frac {\mu ^2}{4}+k^{2}_n - \mu \omega ^{2}_n}. \end{align}

Application of the no-penetration and free-surface boundary conditions yields the dispersion relation and eigenfunctions

(3.5a) \begin{align} D(k_n,\omega _n) &= \lambda ^{(+)}_{n} -\frac {\omega _n^2}{1+\sigma k_n^2} +\frac {\lambda ^{(+)}_{n}}{\lambda ^{(-)}_{n}} \left ( \frac {\omega _n^2}{1+\sigma k_n^2} -\lambda ^{(-)}_{n} \right ) e^{\lambda ^{(-)}_{n}-\lambda ^{(+)}_{n}} =0,\\[-10pt]\nonumber \end{align}

(3.5b) \begin{align} f_n(z) &= e^{\lambda ^{(+)}_{n}z} -\frac {\lambda ^{(+)}_{n}}{\lambda ^{(-)}_{n}}e^{\lambda ^{(-)}_{n}(1+z)-\lambda ^{(+)}_{n}} . \end{align}

3.1. Resonant triads

Consider a triad comprising two exactly oppositely propagating gravity–capillary modes of wavenumbers $k_{\pm }$ and frequencies $\omega _{\pm }$ , and a horizontally uniform vertical acoustic mode of frequency $\omega$ (Longuet-Higgins configuration). Resonant interaction among the triad is possible if the following resonance conditions:

(3.6) \begin{align} k_{+} + k_{-} = 0, \quad \omega _{+} + \omega _{-} = \omega , \end{align}

together with the dispersion relations (3.5a ) are satisfied for each mode. Since the dispersion relation is even in the horizontal wavenumber, the two capillary–gravity modes have the same frequency and vertical structure, with only the sign of the horizontal wavenumber reversed. The resonance conditions then give $\omega _{+}=\omega _{-}= \omega /2$ and $k_{\pm }=\pm \kappa$ , with $\kappa\gt0$ denoting the dimensionless wavenumber magnitude of the capillary-gravity modes. The laboratory regime is characterised by $\mu \ll 1$ . In this limit, the eigensolutions for both the capillary–gravity modes and the acoustic mode can be obtained in closed form. In particular:

1. (i) Capillary–gravity modes: when $\mu \rightarrow 0$ , (3.4) yields $\lambda ^{(\pm )}_{n} = \pm \kappa$ and hence (3.5) reduce to

   (3.7) \begin{align} \frac {\omega ^2}{4} = \kappa (1 + \sigma \kappa ^2)\tanh (\kappa ), \quad f_{\pm }(z) = \frac {\mathrm{cosh}(\kappa (z+1))}{\mathrm{cosh}(\kappa )}. \end{align}

2. (ii) Acoustic mode: for $k_n = 0$ , eigensolutions are still possible provided $\omega _n \sim 1/\sqrt {\mu }$ . Let $\omega _n = \lambda /\sqrt {\mu }$ , substitute in (3.4)–(3.5b ), after taking the limit $\mu \rightarrow 0$ , (3.5) reduce to

   (3.8) \begin{align} \lambda = \pi /2 + m\pi \equiv \lambda _m, \quad f(z) = 2\mathrm{i}\sin (\lambda _m z), \quad m \in \mathbb{Z}. \end{align}

In this limit the acoustic mode has a vertical standing-wave structure while the resonant capillary–gravity waves are exponentially localised near the free surface. The normalisation of the small- $\mu$ eigenfunctions arises naturally and is fixed by requiring $f_{\pm }(0)=1$ , and $f\prime (0)=2\mathrm{i}\lambda _m$ . No additional energy or action normalisation is imposed.

The closed-form eigensolutions can be further simplified by requiring a deep-water capillary–gravity approximation, i.e. $\kappa \gg 1$ , for which the capillary–gravity eigenfunctions become $f_{\pm }(z) = e^{\kappa z}$ and the dispersion relation $\omega ^2/4 = \kappa + \sigma \kappa ^3$ . Although for laboratory water waves, $\mu \ll 1$ and $\sigma \ll 1$ , the combinations $\sigma \kappa ^2$ and $\sigma \kappa ^3$ are retained because they are not small at resonance, in fact allowing the nonlinear interaction between the two type of modes. Such reduction is verified a posteriori against the exact eigenfunctions from (3.5), yielding relative $L_\infty$ -errors of order $10^{-4}$ – $10^{-6}$ for $h\in [0.1,1]\,\mathrm{m}$ , where $L_{\infty}$ denotes the supremum (maximum) norm. Moreover, application of Cardano’s formula (Cooke Reference Cooke2008) to the depressed cubic polynomial $\kappa + \sigma \kappa ^3 - \omega ^2 / 4 = 0$ gives the resonant wavenumber explicitly

