2.1. Fluid equations

The fluid velocity is described by a velocity potential $\phi _l(x,z,t)$ , such that

(2.1) \begin{equation} \boldsymbol{u} = \boldsymbol{\nabla }\phi _l . \end{equation}

The fluid is assumed inviscid and barotropic. Compressibility effects are characterised by the dimensionless parameter

(2.2) \begin{equation} \mu = \frac {g h}{c^2} \ll 1, \end{equation}

which compares the characteristic gravity-wave speed $\sqrt {gh}$ with the sound speed $c$ , which is assumed to be constant; the parameter $g$ is the acceleration due to gravity. This parameter naturally governs acoustic–gravity interactions in surface-wave problems (Longuet-Higgins Reference Longuet-Higgins1950; Kadri & Stiassnie Reference Kadri and Stiassnie2013).

For completeness, we present the fully nonlinear compressible potential-flow equation and the free-surface conditions that underlie the weakly nonlinear triad reduction. Combining continuity with Bernoulli yields the nonlinear interior equation (cf. Longuet-Higgins Reference Longuet-Higgins1950; Kadri & Akylas Reference Kadri and Akylas2016)

(2.3) \begin{equation} \phi _{l,tt}-c^2\boldsymbol{\nabla} ^2\phi _l-\partial _t\left (\frac 12|\boldsymbol{\nabla }\phi _l|^2\right )-\frac 12 \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }\left (|\boldsymbol{\nabla }\phi _l|^2\right )=0, \qquad -h\leqslant z\leqslant 0. \end{equation}

In addition, the standard kinematic and dynamic boundary conditions apply at the free surface $z=\eta (x,t)$ . For the weakly nonlinear analysis that follows, it is sufficient to enforce these conditions consistently through cubic order in the perturbation expansion. Expanding the exact kinematic and dynamic conditions about the mean level $z=0$ and eliminating $\eta$ order by order yields the following combined boundary condition for $\phi _l$ :

(2.4) \begin{align} &\phi _{l,tt}+g\phi _{l,z} +\frac {1}{c^2}\partial _t \left (|\boldsymbol{\nabla }\phi _l|^2\right ) -\partial _z \Bigg [ \frac {1}{c^2}\phi _{l,t}\big (\phi _{l,tt}+g\phi _{l,z}\big ) \Bigg ] +\frac {1}{2c^2} \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }\left (|\boldsymbol{\nabla }\phi _l|^2\right ) \notag\\&\quad -\partial _z \Bigg [ \frac {1}{c^2}\phi _{l,t} \partial _t \left (|\boldsymbol{\nabla }\phi _l|^2\right )\!\Bigg ] -\frac {1}{2}\partial _z \Bigg [\big (\phi _{l,tt}+g\phi _{l,z}\big ) \left (\frac {1}{c^2}|\boldsymbol{\nabla }\phi _l|^2-\frac {1}{c^4}\phi _{l,t}^{2}\right )\! \Bigg ] =0, \!\!\!\qquad z=0. \end{align}

In non-dimensional variables this combined condition (2.4) is given explicitly by (3.2a) of Kadri & Akylas (Reference Kadri and Akylas2016). In the present dimensional variables it can be written compactly as

(2.5) \begin{equation} \phi _{l,tt}+g\phi _{l,z} = \varepsilon \mathcal N_2[\phi _l]+\varepsilon ^2 \mathcal N_3[\phi _l] \qquad (z=0), \end{equation}

where $\varepsilon$ is the small wave-steepness/nonlinearity parameter,  $\mathcal N_2$ and $\mathcal N_3$ are the standard quadratic and cubic free-surface operators arising from the Taylor expansion and elimination of $\eta$ (Phillips Reference Phillips1960; Craik Reference Craik1985), and for the triad ordering adopted here they coincide with those used in Kadri & Akylas (Reference Kadri and Akylas2016).

Under weakly compressible conditions, the compressible potential-flow equations may be expanded asymptotically in $\mu$ . To leading order in the bulk, the velocity potential satisfies the linear acoustic-wave equation

(2.6) \begin{equation} \phi _{l,tt} - c^2 \boldsymbol{\nabla} ^2 \phi _l = 0, \qquad -h \leqslant z \leqslant 0 , \end{equation}

while nonlinear compressibility effects in the interior appear at higher order (Longuet-Higgins Reference Longuet-Higgins1950; Kadri & Akylas 2016). The bulk formulation can be written with or without an added $g\phi _{l,z}$ term depending on the potential convention (see, e.g. Longuet-Higgins Reference Longuet-Higgins1950); we adopt the convention consistent with Kadri & Akylas (Reference Kadri and Akylas2016) and with the elastic-bottom eigenproblem in Eyov et al. (Reference Eyov, Klar, Kadri and Stiassnie2013).

At the free surface $z=\eta (x,t)$ , the exact kinematic and dynamic boundary conditions apply. Linearisation yields the combined free-surface condition

(2.7) \begin{equation} \phi _{l,tt} + g\phi _{l,z} = 0, \qquad z=0 . \end{equation}

In the weakly nonlinear derivation in § 5, the full kinematic and dynamic free-surface conditions at $z=\eta (x,t)$ are expanded consistently through $O(\varepsilon ^3)$ following standard procedures (Phillips Reference Phillips1960; Craik Reference Craik1985; Kadri & Akylas Reference Kadri and Akylas2016). For brevity, we display only the linear combined condition (2.7) here, while the quadratic and cubic contributions enter the $O(\varepsilon ^3)$ problem through the forcing terms introduced in § 5.4.

