2. Preliminaries

Consider a free-surface gravity–acoustic wave disturbance propagating in a slightly compressible water of constant depth $h$ over a rigid bottom ( $z = - h$ ), under the joint effects of gravity and compressibility. The motion is assumed irrotational and the water is treated as an inviscid barotropic fluid with a constant sound speed, $c = (\mathrm {d}p/\mathrm {d}\rho )^{1/2}$ . Under oceanic conditions where $h \sim \mathcal {O}(10^{3})\,\rm m$ and $c=1.5\times 10^3\,\rm m\,s^-{^1}$ , a key parameter that controls the effects of gravity relative to compressibility is $\mu = gh/c^2\ll 1$ , where $g=9.81\,\rm m\,s^{-2}$ is the acceleration due to gravity and $\mu \sim \mathcal {O}(10^{-3})$ . Specifically, we are interested in wave disturbances of characteristic length scales that are comparable to the water depth $h$ . Thus, the acoustic and gravity time scales become $\tau _a \sim h/c$ and $\tau _g\sim (h/g)^{1/2}$ , respectively. Therefore, the ratio between the two time scales is characterised by $\tau _a/\tau _g \sim \mu ^{1/2}$ .

The mathematical problem is formulated in a two-dimensional Cartesian frame of reference, with $x$ denoting the horizontal coordinate and $z$ the vertical coordinate. The velocity, pressure, density and free-surface-elevation fields are given by $\boldsymbol {u} = u(x, z, t)\boldsymbol {\hat {x}} + w(x, z, t)\boldsymbol {\hat {z}}$ , $p = p(x, z, t)$ , $\rho = \rho (x, z, t)$ and $\eta = \eta (x, t)$ .

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

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

From the irrotationality condition the wave motion is described using a velocity potential $\varphi (x, z, t)$ , such that $\boldsymbol {u} = \nabla \varphi$ . The momentum and conservation of mass equations in (2.1), respectively, reduce to

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

(2.3) \begin{align} \frac {1}{\rho }\left( \partial _{t}\rho + \boldsymbol {u}\cdot \nabla \rho \right) = \partial _{t}\left(\log (\rho )\right) + \nabla \varphi \cdot \nabla \left(\log (\rho )\right) = D_{t} \log (\rho ) = -\Delta \varphi , \end{align}

where $\Delta = \partial ^{2}_x + \partial ^{2}_z$ is the Laplacian operator in the two spatial dimensions and $D_{t} = \partial _t + \boldsymbol {u}\cdot \nabla$ is the convective derivative. Applying $D_t$ to (2.2), we get a single equation for the potential:

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

In order for the equation above to admit a unique solution, it has to be subject to proper boundary conditions. Since the flow is a free-surface flow, we need to impose two boundary conditions along the free surface $z = \eta (x, t)$ which itself forms part of the unknowns. These boundary conditions are a kinematic boundary condition and a dynamic boundary condition and read, respectively,

(2.5) \begin{align} \partial _{t}\varphi + \frac {1}{2}|\nabla \varphi |^2 + gz = 0 \quad \text{and} \quad \partial _{t}\eta + \partial _{x}\varphi \partial _{x}\eta = \partial _{z}\varphi , \quad \textrm {both on} \quad z = \eta . \end{align}

In addition, a no-penetration boundary condition holds at the bottom boundary, i.e. $\partial _{z}\varphi = 0$ , at $z = -h$ .

Employing $h$ as the length scale and $\tau _a$ as the time scale, the non-dimensional compressible wave equation (2.4) in the fluid interior $(x, z) \in (-\infty , +\infty )\times (-1, 0)$ becomes (Kadri & Akylas Reference Kadri and Akylas2016)

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

Perturbing about $\eta =0$ , conditions (2.5) become (Benney (Reference Benney1962))