(3.9) \begin{align} \kappa = \sqrt [3]{\frac {\lambda ^{2}_m}{8\mu \sigma } + \sqrt {\frac {\lambda ^{4}_m}{64\mu ^2\sigma ^2} + \frac {1}{27\sigma ^3}}} + \sqrt [3]{\frac {\lambda ^{2}_m}{8\mu \sigma } - \sqrt {\frac {\lambda ^{4}_m}{64\mu ^2\sigma ^2} + \frac {1}{27\sigma ^3}}}. \end{align}

Summarising, the leading-order eigenfunctions and resonance condition are therefore

(3.10a) \begin{align} f_{\pm }(z) & =e^{\kappa z}, \quad f(z)=2\mathrm{i}\sin (\lambda _m z), \quad \lambda _m=\frac {\pi }{2}+m\pi , \quad m=0,1,\ldots ,\\[-10pt]\nonumber \end{align}

(3.10b) \begin{align} \omega _{+} & =\omega _{-}=\frac {\omega }{2}, \quad \omega =\frac {\lambda _m}{\sqrt {\mu }}, \quad \kappa +\sigma \kappa ^3=\frac {\omega ^2}{4}. \end{align}

In figure 1 we show the wavelength (in dimensional form) of the resonant capillary–gravity waves $\lambda _{cg} = 2\pi h / \kappa (h)$ as a function of the water depth $h$ for the first three vertical mode numbers $m = 0, 1, 2$ . Note the dependence of $\kappa$ on the water depth $h$ . The possibility of forming resonant triads highlights the role of surface tension: capillarity increases the surface-wave frequency at large wavenumber, thereby allowing two short capillary–gravity waves to match a vertically oscillating acoustic mode even in shallow laboratory basins. As it will be shown in the next sections, such capillarity-enabled resonance leads to the possibility of measuring the dynamic pressure generated at the bottom of the basin when two surface modes wavepackets collide. This is directly relevant to experiments because it gives a measurable pressure signature of the nonlinear interaction.

Figure 1.

Dimensional resonant capillary–gravity wavelength $\lambda _{cg} = 2\pi h/\kappa$ as a function of water depth for the first three acoustic vertical modes, $m=0,1,2$ . The resonance condition includes surface tension and corresponds to two counter-propagating surface modes forcing a horizontally uniform acoustic mode. The wavenumber $\kappa$ is computed using the analytical expression (3.9) with $\mu = gh/c^2$ , $\sigma = \gamma /(\rho _a g h^2)$ , $\ g=9.81\,\mathrm{m\,s^{-2}}$ , $\ c=1500\,\mathrm{m\,s^{-1}}$ , $\ \gamma =0.072\,\mathrm{N\,m^{-1}}$ and $\ \rho _a=1000\,\mathrm{Kg\,m^{-3}}$ .

It is also useful to contrast this with the gravity-only case. If surface tension is omitted, the resonant condition in the same configuration would reduce to $ \kappa ={\omega ^2}/{4} ={\lambda _m^2}/{4\mu }$ . For laboratory values of $\mu$ , this gives extremely large dimensionless wavenumbers and therefore very short surface wavelengths. The corresponding gravity-wave frequency is incompatible with the gravity-wave dispersion relation at accessible laboratory wavelengths. Surface tension changes this balance by replacing $\kappa$ with $\kappa +\sigma \kappa ^3$ in the capillary–gravity dispersion relation (3.10b ). At the resonant wavenumbers found here, the capillary term is dominant compared with the gravity term $\sigma \kappa ^3 \gg \kappa$ , however, retaining both contributions shows that the experimentally accessible resonance arises through a transition from the gravity-dominated to the capillary-dominated regime. This allows the acoustic frequency to be matched by short capillary–gravity waves at finite laboratory depth. Thus the resonance is not a small perturbation of the gravity–acoustic case; it is enabled by the capillary branch of the dispersion relation.

4. Weakly nonlinear analysis and amplitude equations

Nonlinear contributions in (2.5) become non-negligible at temporal scales of $\mathcal{O}(1/\epsilon )$ . To take into account weak nonlinearities, we apply the method of multiple scales (Nayfeh Reference Nayfeh2008; Charru Reference Charru2011) and introduce a slow temporal variable $T = \epsilon t$ , such that the time derivative transforms according to $\partial _{t}\rightarrow \partial _{t} + \epsilon \partial _{T}$ . Note that one could have chosen the surface waves steepness $ak$ as a measure of nonlinearity (with $k$ denoting their characteristic wavenumber). However, the present nonlinear parameter $\epsilon = a/h$ naturally leads to the possibility of taking into account the deep (although still finite) water regime needed to obtain a standing acoustic resonant mode. Moreover, these two measures are related through $ak = \epsilon \kappa$ , so that the weakly nonlinear regime considered here additionally requires $ak \ll 1$ , which is satisfied for the representative amplitudes used in our laboratory estimates.