2.2. Elastic half-space

The fluid overlies an elastic half-space occupying $z \leqslant -h$ , assumed homogeneous, isotropic and linearly elastic. The solid is characterised by density $\rho _s$ and Lamé parameters $\lambda _s$ and $\mu _s$ , or equivalently by the compressional and shear wave speeds

(2.8) \begin{equation} C_p = \sqrt {\frac {\lambda _s + 2\mu _s}{\rho _s}}, \qquad C_s = \sqrt {\frac {\mu _s}{\rho _s}} . \end{equation}

Elastic coupling at a fluid–solid interface is known to modify both acoustic and gravity-wave propagation (Biot Reference Biot1952). Following standard treatments in hydroacoustics and elastic-wave theory (Achenbach Reference Achenbach1973; Eyov et al. Reference Eyov, Klar, Kadri and Stiassnie2013), we impose at $z=-h$ continuity of vertical velocity and normal traction, together with vanishing shear traction. Eliminating the elastic variables yields an effective boundary condition acting on the fluid potential

(2.9) \begin{equation} \mathcal{B}_e(\omega ,\kappa )[\phi _l]\equiv \phi _{l,z}-\mathcal{Z}(\omega ,\kappa )\phi _l = 0, \qquad z=-h , \end{equation}

where $\mathcal{Z}(\omega ,\kappa )$ is an impedance function determined by the elastic properties of the half-space (see Appendix A for the explicit expression). Here $\omega$ denotes angular frequency and $\kappa$ the horizontal wavenumber of the acoustic–gravity mode.

A central point for what follows is that the bottom operator $\mathcal{B}_e$ depends explicitly on the spectral parameters $(\omega ,\kappa )$ through $\mathcal{Z}(\omega ,\kappa )$ . This explicit dependence is responsible for an additional interfacial contribution in the weakly nonlinear solvability condition derived in § 5. In the rigid-bottom limit one has $\mathcal{Z}\to 0$ , so that (2.9) reduces to the classical no-penetration condition $\phi _{l,z}=0$ at $z=-h$ .

Note that $\mathcal{Z}(\omega ,\kappa )$ depends smoothly on $(\omega ,\kappa )$ in the neighbourhood of the resonant acoustic–gravity mode. Thus, the impedance does not alter the structure of the weakly nonlinear theory; it only affects the coefficient values through $\mathcal{Z}$ and its derivatives evaluated on the resonant branch.

3.1. Eigenvalue problem and explicit eigenfunction

We seek time-harmonic solutions

(3.1) \begin{equation} \phi _l(x,z,t) = \varPhi (z)\mathrm{e}^{\mathrm{i}(\kappa x - \omega t)} +\text{c.c.}, \end{equation}

where $\kappa$ is the horizontal wavenumber, $\omega$ is the (angular) frequency of the acoustic–gravity mode in the coupled fluid–solid system, and c.c. denotes the complex conjugate. For progressive non-evanescent acoustic modes $\kappa$ is both real and positive. The bulk (2.6) yields

(3.2) \begin{equation} \varPhi _{zz} + \alpha ^2 \varPhi = 0, \qquad \alpha ^2 = \frac {\omega ^2}{c^2} - \kappa ^2, \qquad -h\leqslant z\leqslant 0. \end{equation}

We take the branch $\alpha =\sqrt {\omega ^2/c^2-\kappa ^2}$ with $\Re (\alpha )\geqslant 0$ . When $\alpha$ is real the vertical structure in the fluid is oscillatory (propagating acoustic–gravity mode), whereas $\alpha =\mathrm{i}|\alpha |$ corresponds to evanescence in $z$ , i.e. gravity mode. The linear free-surface condition (2.7) becomes

(3.3) \begin{equation} g\varPhi _z - \omega ^2 \varPhi = 0, \qquad z=0, \end{equation}

while the elastic solid interface condition (2.9) becomes

(3.4) \begin{equation} \varPhi _z - \mathcal{Z}(\omega ,\kappa )\varPhi = 0, \qquad z=-h. \end{equation}

Solving (3.2) subject to (3.4) gives the explicit representation

(3.5) \begin{equation} \varPhi _e(z) = C\Bigg [ \cos \alpha (z+h) + \frac {\mathcal Z(\omega ,\kappa )}{\alpha } \sin \alpha (z+h) \Bigg ], \qquad -h\leqslant z\leqslant 0, \end{equation}

so that $\varPhi _e(-h)=C$ . A convenient normalisation is $\varPhi _e(0)=1$ , giving

(3.6) \begin{equation} C= \Bigg [ \cos (\alpha h)+\frac {\mathcal Z(\omega ,\kappa )}{\alpha }\sin (\alpha h) \Bigg ]^{-1}. \end{equation}

3.2. Elastic solid dispersion relation and rigid limit

Imposing the free-surface condition (3.3) on (3.5) yields the elastic solid dispersion relation (see Appendix A)