(2.7) \begin{align} \begin{aligned} &\partial ^{2}_{t}\varphi + \mu \partial _{z}\varphi = - \partial _{t}|\nabla \varphi |^{2} + \frac {1}{\mu }\partial _{t}\left(\partial _{t}\varphi \partial ^{2}_{tz}\varphi \right) - \partial _{x}\varphi \partial ^{2}_{xt}\varphi + \partial ^{2}_{z}\varphi \partial _{t}\varphi \\ & + \frac {1}{2\mu }\partial ^{2}_{tz}\left(\partial _{t}\varphi |\nabla \varphi |^2\right) - \frac {1}{2}\partial _{x}\varphi \partial _{x}|\nabla \varphi |^{2} + \frac {1}{2}\partial ^{2}_{z}\varphi |\nabla \varphi |^2 - \frac {1}{\mu ^2}\partial _{t}\left(\partial _{t}\varphi (\partial ^{2}_{tz}\varphi )^2\right) \\ & - \frac {1}{2\mu ^2}\partial _{t}\left(\partial ^{3}_{tzz}\varphi (\partial _{t}\varphi )^2\right) + \frac {1}{\mu }\partial _{x}\varphi \partial _{x}\left(\partial _{t}\varphi \partial ^{2}_{tz}\varphi \right) + \frac {1}{\mu }\partial _{t}\varphi \partial ^{2}_{tx}\varphi \partial ^{2}_{xz}\varphi \\ & - \frac {1}{\mu }\partial _{t}\varphi \partial ^{2}_{z}\varphi \partial ^{2}_{tz}\varphi - \frac {1}{2\mu }\partial ^{3}_{z}\varphi \left(\partial _{t}\varphi \right)^2, \qquad \textrm {on} \quad z = 0. \end{aligned} \end{align}

Equation (2.7) is the cubic truncation of the fully nonlinear free-surface boundary condition expressed in terms of the velocity potential only. Thus, the wave amplitudes need to be sufficiently small so that the remainder is negligible relative to the terms in (2.7).

Finally, the bottom boundary condition is

(2.8) \begin{align} \partial _{z}\varphi = 0, \quad \textrm {on} \quad z = -1. \end{align}

Unlike the deep-water case studied by Kadri & Akylas (Reference Kadri and Akylas2016), here in the limit $\mu \ll 1$ (2.6) pertains to water compressibility which reflects its importance on gravity waves for the specific choice of the length scale $h$ .

3. Linear solution

Neglecting the nonlinear terms in (2.6) and (2.7), we look for a solution in terms of travelling waves in the $x$ -direction with amplitudes modulated in $z$ , namely

(3.1) \begin{align} \varphi (x, z, t) = \sum _{n = 1}^{\infty } A_{n}f_{n}(z){\mathrm {e}}^{{\mathrm {i}} (k_{n}x - \omega _{n} t)} + \textrm {c.c.}, \end{align}

where c.c. stands for complex conjugate, and the mode number $n$ is omitted in the following, for brevity, unless necessary. Substituting (3.1) into (2.6)–(2.8) yields a generalised eigenvalue problem for the unknowns modes $f_{n}(z)$ of the form

(3.2) \begin{align} \begin{gathered} f'' - \mu f' + (\omega ^{2} - k^{2})f = 0 \quad (-1 \lt z \lt 0), \\ \mu f' - \omega ^{2}f=0 \quad (z = 0), \\ f' = 0 \quad (z = -1). \end{gathered} \end{align}

Equations (3.2) are solved asymptotically for $\mu \ll 1$ .

3.1. Linear gravity modes

To examine the effects of gravity we rescale the frequency as $\omega \rightarrow \mu ^{1/2}\omega$ and take the limit $\mu = 0$ of system (3.2), obtaining the eigensolutions

(3.3) \begin{align} f(z) = \cosh k(z+1)+\mathcal {O}(\mu ), \qquad k \gt 0, \end{align}

with dispersion relation

(3.4) \begin{align} \omega ^{2} = k\tanh {k}+\mathcal {O}(\mu ). \end{align}

3.2. Linear acoustic modes

For the acoustic modes, the leading order in (3.2) yields

(3.5) \begin{align} f(z) = \sin \lambda z + \mathcal {O}(\mu ), \end{align}

where

(3.6) \begin{align} \omega ^{2} = k^{2} + \lambda ^{2} + \mathcal {O}(\mu ) \quad \textrm {and} \quad \lambda ^{(n)} = (n+1/2)\pi , \quad n=0,1,2,\dots , \end{align}

where superscript $(n)$ indicates the eigenvalue mode number, which is omitted for brevity.