After expanding the velocity potential and the surface elevation in powers of $\epsilon$ , at leading order the system returns the linear problem. At $\mathcal{O}(\epsilon )$ , the differential problem can be entirely written in terms of the perturbation potential $\varphi _1$ and the linear solution $(\varphi _0, \eta _0)$ , yielding

(4.1a) \begin{align} \mathcal{L}\varphi _1 =& -2\partial _{t}\partial _{T}\varphi _{0} - \partial _{t}|\boldsymbol{\nabla }\varphi _{0}|^{2}, \quad -1\lt z\lt 0, \end{align}

(4.1b) \begin{align} \partial ^{2}_{t}\varphi _1 + \big(1-\sigma \partial ^2_x\big)\partial _{z}\varphi _1 &= -\big(1-\sigma \partial ^{2}_x\big)\big(\partial ^{2}_z\varphi _0\eta _0 - \partial _T\eta _0 - \partial _x\varphi _0\partial _x\eta _0\big) \nonumber \\ & - \partial _{t}\partial _{T}\varphi _{0} -\frac {1}{2}\partial _{t}|\boldsymbol{\nabla }\varphi _{0}|^{2} - \partial _t(\eta _0\partial _t\partial _z\varphi _0), \quad \mathrm{on} \quad z = 0,\\[-10pt]\nonumber \end{align}

(4.1c) \begin{align} \partial _{z}\varphi _1 &=\, 0, \quad \mathrm{on} \quad z = -1, \end{align}

where we have introduced the linear operator $\mathcal{L} = \partial ^{2}_t - \mu ^{-1}\Delta + \partial _z$ for notational brevity. As can be noted, all the linear terms in (4.1) are on the left-hand side; all the nonlinear terms are on the right-hand side. The nonlinear terms in (4.1) can give rise to secular contributions due to the resonant interaction (3.6) which need to be removed by imposing a solvability condition. To compute the resonant terms, the linear potential and free-surface elevation are expressed as sum of two capillary–gravity modes and a standing acoustic mode

(4.2) \begin{align} & \varphi _0(x, z, t, T) = A_{+}(T)f_{+}(z)e^{\mathrm{i}(k_{+}x-\omega _{+}t)} + A_{-}(T)f_{-}(z)e^{\mathrm{i}(k_{-}x-\omega _{-}t)} + A(T)f(z)e^{-\mathrm{i}\omega t} + \mathrm{c.c.}, \nonumber\\[3pt] & \eta _0(x, t, T) = B_{+}(T)e^{\mathrm{i}(k_{+}x-\omega _{+}t)} + B_{-}(T)e^{\mathrm{i}(k_{-}x-\omega _{-}t)} + B(T)e^{-\mathrm{i}\omega t} + \mathrm{c.c.}, \end{align}

with

(4.3) \begin{align} \begin{aligned} & B_{\pm }(T) = \frac {\mathrm{i}}{\omega _{\pm }}\frac {\mathrm{d} f_{\pm }(0)}{\mathrm{d} z}A_{\pm }(T), \quad \mathrm{and} \quad B(T) = \frac {\mathrm{i}}{\omega }\frac {\mathrm{d} f(0)}{\mathrm{d} z}A(T). \end{aligned} \end{align}

Let $F_n(z, T)$ and $G_n(0, T)$ denote the generic resonant Fourier coefficients of the bulk and free-surface forcing forcing in (4.1) for any modes. Let $F_{0_{\pm }}(z, T)$ and $G_{0_{\pm }}(0, T)$ be the aforementioned resonant Fourier coefficients for the two capillary–gravity modes, and $F_{0}(z, T)$ and $G_{0}(0, T)$ be those for the acoustic mode. In general terms, these are given by

(4.4a) \begin{align} G_{0_{\pm }}(0, T) &= 2\mathrm{i}\omega _{\pm }f_{\pm }(0)\dot {A}_{\pm }(T)\nonumber\\[6pt] & \quad + \left[\mathrm{i}(1+\sigma k^{2}_{\pm })\left(\frac {1}{\omega _{\mp }}\frac {\mathrm{d}f^{\ast }_{\mp }(0)}{\mathrm{d}z}\frac {\mathrm{d}^2f(0)}{\mathrm{d}z^2} - \frac {1}{\omega }\frac {\mathrm{d}^2f^{\ast }_{\mp }(0)}{\mathrm{d}z^2}\frac {\mathrm{d}f(0)}{\mathrm{d}z}\right) \right.\nonumber \\[6pt] & \quad \left. +\, \mathrm{i}\omega _{\pm }\left(1 - \frac {\omega ^{2}_{\mp }+\omega ^2}{\omega _{\mp }\omega }\right)\frac {\mathrm{d}f^{\ast }_{\mp }(0)}{\mathrm{d}z}\frac {\mathrm{d}f(0)}{\mathrm{d}z}\right]A^{\ast }_{\mp }(T)A(T),\\[-5pt]\nonumber \end{align}