(3.7) \begin{align} D_e(\omega ,\kappa ;h)&\equiv g\Big [-\alpha \sin (\alpha h)+\mathcal Z(\omega ,\kappa )\cos (\alpha h)\Big ] -\omega ^2\Big [\cos (\alpha h)+\frac {\mathcal Z(\omega ,\kappa )}{\alpha }\sin (\alpha h)\Big ]\notag\\& =0. \end{align}

In the rigid-bottom limit $\mathcal Z\to 0$ , (3.7) reduces to

(3.8) \begin{equation} D_r(\omega ,\kappa ;h)\equiv -g\alpha \sin (\alpha h)-\omega ^2\cos (\alpha h)=0, \end{equation}

which is the classical rigid-bottom acoustic–gravity dispersion relation used in previous triad analyses. It will be useful later to write

(3.9) \begin{equation} D_e(\omega ,\kappa ;h)=D_r(\omega ,\kappa ;h)+\mathcal Z(\omega ,\kappa )Q(\omega ,\kappa ;h), \end{equation}

where the multiplier $Q$ is explicitly

(3.10) \begin{equation} Q(\omega ,\kappa ;h) = g\cos (\alpha h)-\frac {\omega ^2}{\alpha }\sin (\alpha h). \end{equation}

3.3. Cutoff and admissibility: rigid versus elastic bottoms

In the present lossless setting we focus on real $(\omega ,\kappa )$ solutions of the dispersion relations. Under rigid-bottom conditions, admissible propagating acoustic–gravity modes correspond to real roots of $D_r(\omega ,\kappa ;h)=0$ with real $\alpha$ (equivalently $\omega ^2/c^2\gt \kappa ^2$ ), while $\alpha =\mathrm{i}|\alpha |$ yields evanescent vertical structure. In shallow configurations, the rigid-bottom acoustic–gravity branch may terminate in the sense that $D_r(\omega ,\kappa ;h)=0$ admits no real propagating roots for given $(\omega ,\kappa )$ ; this is the rigid cutoff referred to in earlier triad analyses.

A convenient mathematical characterisation of a cutoff point $(\omega _c,\kappa _c;h_c)$ is a turning-point (degeneracy) condition

(3.11) \begin{equation} D_r(\omega _c,\kappa _c;h_c)=0, \qquad \partial _\omega D_r(\omega _c,\kappa _c;h_c)=0, \end{equation}

at which the propagating branch ceases to continue as a real-valued function of depth (or frequency). Physically, the rigid no-penetration condition inhibits vertical motion at $z=-h$ and can prevent the existence of a trapped propagating acoustic–gravity mode when the water column is too shallow, so that the acoustic member of the triad becomes evanescent and resonance cannot be satisfied with real $(\omega ,\kappa )$ .

For the first rigid acoustic–gravity branch, a simple way to view the cutoff is to note that in the rigid limit the vertical structure must satisfy a node/antinodal constraint imposed by $\varPhi _z=0$ at $z=-h$ together with the free-surface condition at $z=0$ . When the water column is too shallow, these constraints cannot be satisfied by a real (oscillatory) vertical wavenumber $\alpha$ at a prescribed frequency, so the admissible propagating root disappears and the mode becomes vertically evanescent. In particular, for $\kappa \approx 0$ the first cutoff occurs near $\alpha h=\pi /2$ , giving the classical estimate $\omega _c\approx \pi c/(2h)$ used in figure 1 (also see Stiassnie Reference Stiassnie2010).

Figure 1.

Rigid-bottom versus elastic-bottom acoustic–gravity dispersion at $h=0.30,\mathrm{m}$ (parameters in § 3.4). (a) Phase speed $\omega /\kappa$ versus $\omega$ for the first acoustic–gravity branch: the rigid-bottom mode (dashed) exists only for $\omega \gt \omega _c=\pi c/(2h)$ (vertical dotted line), whereas elastic coupling (solid) supports an interface-guided branch below $\omega _c$ . (b) Elastic branch shown as $\kappa (\omega )$ on logarithmic axes in the laboratory-relevant low-frequency range; the vertical dash-dotted line marks the gravity–capillary cross-over $\omega _{gc}$ .

Figure 1. Long description

Two line graphs depict phase speed and wavenumber as functions of angular frequency, comparing rigid and elastic bottom conditions. Panel A: A line graph shows phase speed (c) in meters per second (m/s) on the vertical axis versus angular frequency (ω) in radians per second (rad/s) on the horizontal axis. The graph includes four lines: a solid line for elastic bottom, a dashed line for rigid bottom, a dotted line for rigid cut-off, and a dash-dotted line for the gravity-capillary limit. The elastic bottom line shows a distinct behavior below a certain frequency, while the rigid bottom line exists only above a specific frequency. Panel B: Another line graph shows wavenumber (κ) in per meter (1/m) on the vertical axis versus angular frequency (ω) in radians per second (rad/s) on the horizontal axis, both on logarithmic scales. The graph includes a solid line representing the elastic branch and a vertical dash-dotted line marking the gravity-capillary cross-over. The elastic branch shows a clear trend in the low-frequency range relevant to laboratory settings.

Elastic solid coupling modifies admissibility through the impedance condition (3.4), which changes the boundary-value problem and hence the dispersion relation (3.7). In particular, the coupled system may admit propagating or interface-guided acoustic–gravity modes (with energy shared between the fluid and the solid) in parameter regimes where the rigid problem supports only evanescence. This extension of the admissible spectrum is the primary mechanism by which elasticity enlarges the parameter space of resonant triads and, in the examples below, removes the rigid cutoff in frequency (or depth) relevant to shallow-water and laboratory-scale configurations.