3.3. Resonant triads

Consider a triad comprising two acoustic modes characterised by wavenumbers $(k_1, k_2)$ and frequencies $(\omega _1, \omega _2)$ , satisfying the dispersion relation (3.6) and a gravity mode with wavenumber $k_3$ and frequency ${\mu ^{1/2}}\omega _3$ , with $\omega _3$ observing the gravity dispersion relation (3.4). Resonant interaction among the triad is possible if the following resonance conditions are met:

(3.7) \begin{align} k_1 + k_2 = k_3, \qquad \omega _1 + \omega _2 = \mu ^{1/2}\omega _{3}. \end{align}

Note that numerically $\omega _2$ in (3.7) must have an opposite sign compared with $\omega _1$ , and the current convention is employed for convenience. In order for resonant triad interaction to occur, (3.7), together with the dispersion relations for the gravity and acoustic modes, must be satisfied.

For the triad resonance to be satisfied there are two scenarios. The first scenario comprises two acoustic modes of almost identical wavenumbers and frequencies, and thus we can write

(3.8) \begin{align} |k_1|+|k_2|=k, \quad |\omega _1|-|\omega _2|=\varOmega , \end{align}

where $|k_1| = (k +\mu ^{1/2})/2$ , $|k_2| = (k - \mu ^{1/2})/2$ and $\varOmega$ is the gravity-mode frequency which must be $\sim \mathcal {O}(\mu ^{1/2})$ . Under this setting, we can approximate the frequencies in the form

(3.9) \begin{align} |\omega _j| \simeq |k_j|/2 +\lambda _j^2/(2|k_j|), \qquad j=1,2, \end{align}

which requires

(3.10) \begin{align} \varOmega \sim \mu ^{1/2} + (\lambda _1^2-\lambda _2^2)/{k}. \end{align}

Therefore, for (3.8) to be satisfied the interaction must comprise the same acoustic mode numbers $n$ , i.e. $\lambda _1 = \lambda _2$ , and thus the second term of the right-hand side of (3.10) must be zero.

The second scenario requires the conditions

(3.11) \begin{align} |k_1|-|k_2|=|k_3|, \quad |\omega _1|-|\omega _2|=\mu ^{1/2}|\omega _3|. \end{align}

Under this setting, making use of (3.9) and after some algebra we arrive at the condition

(3.12) \begin{align} |k_3|^2 = \frac {|k_1|\lambda _2^2-|k_2|\lambda _1^2}{|k_1k_2|}. \end{align}

Condition (3.12) indicates that interaction between acoustic modes with different $n$ is possible. In the case that $\lambda _1=\lambda _2\equiv \lambda$ , the condition reduces to $|k_1k_2k_3|=\lambda ^2$ .

4. Amplitude equations

In this section we derive the amplitude evolution equations of the triad discussed in the previous section. Specifically, we apply the method of multiple scales as employed by Kadri & Akylas (Reference Kadri and Akylas2016). Initially, we derive the amplitude equations with no spatial modulation, hence assuming that the amplitude of each mode is a function of a slow temporal variable $T = \mu t$ .

Building on the scaling arguments presented above, the velocity potential for the triad is expressed as follows:

(4.1) \begin{align} \begin{aligned} \varphi &= \epsilon \left[ A_{1}(T)\sin (\lambda _1 z) {\mathrm {e}}^{{\mathrm {i}}\varTheta _1} + \text{c.c.} \right] + \epsilon \left[ A_{2}(T)\sin (\lambda _2 z){\mathrm {e}}^{{\mathrm {i}}\varTheta _2} + \text{c.c.} \right] \\ & + \alpha \left[ S(T)\mathrm {cosh} k_3(z+1) {\mathrm {e}}^{{\mathrm {i}}\varTheta _3} + \text{c.c.} \right] +\cdots \end{aligned} \end{align}

where $\varTheta _j = k_j x - \omega _j t$ , with $j = \{1, 2\}$ , $\varTheta _{3} = k_{3}x - \mu ^{1/2}\omega _{3} t$ and $\epsilon$ and $\alpha$ are coefficients depending on $\mu$ . Their dependence is specified in the next section.