(4.4b) \begin{align} G_{0}(0, T) &= 2\mathrm{i}\omega f(0)\dot {A}(T) + \left[\mathrm{i}\omega \left(\frac {\mathrm{d} f_{+}(0)}{\mathrm{d} z}\frac {\mathrm{d} f_{-}(0)}{\mathrm{d} z} - k_{+}k_{-}f_{+}(0)f_{-}(0)\right)\right.\nonumber \\[6pt] & \quad -\mathrm{i} k_{+}k_{-}\left(\frac {1}{\omega _{+}}\frac {\mathrm{d} f_{+}(0)}{\mathrm{d} z}f_{-}(0) + \frac {1}{\omega _{-}}\frac {\mathrm{d} f_{-}(0)}{\mathrm{d} z}f_{+}(0)\right) \nonumber \\[6pt] & \quad -\mathrm{i}\left(\frac {1}{\omega _{+}}\frac {\mathrm{d} f_{+}(0)}{\mathrm{d} z}\frac {\mathrm{d}^2 f_{-}(0)}{\mathrm{d} z^2} + \frac {1}{\omega _{-}}\frac {\mathrm{d}^2 f_{+}(0)}{\mathrm{d} z^2}\frac {\mathrm{d} f_{-}(0)}{\mathrm{d} z}\right) \nonumber \\[6pt] & \quad \left. +\, \mathrm{i}\frac {\omega }{\omega _{+}\omega _{-}}\big(\omega ^{2}_{+}+\omega ^{2}_{-}\big)\frac {\mathrm{d} f_{+}(0)}{\mathrm{d} z}\frac {\mathrm{d} f_{-}(0)}{\mathrm{d} z}\right]A_{+}(T)A_{-}(T). \end{align}

and

(4.5a) \begin{align} F_{0_{\pm }}(z, T) &= 2\mathrm{i}\omega _{\pm }f_{\pm }(z)\dot {A}_{\pm }(T) + 2\mathrm{i}\omega _{\pm }\frac {\mathrm{d} f^{\ast }_{\mp }(z)}{\mathrm{d} z}\frac {\mathrm{d} f(z)}{\mathrm{d} z},\\[-6pt]\nonumber \end{align}

(4.5b) \begin{align} F_{0}(z, T) &= 2\mathrm{i}\omega f(z)\dot {A}(T) + 2\mathrm{i}\omega \left(\frac {\mathrm{d} f_{+}(z)}{\mathrm{d} z}\frac {\mathrm{d} f_{-}(z)}{\mathrm{d} z} - k_{+}k_{-}f_{+}(z)f_{-}(z)\right). \end{align}

To make the solvability calculation explicit, leading to a set of nonlinear equations in the modes amplitudes $A_{\pm }$ and $A$ , we need to project the resonant contributions (4.4)–(4.5) onto the adjoint eigenfunctions associated with (3.3). If $f_n(z)$ is a direct eigenfunction, the corresponding adjoint eigenfunction is $f_n^{(a)}(z)=e^{-\mu z}f_n(z)$ , satisfying

(4.6) \begin{align} \begin{aligned} &{f_n^{(a)}}''+\mu {f_n^{(a)}}'+\big(\mu \omega _n^2-k_n^2\big)f_n^{(a)}=0, \\[4pt] &{f_n^{(a)}}'(0)-\left (\frac {\omega _n^2}{1+\sigma k_n^2}-\mu \right )f_n^{(a)}(0)=0, \quad {f_n^{(a)}}'(-1)+\mu f_n^{(a)}(-1)=0. \end{aligned} \end{align}

The solvability condition is given by

(4.7) \begin{align} -f_n^{(a)}(0)G_n(0,T)=\mu (1+\sigma k_n^2)\int _{-1}^{0}f_n^{(a)}(z)F_n(z,T)\mathrm{d}z. \end{align}

More details on the derivation of (4.7) are given in Appendix A.