3.4. Illustrative shallow-water parameters and dispersion comparison

To quantify frequency scales relevant to admissibility at laboratory depths, we take $h=0.30\,\mathrm{m}$ and the water–elastic solid half-space parameters used by Eyov et al. (Reference Eyov, Klar, Kadri and Stiassnie2013): $c=1480\,\mathrm{m\,s^{-1}}$ and $\rho _l=1000\,\mathrm{kg\,m^{-3}}$ in the fluid, over a stiff (granite-like) elastic solid half-space with $\rho _s=2750\,\mathrm{kg\,m^{-3}}$ , $C_s=3550\,\mathrm{m\,s^{-1}}$ and $C_p=6300\,\mathrm{m\,s^{-1}}$ . For these values, the weak-compressibility parameter is $\mu =gh/c^2\approx 1.3\times 10^{-6}$ .

At this depth the rigid-bottom acoustic–gravity branch is effectively inaccessible in the laboratory-frequency band: the classical rigid cutoff $\omega _c=\pi c/(2h)\approx 7.75\times 10^3\,\mathrm{rad\,s^{-1}}$ (equivalently $f_c\approx 1.23\,\mathrm{kHz}$ ) lies far above typical gravity-wave forcing frequencies $O(1)$ – $O(10)\,\mathrm{Hz}$ . This is evident in figure 1(a), where the rigid mode (dashed) appears only for $\omega \gt \omega _c$ , whereas elastic solid coupling supports an interface-guided branch (solid) that continues smoothly below $\omega _c$ and therefore admits a propagating acoustic–gravity participant in the laboratory regime. The low-frequency behaviour of this elastic solid branch is highlighted in figure 1(b), which plots $\kappa (\omega )$ on logarithmic axes over the experimentally relevant range. In particular, the elastic solid branch persists below the gravity–capillary cross-over $\omega _{gc}\approx 85.1\,\mathrm{rad,s^{-1}}$ ( $f_{gc}\approx 13.5\,\mathrm{Hz}$ ), so the surface-wave components can remain ordinary gravity waves while the acoustic–gravity member remains admissible. Together, figures 1(a) and 1(b) illustrate the key experimental implication: elasticity removes the rigid-bottom cutoff obstruction and makes triad configurations feasible at laboratory scales, without the need to enter the gravity-capillary regime.

4.1. Triad structure

We consider resonant interactions involving two free-surface gravity waves and one acoustic–gravity wave. The two surface wavetrains are taken to be nearly monochromatic, with (possibly distinct) horizontal wavenumbers $k_+$ and $k_-$ and corresponding frequencies $\omega _+$ and $\omega _-$ . The third member of the triad is an acoustic–gravity mode with horizontal wavenumber $\kappa$ and frequency $\omega$ .

Resonance requires approximate satisfaction of the wavenumber- and frequency-matching conditions

(4.1) \begin{equation} k_+ + k_- = \kappa , \qquad \omega _+ + \omega _- = \omega - \mu \beta , \end{equation}

where $\beta =O(1)$ measures detuning, where $\beta =0$ corresponds to exact resonance. The resonance conditions (4.1) are written for a two-dimensional, long-crested setting in which all waves share the same horizontal direction $x$ ; hence only the horizontal wavenumbers appear. Energy exchange with the elastic half-space does not require an additional horizontal component: it is mediated by the vertical eigenstructure and the impedance condition at $z=-h$ , which couples the fluid pressure field to evanescent (or guided) elastic fields in the substrate.

The detuning is written as $\mu \beta$ because the interaction is studied on the slow time $T=\mu t$ (with $\varepsilon \sim \mu ^{1/2}$ ), so only an $O(\mu )$ frequency mismatch produces an $O(1)$ phase drift on the modulation time scale. In the nearly counter-propagating configuration of primary interest, $k_+\approx -k_-$ so that $\kappa =k_++k_-$ is small compared with $|k_\pm |$ , consistent with the long horizontal scale of the acoustic–gravity member. Other triad geometries are possible in principle, but the near counter-propagating case is the most relevant for laboratory forcing and for the classic microseism setting where two opposing surface wavetrains generate the compressional response.

4.2. Admissibility under rigid and elastic bottoms

Under rigid-bottom conditions, the admissibility of acoustic–gravity triads is constrained by the dispersion relation of the acoustic mode. In shallow water, the acoustic–gravity branch may become evanescent, or cease to exist altogether as a propagating mode. In such cases, the resonance conditions (4.1) cannot be satisfied with real $\omega$ and $\kappa$ , and the triad interaction is suppressed.

By altering the dispersion relation $\omega (\kappa )$ of the acoustic–gravity mode, elastic coupling modifies the above picture in two distinct ways. First, it allows for new propagating elastic or interface-guided modes that become admissible in parameter regimes where the rigid-bottom problem supports only evanescent solutions. Second, it changes the set of wavenumber pairs $(k_+,k_-)$ for which resonance is possible.

As a result, elasticity enlarges the admissible parameter space for resonant triads, particularly in shallow-water configurations. This observation has important implications for both laboratory experiments and field-scale applications, as it suggests that elastic substrates can be used to tune the system into resonance without requiring large fluid depths.

5.1. Multiple-scale expansion

We introduce a small wave-steepness parameter $\varepsilon \ll 1$ and expand the velocity potential as

(5.1) \begin{equation} \phi _l = \varepsilon \phi ^{(1)} +\varepsilon ^2 \phi ^{(2)} +\varepsilon ^3 \phi ^{(3)} +\cdots . \end{equation}

Slow modulation associated with near-resonant interactions is captured by the slow time $T$ , and we adopt the distinguished scaling $\varepsilon \sim \mu ^{1/2}$ , which ensures that quadratic nonlinear forcing balances slow-time modulation at leading order (Kadri & Akylas Reference Kadri and Akylas2016). Under this ordering, nonlinear terms in the bulk equations are asymptotically smaller than the nonlinear contributions arising from the free-surface boundary condition. Consequently, at the order relevant to triad resonance the interior correction is governed by the homogeneous linear equation associated with the acoustic–gravity eigenproblem, while nonlinearity enters through the free-surface boundary conditions and appears as resonant forcing at $O(\varepsilon ^3)$ . To allow for weak spatial modulation, we also introduce the slow horizontal variable $X=\mu ^{2}x$ , and hence take the wave envelopes to depend on $(X,T)$ . Accordingly, derivatives act as

(5.2) \begin{equation} \partial _t \mapsto \partial _t + \mu \partial _T, \qquad \partial _x \mapsto \partial _x + \mu ^{2}\partial _X, \end{equation}

where $\partial _t,\partial _x$ on the right-hand side denote differentiation with respect to fast scales, and $\partial _T,\partial _X$ represent differentiation with respect to slow scales.

5.2. Leading-order solution

At $O(\varepsilon )$ , the solution is taken as a superposition of two surface gravity wavetrains and one acoustic–gravity mode

(5.3) \begin{equation} \phi ^{(1)}=\phi _g^+ + \phi _g^- + \phi _e . \end{equation}

The gravity-wave components are written as

(5.4) \begin{equation} \phi _g^\pm = S_\pm (X,T)\varPhi _g^\pm (z) \mathrm{e}^{\mathrm{i}(k_\pm x-\omega _\pm t)} +\text{c.c.}, \end{equation}

where $S_\pm (X,T)$ are slowly varying complex amplitudes. For each wavetrain, the vertical structure $\varPhi _g^\pm (z)$ is the standard linear gravity-wave eigenfunction associated with $(k_\pm ,\omega _\pm )$ for the chosen depth and bottom model. In the nearly counter-propagating configuration of interest, $k_+\approx -k_-$ and the two gravity-wave frequencies are close, but they are not assumed identical a priori. In the weakly compressible regime $\mu \ll 1$ , and following Kadri & Akylas (Reference Kadri and Akylas2016), we take the two surface gravity wavetrains to be the classical incompressible deep-water modes. Accordingly, $\varPhi _g^\pm$ satisfy the standard Laplace eigenproblem in $z\lt 0$ with the linear free-surface condition at $z=0$ and decay as $z\to -\infty$ . For consistency with the finite-depth coupled problem, we assume $k_\pm h\gg 1$ , so that the gravity-wave eigenfunctions are exponentially small at $z=-h$ and bottom compliance affects $\phi _g^\pm$ only beyond the order retained; elasticity is therefore retained only in the acoustic–gravity eigenproblem that closes the triad. More precisely, taking $k_\pm h=O(1/\mu )\gg 1$ gives $\varPhi _g^\pm (-h)=O (\mathrm{e}^{-|k_\pm |h} )=O (\mathrm{e}^{-1/\mu } )$ , so seabed (bottom boundary) elasticity modifies the gravity-wave dispersion/structure only at exponentially small order, beyond the $O(\mu )$ accuracy retained here. Accordingly, the surface-wave frequencies satisfy the deep-water dispersion relation

(5.5) \begin{equation} \omega _\pm ^2 = g|k_\pm |, \end{equation}

and the corresponding vertical structure may be taken as $\varPhi _g^\pm (z)\propto \mathrm{e}^{|k_\pm |z}$ in the fluid region.

The acoustic–gravity component is written as

(5.6) \begin{equation} \phi _e = A(X,T)\varPhi _e(z) \mathrm{e}^{\mathrm{i}(\kappa x-\omega t)} +\text{c.c.}, \end{equation}

where $A(X,T)$ is the slowly varying amplitude of the acoustic–gravity mode and $\varPhi _e(z)$ satisfies the elastic eigenvalue problem derived in § 3. Here, the complex envelopes $S_\pm$ and $A$ vary on the slow scales $(X,T)$ .

5.3. Second-order fields

At $O(\varepsilon ^2)$ , the governing equations are forced by quadratic products of the leading-order fields. The resulting solution consists of bound waves and non-resonant components. No secular terms arise at this order, and the explicit form of $\phi ^{(2)}$ is not required; only its contribution to resonant forcing at $O(\varepsilon ^3)$ enters the amplitude equations.