Substituting the small- $\mu$ eigensolutions (3.10) into (4.4)–(4.5) and (4.7) gives the three-wave interaction equations

(4.8) \begin{align} \begin{aligned} & \dot {A}_{\pm } = \mathrm{i}\alpha _{s}A^{\ast }_{\mp }A, \quad \dot {A} = \mathrm{i}\alpha _{a}A_{+}A_{-}, \end{aligned} \end{align}

where $\alpha _s$ and $\alpha _a$ are the surface-wave and acoustic-wave coupling coefficients. With the eigenfunction normalisation used in (3.10), these coefficients are

(4.9) \begin{align} \begin{aligned} & \alpha _s = \frac {8\lambda _m \kappa ^3}{\lambda ^2_m + 4\kappa ^2}, \quad \alpha _a = \frac {8\kappa ^4}{\lambda ^2_m + 4\kappa ^2}. \end{aligned} \end{align}

The dependence of the coupling coefficients on the small physical parameters is important in assessing the range of validity of the weakly nonlinear description. In the resonant regime $\kappa \gg 1$ , (4.9) gives

(4.10) \begin{align} \alpha _s \sim 2\lambda _m\kappa , \quad \alpha _a \sim 2\kappa ^2 . \end{align}

Moreover, when the capillary contribution dominates the resonant balance

(4.11) \begin{align} \sigma \kappa ^3\gg \kappa , \quad \kappa \sim \left (\frac {\lambda _m^2}{4\mu \sigma }\right )^{1/3}, \end{align}

so that $\alpha _s$ and $\alpha _a$ are large and inherit an explicit dependence on $\mu$ and $\sigma$ . Thus they cannot in general be regarded as $O(1)$ constants in a simultaneous limit $\mu ,\sigma \rightarrow 0$ .

Table 1.

Representative dimensional and dimensionless parameters for the $m=0$ capillary–gravity–acoustic resonant triad. The attenuation estimates use $\nu =10^{-6}\,\mathrm{m^2\,s^{-1}}$ , $T_\nu =(2\nu k^2)^{-1}$ and $L_\nu =c_{g,{dim}}T_\nu$ , where $c_{g,{dim}}$ is the dimensional capillary–gravity group velocity.

Table 1. Long description

A table listing dimensional and dimensionless parameters for a capillary-gravity-acoustic resonant triad. The table has two main sections: Dimensional quantities and Dimensionless quantities. Each section lists various parameters with their symbols and values. Dimensional quantities include Water depth (h), Capillary-gravity frequency (fcg), Surface-wave amplitude (a), Capillary-gravity wavelength (λcg), Envelope width (λd), Viscous attenuation length (Lv), Viscous attenuation time (Tv), and Maximum acoustic pressure (Qb). Dimensionless quantities include Amplitude ratio (ε), Resonant wavenumber (κ), Surface-wave slope (ak), Compressibility parameter (μ), Capillarity parameter (σ), Attenuation time in carrier periods (Tv*fcg), Normalised acoustic pressure (Qb/(ρagh)), and Envelope/carrier scale ratio (λd/λcg). Each row lists the quantity, its symbol, and its value.

The asymptotic limit adopted here is instead $\epsilon \rightarrow 0$ with the physical parameters $\mu$ and $\sigma$ fixed. The effect of the large coupling coefficients is then to shorten the nonlinear interaction time scale. This can be made explicit by defining

(4.12) \begin{align} C=\sqrt {\alpha _s\alpha _a},\quad B=\sqrt {\frac {\alpha _s}{\alpha _a}}\,A,\quad \tau =CT . \end{align}

In these variables equation (4.8) becomes

(4.13) \begin{align} \frac {\mathrm{d}A_{\pm }}{\mathrm{d}\tau }=\mathrm{i}A_{\mp }^{\ast }B, \quad \frac {\mathrm{d}B}{\mathrm{d}\tau } =\mathrm{i}A_{+}A_{-}. \end{align}

Hence, the characteristic nonlinear modulation rate in the original dimensionless time is of order $\epsilon \sqrt {\alpha _s\alpha _a}$ . For surface-mode amplitudes of order unity, separation between the carrier oscillation and the nonlinear modulation therefore requires

(4.14) \begin{align} \frac {\epsilon \sqrt {\alpha _s\alpha _a}}{\omega _{+}}\ll 1, \quad \epsilon \kappa =ak\ll 1 . \end{align}

For the representative laboratory configuration reported in table 1, $\alpha _s=8.37\times 10^3$ and $\alpha _a=1.42\times 10^7$ , giving $ak=1.33\times 10^{-2}$ and $\epsilon \sqrt {\alpha _s\alpha _a}/\omega _+=4.58\times 10^{-3}$ . Thus, both the weak-slope and slow-modulation conditions are satisfied. If $\mu$ and $\sigma$ were allowed to approach zero together with $\epsilon$ in such a way that (4.14) failed, a different distinguished asymptotic scaling would be required.