5.4. Third-order problem and solvability condition

At $O(\varepsilon ^3)$ the governing equations admit resonant forcing at the acoustic–gravity phase $\mathrm{e}^{\mathrm{i}(\kappa x-\omega t)}$ . The corresponding correction is written as

(5.7) \begin{equation} \phi ^{(3)} = F(z)\mathrm{e}^{\mathrm{i}(\kappa x-\omega t)} + \text{c.c.}, \end{equation}

where $F(z)$ satisfies the forced boundary-value problem

(5.8) \begin{equation} F_{zz}+\alpha ^2 F = R_1(z), \qquad -h\lt z\lt 0. \end{equation}

Here, $R_1(z)$ collects quadratic products of the leading-order fields and contributions arising from slow-time modulation.

At the free surface $z=0$ , the inhomogeneous boundary condition takes the form

(5.9) \begin{equation} \big (gF_z-\omega ^2 F\big )\big |_{z=0}=R_2, \end{equation}

where $R_2$ arises from quadratic free-surface nonlinearities and from slow-time derivatives.

Since $\mathcal B_e$ depends explicitly on the spectral parameters, the boundary operator acquires an $O(\mu )$ correction from the same slow detuning already introduced in the resonance condition (4.1). Thus, rather than introducing a separate corrected frequency, we let the mismatch enter through the acoustic-phase forcing at frequency $\omega -\mu \beta$ , where $\beta =O(1)$ is the detuning parameter. Linearising the elastic boundary operator about the resonant acoustic–gravity mode then gives

(5.10) \begin{align} \mathcal B_e(\omega -\mu \beta ,\kappa )[\varPhi _e+\mu F] &=\mathcal B_e(\omega ,\kappa )[\varPhi _e] +\mu \mathcal B_e(\omega ,\kappa )[F] -\mu \beta \left (\partial _\omega \mathcal B_e\right )(\omega ,\kappa )[\varPhi _e]\notag\\&\quad +O(\mu ^2). \end{align}

Since $\mathcal B_e(\omega ,\kappa )[\varPhi _e]=0$ by the linear eigenproblem, the $O(\mu )$ balance yields the effective boundary condition for $F$

(5.11) \begin{equation} F_z-\mathcal Z(\omega ,\kappa )F=-\beta \,\partial _\omega \mathcal Z(\omega ,\kappa )\varPhi _e, \qquad z=-h, \end{equation}

or equivalently

(5.12) \begin{equation} \mathcal B_e(\omega ,\kappa )[F] =\beta \left (\partial _\omega \mathcal B_e(\omega ,\kappa )\right )[\varPhi _e], \end{equation}

since $\mathcal B_e(\omega ,\kappa )[\phi ]=\phi _z-\mathcal Z(\omega ,\kappa )\phi$ .

For fixed $(\omega ,\kappa )$ and real (lossless) impedance, the vertical Sturm–Liouville operator with the boundary condition (3.4) is self-adjoint under the standard inner product, so the adjoint eigenfunction is proportional to $\varPhi _e$ . Multiplying (5.8) by $\varPhi _e$ , subtracting the homogeneous equation multiplied by $F$ and integrating by parts over $-h\lt z\lt 0$ yields the Green-type identity

(5.13) \begin{equation} \int _{-h}^{0} \varPhi _e R_1\mathrm{d} z = \Big [\varPhi _e F_z-\varPhi _{e,z}F\Big ]_{-h}^{0}. \end{equation}

The contribution at the free surface reduces to

(5.14) \begin{equation} \Big (\varPhi _e F_z-\varPhi _{e,z}F\Big )\Big |_{z=0} = \frac {1}{g}\varPhi _e(0)R_2, \end{equation}

which coincides with the rigid-bottom case.

Using (5.11) in the boundary term of the Green identity shows that the interface contribution does not vanish and is proportional to the slow detuning

(5.15) \begin{equation} -\Big (\varPhi _e F_z-\varPhi _{e,z}F\Big )\Big |_{z=-h} = \mathcal I_e, \end{equation}

where

(5.16) \begin{equation} \mathcal I_e = \beta \big (\partial _\omega \mathcal{B}_e(\omega ,\kappa )\big )[\varPhi _e] \varPhi _e(-h), \end{equation}

where $\beta =O(1)$ is the same detuning parameter introduced in the resonance condition (4.1), so that the actual frequency mismatch enters at order $\mu \beta$ on the slow modulation time scale. Any contribution proportional to $\partial _\kappa \mathcal{B}_e$ can arise only through spatial modulation of the phase, i.e. through $\kappa =\kappa _0+O(\mu ^{2})$ induced by dependence on $X=\mu ^{2}x$ . Under this distinguished scaling, the resulting $\partial _\kappa \mathcal{B}_e$ term enters one order higher than the $O(\mu )$ detuning contribution and is therefore neglected at the present order.

The solvability condition governing resonant-triad interactions over an elastic half-space thus takes the form

(5.17) \begin{equation} \int _{-h}^{0} \varPhi _e R_1\mathrm{d} z = \frac {1}{g}\varPhi _e(0)R_2 + \mathcal I_e. \end{equation}

5.5. Triad equations and elastic normalisation

The solvability condition (5.17) provides the slow-time evolution of the acoustic–gravity amplitude $A(X,T)$ and the two gravity-wave envelopes $S_\pm (X,T)$ . The passage from (5.17) to the modulation system is standard: after extracting the coefficients of $\mathrm{e}^{\mathrm{i}(\kappa x-\omega t)}$ in $R_1$ and $R_2$ , the slow-time terms generated by $\partial _t\mapsto \partial _t+\mu \partial _T$ produce the factor $\mathcal N_e\partial _T A$ , while the quadratic interaction of the two gravity waves produces the forcing term $\mathcal P S_+S_-$ . Carrying out the analogous projection for the gravity-wave phases yields the coupled system (5.18). The detuning parameter $\beta$ is the same $O(\mu )$ frequency mismatch introduced in (4.1). The modulation equations take the canonical resonant-triad form with weak spatial modulation

(5.18a) \begin{align} \partial _T A + c_e\partial _X A &= \mathrm{i} \gamma _e A + \delta _e S_+S_-, \end{align}

(5.18b) \begin{align} \partial _T S_\pm + c_{g\pm }\partial _X S_\pm &= -\frac {\delta _e}{2}AS_\mp ^\ast . \end{align}

Here, $c_{g\pm }=\partial \omega _\pm /\partial k_\pm$ are the group velocities of the two surface gravity wavetrains (for the chosen depth and bottom model), and $c_e=\partial \omega /\partial \kappa$ is the group velocity of the elastic acoustic–gravity mode, with $\omega =\omega (\kappa )$ determined implicitly by the elastic dispersion relation $D_e(\omega ,\kappa ;h)=0$ . In particular, differentiating $D_e(\omega (\kappa ),\kappa ;h)=0$ yields

(5.19) \begin{equation} c_e=\frac {\partial \omega }{\partial \kappa } =-\frac {\partial _\kappa D_e}{\partial _\omega D_e}, \end{equation}

evaluated on the acoustic–gravity branch.

The detuning parameter $\beta$ is the same $O(\mu )$ frequency mismatch introduced in (4.1). It enters the acoustic-mode equation as an additive phase-mismatch contribution, so that

(5.20) \begin{equation} \delta _e=\frac {\mathcal P}{\mathcal N_e}, \qquad \gamma _e=\frac {\mathcal D}{\mathcal N_e}+\beta , \end{equation}

where $\mathcal P$ is the quadratic projection of the gravity–gravity forcing onto the acoustic mode and $\mathcal D$ collects linear dispersive corrections. For completeness, the coefficients $\mathcal P$ and $\mathcal D$ can be written directly in terms of the resonant forcing in (5.8)–(5.9). In the lossless case considered here the vertical operator is self-adjoint, so the adjoint mode may be taken as $\varPhi _e^\ast$ (and $\varPhi _e^\ast =\varPhi _e$ for real eigenfunctions). Denoting by $R_1^{(\textit{res})}(z)$ and $R_2^{(\textit{res})}$ the components of $(R_1,R_2)$ proportional to $\mathrm e^{\mathrm{i}(\kappa x-\omega t)}$ , we define

(5.21) \begin{align} \mathcal P &= \int _{-h}^{0} \varPhi _e^\ast (z)R_1^{(\mathrm{res})}(z)\mathrm{d} z +\frac {1}{g}\varPhi _e^\ast (0)R_2^{(\mathrm{res})}, \notag\\ \mathcal D &= \int _{-h}^{0} \varPhi _e^\ast (z)R_{1}^{(\textit{disp})}(z)\mathrm{d} z +\frac {1}{g}\varPhi _e^\ast (0)R_{2}^{(\textit{disp})}, \end{align}

where $(R_{1}^{(\textit{disp})},R_{2}^{(\textit{disp})})$ denote the parts of the $O(\varepsilon ^3)$ forcing generated by slow modulation (including the $T$ -derivative terms) and other linear dispersive corrections on the chosen scaling. Elasticity enters the triad dynamics only through the acoustic-mode normalisation $\mathcal N_e$ . With the eigenfunction $\varPhi _e(z)$ normalised by $\varPhi _e(0)=1$ , the normalisation decomposes as

(5.22) \begin{equation} \mathcal N_e=\mathcal N_{\textit{fs}}+\mathcal N_{\textit{int}}, \end{equation}

where the classical free-surface contribution is

(5.23) \begin{equation} \mathcal N_{\textit{fs}} = \int _{-h}^{0}\frac {2\omega }{c^2}|\varPhi _e(z)|^2\mathrm{d} z + \frac {2\omega }{g} |\varPhi _e(0)|^2 = \int _{-h}^{0}\frac {2\omega }{c^2}|\varPhi _e(z)|^2\mathrm{d} z + \frac {2\omega }{g}, \end{equation}

and the elastic interface contribution, arising from the explicit $(\omega ,\kappa )$ -dependence of the impedance boundary operator (2.9), is

(5.24) \begin{equation} \mathcal N_{\textit{int}} = 2\omega \frac {\partial \mathcal Z}{\partial \omega }(\omega ,\kappa ) |\varPhi _e(-h)|^2. \end{equation}

The term $\mathcal N_{\textit{int}}$ is the direct counterpart of the interface contribution $\mathcal I_e$ in (5.16) for impedance-type boundary conditions. In the rigid-bottom limit $\mathcal Z\to 0$ smoothly (hence $\partial _\omega \mathcal Z\to 0$ ), so $\mathcal N_{\textit{int}}\to 0$ and $\mathcal N_e\to \mathcal N_{\textit{fs}}$ .