The system (4.8) has the standard three-wave interaction structure. With the present phase convention, the Manley–Rowe-type invariants may be written as

(4.15) \begin{align} |A_{+}|^2-|A_{-}|^2=\mathrm{const.}, \quad |A_{+}|^2+\frac {\alpha _s}{\alpha _a}|A|^2=\mathrm{const.}, \end{align}

together with the corresponding invariant obtained by interchanging $A_{+}$ and $A_{-}$ . These conservation laws provide a useful check on the signs and normalisation of the coupling coefficients. They also imply a bounded temporal dynamics for the spatially homogeneous triad.

For equal initial surface-wave amplitudes it is useful to set $p=A_{+}+A_{-}$ , $q=A_{+}-A_{-}$ . Then

(4.16) \begin{align} \dot {p}=\mathrm{i}\alpha _s p^\ast A,\quad \dot {q}=\mathrm{i}\alpha _s q^\ast A,\quad \dot {A}=\mathrm{i}\frac {\alpha _a}{4}(p^2-q^2). \end{align}

If $q(0)=0$ , then $q(T)=0$ , and the dynamics reduces to the standard two-amplitude exchange problem. Writing $p=se^{\mathrm{i}\varphi _s}$ and $A=re^{\mathrm{i}\varphi _a}$ gives the invariant

(4.17) \begin{align} s^2+4\frac {\alpha _s}{\alpha _a}r^2 = s_0^2+4\frac {\alpha _s}{\alpha _a}r_0^2, \end{align}

so the reduced temporal trajectories are elliptical and bounded (Bretherton Reference Bretherton1964; Craik Reference Craik1988). General initial conditions are described by the standard three-wave solution in terms of Jacobi elliptic functions (Craik Reference Craik1988, p. 138).

5. Spatial modulation and pressure estimate

When spatial modulation is allowed, the system of amplitude equations (4.8) modify into

(5.1) \begin{align} \begin{aligned} & \partial _T S_{\pm } \pm c_g\partial _X S_{\pm } = \mathrm{i}\alpha _{s}S^{\ast }_{\mp }A, \quad \partial _T A = \mathrm{i}\alpha _{a}S_{+}S_{-}, \end{aligned} \end{align}

where $X=\epsilon x$ , $S_\pm$ are the surface-wave envelopes, $A$ is the acoustic envelope and

(5.2) \begin{align} c_g=\frac {\mathrm{d}\omega _{+}}{\mathrm{d}\kappa } =\frac {1}{2}(1+3\sigma \kappa ^2)(\kappa +\sigma \kappa ^3)^{-1/2} \end{align}

is the surface-wave group velocity. We consider the equations above subject to the initial conditions

(5.3) \begin{align} S_{\pm }(X,0)=S_0e^{-\beta (X\pm \ell )^2}, \quad A(X,0)=0. \end{align}

Here, $S_0$ is the maximum initial amplitude of each capillary–gravity mode, $\beta$ measures the inverse squared of the Gaussian envelope and $\ell$ is the initial distance of each wavepacket centre from the collision point. The dimensional envelope width $\lambda _d$ is related to $\beta$ by $\lambda _d=h^2/(a\sqrt {\beta })$ .

The Gaussian envelopes are assumed to be slowly varying relative to the carrier wavelength, so that $\lambda _d\gg 2\pi h/\kappa$ . For the representative laboratory configuration considered below, $\lambda _{cg}=2.36\times 10^{-3}\,\mathrm{m}$ and $\lambda _d=5.00\times 10^{-2}\,\mathrm{m}$ , giving $\lambda _d/\lambda _{cg}=21.2$ ; the wavepackets therefore remain narrow-band relative to the carrier. If $\lambda_d$ were reduced to only a few carrier wavelengths, detuning and other near-resonant interactions would need to be retained.

We next non-dimensionalise the reduced envelope problem using the wavepacket collision length and time. This is an algebraic rescaling of the already-derived envelope equations and does not introduce a new asymptotic ordering. We set

(5.4a) \begin{align} X=\ell \tilde {X}, \quad T=\frac {\ell }{c_g}\tilde {T}, \quad S_{\pm }=\frac {c_g}{\ell \sqrt {\alpha _s\alpha _a}}\tilde {S}_{\pm }, \quad A=\frac {c_g}{\ell \alpha _s}\tilde {A}, \end{align}

(5.4b) \begin{align} \tilde {G}_0=\frac {S_0\ell \sqrt {\alpha _s\alpha _a}}{c_g}, \quad \tilde {\beta }=\beta \ell ^2. \end{align}

Dropping the tildes from the rescaled variables and parameters for readability, the previous system of equations becomes

(5.5) \begin{align} & \partial _{T} S_{\pm } \pm \partial _{X} S_{\pm } = \mathrm{i} S^{\ast }_{\mp }A, && S_{\pm }(X, 0) = G_0e^{-\beta (X\pm 1)^2},\nonumber \\[3pt] & \partial _{T}A = \mathrm{i} S_{+} S_{-}, && A(X, 0) = 0. \end{align}

We now seek a leading-order solution in the weak-generation regime. During the short collision interval near $T=1$ , pump depletion is neglected and the surface-wave amplitudes are approximated by

(5.6) \begin{align} S_{+}=G_0e^{-\beta (X-T+1)^2}, \quad S_{-}=G_0e^{-\beta (X+T-1)^2}. \end{align}

The acoustic amplitude generated by the undepleted packets is $\max |A|=G_0^2\sqrt {\pi /(2\beta )}$ . Neglecting pump depletion in the $S_\pm$ -equations is therefore self-consistent when this generated acoustic amplitude is small compared with the incident surface-wave amplitude $G_0$ , namely

(5.7) \begin{align} G_0\sqrt {\frac {\pi }{2\beta }}\ll 1 . \end{align}

This is a sufficient weak-generation condition; otherwise the full spatial system (5.1) must be integrated numerically. Eventually, we are left only with the amplitude equation for the acoustic mode which reads

(5.8) \begin{align} \partial _T A=\mathrm{i} G_0^2e^{-2\beta X^2}e^{-2\beta (T-1)^2}, \end{align}

(5.9) \begin{align} A(X,T)=\mathrm{i}\frac {G_0^2}{2}\sqrt {\frac {\pi }{2\beta }}e^{-2\beta X^2}[\mathrm{erf}(\sqrt {2\beta }(T-1))+1]. \end{align}

The maximum value predicted by this leading-order approximation is therefore $\max _{X,T}|A|=G_0^2\sqrt {\pi /(2\beta )}$ . An example of the three-mode amplitude closed-form solution is shown in figure 2. Parameters are chosen as $G_0 = 0.75$ and $ \beta = 20$ , giving $G_0\sqrt {({\pi }/({2\beta }))} = 0.21$ . For these illustrative parameters, $G_0\sqrt {\pi /(2\beta )}=0.21$ , so the weak-generation condition is only moderately satisfied. The example is therefore intended primarily to illustrate the spatio-temporal structure of the interaction; more stringent quantitative applications of the undepleted-pump approximation require $G_0\sqrt {\pi /(2\beta )}\ll 1$ .

Figure 2.

Spatio-temporal representation of the slowly varying mode amplitudes according to the approximate closed-form solutions (5.6)–(5.9). Parameters are $G_0=0.75$ and $\beta =20$ . Right-going capillary–gravity envelope $|S_{+}(X,T)|$ ; (b) left-going capillary–gravity envelope $|S_{-}(X,T)|$ ; (c) generated acoustic-mode amplitude $|A(X,T)|$ ; (d) initial spatial profiles of the two surface-mode envelopes. The rapidly oscillating capillary–gravity carrier waves are shown: the physical surface short-wavelength carriers are modulated by the plotted Gaussian envelopes, with $\lambda _{cg}\ll \lambda _d$ .

5.1. Dynamic pressure signature at the bottom of the tank

The most direct experimental signature of the interaction is the pressure perturbation at the bottom of the tank. We define $Q_b$ as the maximum acoustic pressure perturbation, relative to the hydrostatic background, produced by the resonantly generated acoustic mode at $z=-h$ . From the linearised Bernoulli relation, the dimensional pressure perturbation associated with the potential is $p'=-\rho _a\partial _t\varphi$ . Using the acoustic eigenfunction in (3.8), the corresponding acoustic pressure perturbation has the vertical dependence

(5.10) \begin{align} p'_a(X,z,t,T)=\mathcal{P}(X,T,t)\sin \left (\lambda _m\frac {z}{h}\right ), \end{align}

where $\mathcal{P}(X,T,t)$ contains the fast acoustic oscillation and its slowly varying envelope. Hence, the acoustic pressure perturbation vanishes at the free surface and has an antinode at the rigid bottom, since $|\sin \lambda _m|=1$ . For $m\gt 0$ , additional pressure nodes occur within the water column. In terms of the maximum bottom pressure $Q_b$ , the maximum pressure amplitude at depth $z$ is

(5.11) \begin{align} \max |p'_a(z)|=Q_b\left |\sin \left (\lambda _m\frac {z}{h}\right )\right |. \end{align}

In the deep-water resonant regime, the capillary–gravity eigenfunctions are exponentially small at the bottom, whereas the acoustic eigenfunction is $O(1)$ there. Thus, retaining only the acoustic contribution gives