5.6. Dispersion-derivative identity

An equivalent characterisation of the normalisation follows directly from the elastic dispersion relation $D_e(\omega ,\kappa ;h)=0$ . Differentiating $D_e$ with respect to $\omega$ , and using the homogeneous eigenproblem for $\varPhi _e$ , yields the identity

(5.25) \begin{equation} \mathcal N_e = \frac {1}{2\omega }\frac {\partial D_e}{\partial \omega }(\omega ,\kappa ;h). \end{equation}

Here, $\partial _\omega D_e$ denotes differentiation with respect to $\omega$ holding $\kappa$ fixed, and the result is evaluated on the acoustic–gravity branch satisfying $D_e(\omega ,\kappa ;h)=0$ . This is convenient for analysis and confirms that elasticity modifies the triad coefficients through the same object that controls the sensitivity of the acoustic branch to frequency.

6.1. A conserved quadratic invariant

With weak spatial modulation, the triad system (5.18) possesses a local conservation law. To see this, multiply the acoustic equation by $A^\ast$ , add its complex conjugate, and use

(6.1) \begin{align} \partial _T(A^\ast A) + c_e \partial _X(A^\ast A) = A^\ast \big (\mathrm{i}\gamma _e A + \delta _e S_+S_-\big ) + A\big (-\mathrm{i}\gamma _e^\ast A^\ast + \delta _e^\ast S_+^\ast S_-^\ast \big ). \end{align}

This gives

(6.2) \begin{equation} \partial _T\big(|A|^2\big) + c_e \partial _X\big(|A|^2\big) = 2\Re \left (\delta _e A^\ast S_+S_-\right ), \end{equation}

where the detuning term contributes only an imaginary part and hence does not affect the balance. Similarly, multiplying the gravity-wave equations by $S_\pm ^\ast$ and adding complex conjugates yields

(6.3) \begin{equation} \partial _T\big(|S_\pm |^2\big) + c_{g\pm } \partial _X\big(|S_\pm |^2\big) = -\Re \left (\delta _e A^\ast S_+S_-\right ). \end{equation}

Summing (6.2) and the two relations (6.3) gives the local conservation law

(6.4) \begin{equation} \partial _T\left (|A|^2+|S_+|^2+|S_-|^2\right ) + \partial _X\left (c_e|A|^2+c_{g+}|S_+|^2+c_{g-}|S_-|^2\right ) =0. \end{equation}

In particular, if the envelopes are periodic in $X$ or decay sufficiently rapidly as $|X|\to \infty$ , then integration over $X$ shows that $\int (|A|^2+|S_+|^2+|S_-|^2 )\mathrm{d} X$ is conserved. This ‘action-like’ conservation law is a standard property of resonant triads and follows from the canonical structure of the modulation equations. It does not, by itself, specify how the acoustic-mode physical energy is distributed between the fluid and the solid.

6.2. Physical acoustic-mode energy and elastic partition

The physical quadratic energy associated with the acoustic–gravity mode is proportional to the normalisation arising from the solvability condition. With the eigenfunction normalised by $\varPhi _e(0)=1$ , the mode energy may be written (up to a constant prefactor) as

(6.5) \begin{equation} E_e^{\textit{phys}} \propto ( \mathcal N_{\textit{fs}}+\mathcal N_{\textit{int}}) |A|^2, \end{equation}

where $\mathcal N_{\textit{fs}}$ and $\mathcal N_{\textit{int}}$ are given explicitly by (5.23)–(5.24). For impedance conditions of the form $\phi _{l,z}-\mathcal Z(\omega ,\kappa )\phi _l=0$ , the interface contribution to the frequency normalisation arises from the $\omega$ -sensitivity of the boundary operator (equivalently, from $\partial _\omega D_e$ ), while $\kappa$ -dependence enters the detuning/dispersion corrections through the elastic dispersion relation $D_e(\omega ,\kappa ;h)$ and is accounted for in $\mathcal D$ and $\beta$ . This decomposition provides a natural energetic interpretation

(6.6) \begin{equation} E_{e,f}^{\textit{phys}} \propto \mathcal N_{\textit{fs}}|A|^2, \qquad E_{e,s}^{\textit{phys}} \propto \mathcal N_{\textit{int}}|A|^2, \end{equation}

where $E_{e,f}^{\textit{phys}}$ is the fluid-associated contribution (bulk plus free surface), while $E_{e,s}^{\textit{phys}}$ measures the portion of the acoustic-mode energy stored in (or transmitted into) the elastic half-space through the impedance operator.

In the rigid-bottom limit $\mathcal Z\to 0$ , $\mathcal N_{\textit{int}}\to 0$ and the acoustic-mode energy is entirely contained in the fluid. For an elastic half-space, $\mathcal N_{\textit{int}}$ is generally non-zero and quantifies the degree of elastic participation in the acoustic–gravity mode.

6.3. Energetic meaning of ‘elastic leakage’

The present model is conservative: in the absence of attenuation, energy transferred into the elastic half-space is not lost but becomes part of the coupled fluid–solid mode. In that sense, elasticity does not dissipate triad energy; it redistributes the acoustic-mode energy between the fluid and the solid according to (6.6). If weak attenuation in the solid is included (not considered here), the same partition determines the fraction of triad-fed acoustic energy that is subsequently dissipated in the solid bottom boundary.

In the lossless elastic half-space considered here, the impedance is taken to be real on the branches analysed, so the triad system remains conservative. If attenuation in the solid is included, the effective impedance becomes complex and the modulation equations would acquire genuine decay on the slow time scale.