(5.12) \begin{align} Q_b=\rho _a\left (a^2g\lambda _d\sqrt {\frac {\pi }{2}}\right )\left (\frac {8\alpha _a}{c_gh^2}\frac {\sqrt {\mu }}{\lambda _m}(1+\sigma \kappa ^2)^2\right )\!. \end{align}

The estimate is independent of the arbitrary eigenfunction normalisation: a rescaling of the acoustic eigenfunction rescales $A$ , $\alpha _a$ and $G_0$ , but leaves the dimensional pressure $Q_b$ unchanged. Equation (5.12) shows explicitly that, for fixed fluid depth and the axial acoustic mode, $Q_b\propto a^2\lambda _d$ . This quadratic dependence on the initial surface amplitude should be useful experimentally, because it distinguishes the resonant acoustic signal from linear pressure fluctuations.

Figure 3.

Maximum acoustic pressure perturbation at the bottom of the tank, $Q_b$ , as a function of the dimensional initial envelope width $\lambda _d$ and water depth $h$ . The calculation uses $a=5\,{\unicode[Arial]{x03BC}} \mathrm{m}$ and $m=0$ (a), $m=1$ (b), $m=2$ (c). For other amplitudes in the weakly nonlinear regime, the pressure rescales as $Q_b\propto m=1$ (b), $m=2$ (c). For other amplitudes in the weakly nonlinear regime, the pressure rescales as $Q_b\propto a^2\lambda _d$ .

In figure 3 the maximum dynamic pressure achievable at the bottom of the tank is shown for $a=5 \ {\unicode[Arial]{x03BC}} \mathrm{m}$ and vertical mode numbers $m=0, 1, 2$ . This conservative choice keeps $\epsilon =a/h$ very small throughout the plotted range. Larger surface-wave amplitudes increase the generated acoustic pressure according to $Q_b\propto a^2$ . However, the admissible increase in $a$ is constrained not only by $\epsilon =a/h\ll 1$ , but also by the surface-wave slope condition $ak=\epsilon \kappa \ll 1$ , which is more restrictive for the short resonant capillary–gravity waves considered here. We therefore retain $a=5\,{\unicode[Arial]{x03BC}} \mathrm{m}$ in the representative quantitative estimates. A significant change in the orders of magnitude of the dynamic pressure is appreciable even for different mode numbers, as shown by the three cases in figure 3. The direct linear pressure from the capillary–gravity waves at the bottom is exponentially small, proportional to $e^{-\kappa }$ , so the bottom signal should be dominated by the generated acoustic mode.

The required carrier frequencies are high for conventional wavemakers. At resonance, each capillary–gravity wave has dimensional frequency

(5.13) \begin{align} f_{cg}=\frac {\lambda _m c}{4\pi h}, \end{align}

while the horizontally uniform acoustic mode has $f_a=2f_{cg}$ . For $m=0$ , this gives $f_{cg}\approx 938$ , $375$ and $188\,\mathrm{Hz}$ at $h=0.2$ , $0.5$ and $1.0\,\mathrm{m}$ , respectively, with corresponding acoustic frequencies of approximately $1875$ , $750$ and $375\,\mathrm{Hz}$ .

However, since the objective of the present work is to establish the leading-order inviscid resonant interaction and quantify the acoustic pressure generated by this mechanism only, incorporating viscous damping into the amplitude equations constitutes an interesting extension of the present work, but lies beyond the scope of the current theoretical development. Therefore, in keeping with the weakly nonlinear asymptotic formulation, viscous effects are neglected at leading order.

High-frequency localised or parametric forcing would lead to an externally forced problem and, depending on the forcing mechanism, may introduce a Mathieu-type dynamics associated with Faraday excitation. Such forcing could also provide a practical route for seeding or enhancing the horizontally uniform acoustic mode participating in the present triad. The resulting system would, however, contain explicit forcing and damping terms in addition to the self-consistent nonlinear coupling considered here. Analysing that externally forced extension lies beyond the scope of the present study, but constitutes a natural direction for experimental implementation.

The dimensional and dimensionless parameters for a representative laboratory configuration are summarised in table 1. We take $h=1\,\mathrm{m}$ , $a=5\,{\unicode[Arial]{x03BC}} \mathrm{m}$ , $\lambda _d=0.05\,\mathrm{m}$ and the first vertical acoustic mode $m=0$ . For this representative configuration, the viscous attenuation length $L_\nu \simeq 0.046\,\mathrm{m}$ is comparable to the envelope width $\lambda _d=0.050\,\mathrm{m}$ . Viscous attenuation may therefore be quantitatively important in a laboratory realisation, and the pressure estimate should be interpreted as an inviscid leading-order value.