4.1. Interaction time scale

To estimate the appropriate interaction time scale, we note that the identified triads consist of two acoustic modes and a gravity wave. Let the velocity potential of each acoustic mode be $\mathcal {O}(\epsilon )$ , where $0 \lt \epsilon \ll 1$ . The nonlinear interaction between these acoustic modes, governed by the quadratic terms in (2.6), (2.7) and (2.8), excites the gravity wave, whose velocity potential grows to $\mathcal {O}(\alpha )$ , where $\alpha$ will be specified later. Based on previous analyses of triad interactions (e.g. Bretherton Reference Bretherton1964), this energy transfer is expected to occur on a time scale of $\mathcal {O}(\alpha /\epsilon ^2)$ .

Next, consider the interaction between the $\mathcal {O}(\alpha )$ gravity wave and one of the $\mathcal {O}(\epsilon )$ acoustic modes. Accounting for the fact that the velocity field of the gravity wave is $\mathcal {O}(\alpha /\mu ^{1/2})$ , as indicated in (3.7), the time scale for energy flow to the other acoustic mode is expected to be $\mathcal {O}(\mu ^{1/2}/\alpha )$ .

For a fully coupled three-wave interaction resulting in an equitable energy distribution among all triad members, these two separate interactions must occur on the same time scale. This condition requires

(4.2) \begin{align} \alpha = \epsilon \mu ^{1/4}. \end{align}

Now it becomes clear that cubic self-interaction of acoustic modes must take a much longer time scale. Therefore, cubic terms can be neglected in the derivation of the amplitude equations.

4.2. Acoustic modes

We introduce a small-order correction term to the background potential (4.1) in the form $G_j(z, T){\mathrm {e}}^{{\mathrm {i}}\varTheta _j}$ . Substituting this into (2.6), (2.7) and (2.8), and isolating the resonant acoustic terms, we obtain the following equations for $G_j$ :

(4.3a) \begin{align} \partial ^{2}_{z}G_{j} + \lambda ^{2}_{j}G_j & = -2\epsilon \mu {\mathrm {i}}\omega _{j}\sin (\lambda _{j} z)\dot {A}_{j} + \epsilon \mu \lambda _{j}\cos (\lambda _{j} z)A_{j} \quad (-1 \lt z \lt 0), \\[-4pt] \nonumber \end{align}

(4.3b) \begin{align} G_j & = \epsilon \mu \frac {\lambda _j}{\omega ^{2}_{j}}A_{j} + \mu ^{-1/4}\epsilon ^2{\mathrm {i}}\frac {\omega _{j}\omega _3}{\omega _{i}}\lambda _{j}\cosh (k_3)A^{\ast }_{i}S \quad (z = 0), \\[-4pt] \nonumber \end{align}

(4.3c) \begin{align} & \qquad \qquad \qquad \partial _{z}G_{j} = 0 \quad (z = -1), \\[6pt] \nonumber \end{align}

where the over dot stands for time derivative. Application of the Fredholm alternative as a solvability condition returns an amplitude equation of the form

(4.4) \begin{align} \mu ^{5/4}\dot {A}_{j} = \mu ^{5/4}\frac {{\mathrm {i}}}{\omega _j}\left(\frac {1}{2}-\frac {\lambda ^{2}_j}{\omega ^{2}_j}\right)A_j + \epsilon \lambda _i\lambda _j\cosh (k_3)\frac {\omega _i\omega _3}{\omega ^{2}_j}A^{\ast }_i S. \end{align}

In order to balance all terms in (4.4), we require that $\epsilon = \mu ^{5/4}$ , so the final form of the amplitude equations for the two acoustic modes reads

(4.5) \begin{align} \dot {A}_{j} = \frac {{\mathrm {i}}}{\omega _j}\left(\frac {1}{2}-\frac {\lambda ^{2}_j}{\omega ^{2}_j}\right)A_j + \lambda _i\lambda _j\cosh (k_3)\frac {\omega _i\omega _3}{\omega ^{2}_j}A^{\ast }_i S. \end{align}

4.3. Gravity mode

Similarly to the approach taken previously for the acoustic perturbations, we introduce a correction term for the gravity contribution in the form $G_3(z, T){\mathrm {e}}^{{\mathrm {i}}\varTheta _3}$ . Substituting this into the field (2.6) and the boundary conditions (2.7)–(2.8), we derive the following system:

(4.6a) \begin{align} k^{2}_{3}G_{3} - \partial ^{2}_{z}G_{3} & = 0 \quad (-1 \lt z \lt 0), \\[-2pt] \nonumber \end{align}

(4.6b) \begin{align} \partial _{z}G_{3} = \omega ^{2}_{3} G_3 + 2{\mathrm {i}}\omega _3 & \cosh (k_3)\dot {A}_3 + 2{\mathrm {i}}\lambda _{i}\lambda _{j} A_{i}A_{j} \quad (z = 0), \\[-2pt] \nonumber \end{align}

(4.6c) \begin{align} \partial _{z}G_{3} = 0 & \quad (z = -1). \\[6pt] \nonumber \end{align}

Applying the solvability condition, we obtain the amplitude equation for $S(T)$ :

(4.7) \begin{align} \dot {S} = -\frac {\lambda _{1}\lambda _{2}}{\cosh (k_3)}A_1A_2. \end{align}

4.4. Spatial modulation

When spatial modulation is allowed, a slowly varying spatial coordinate $X = \mu x$ has to be introduced, leading to the appearance of a term of the form $2\epsilon \mu {\mathrm {i}} k_j\sin (\lambda _j z)\partial _X A_j$ on the right-hand side of equation (4.3a ). The solvability condition then yields a generalised set of amplitude equations incorporating spatial dependence:

(4.8a) \begin{align} \partial _{T}A_1 + \frac {k_1}{\omega _1}\partial _X A_1 & = \frac {{\mathrm {i}}}{\omega _1}\left(\frac {1}{2}-\frac {\lambda ^{2}_1}{\omega ^{2}_1}\right)A_1 + \lambda _1\lambda _2\cosh (k_3)\frac {\omega _2\omega _3}{\omega ^{2}_1}A^{\ast }_2 S, \\[-2pt] \nonumber \end{align}

(4.8b) \begin{align} \partial _{T}A_2 + \frac {k_2}{\omega _2}\partial _X A_2 & = \frac {{\mathrm {i}}}{\omega _2}\left(\frac {1}{2}-\frac {\lambda ^{2}_2}{\omega ^{2}_2}\right)A_2 + \lambda _1\lambda _2\cosh (k_3)\frac {\omega _1\omega _3}{\omega ^{2}_2}A^{\ast }_1 S, \\[-2pt] \nonumber \end{align}

(4.8c) \begin{align} & \qquad \qquad \partial _T S = -{\lambda _1\lambda _2}{{sech}(k_3)}A_1A_2, \\[12pt] \nonumber \end{align}

where $k_j/\omega _j$ , with $j=1,2$ , represent the group velocities of the two acoustic modes. It is worth noting that the equation for $S$ remains unchanged compared with the case without spatial dependence. This is because spatial variations for gravity modes occur on a much larger spatial scale than those for the acoustic modes.

5. Energy considerations: non-spatial equations

In this section, we aim to provide deeper insights into (4.5) and (4.7) and their solutions. We begin by reducing the system to a single ordinary differential equation, enabling us to construct the phase portrait of the corresponding dynamical system. Subsequently, we present analytical results derived under suitable approximations and compare them with numerical simulations. Thus, we can write

(5.1a) \begin{align} \dot {A}_{1} = {\mathrm {i}}\beta _1 A_1 + \alpha _1 A^{\ast }_{2}S, \\[-3pt] \nonumber \end{align}

(5.1b) \begin{align} \dot {A}_2 = {\mathrm {i}}\beta _2 A_2 + \alpha _2 A^{\ast }_{1}S, \\[-3pt] \nonumber \end{align}

(5.1c) \begin{align} \dot {S} = -\alpha _3 A_1A_2, \\[12pt] \nonumber \end{align}

where

(5.2) \begin{align} \begin{gathered} \alpha _1 = \lambda _1\lambda _2\cosh (k_3)\frac {\omega _2\omega _3}{\omega ^2_1}, \quad \alpha _2 = \lambda _1\lambda _2\cosh (k_3)\frac {\omega _1\omega _3}{\omega ^2_2}, \quad \alpha _3 = \frac {\lambda _1\lambda _2}{\cosh (k_3)}, \\ \beta _1 = \frac {1}{\omega _1}\left(\frac {1}{2} - \frac {\lambda ^2_1}{\omega ^2_1}\right), \quad \beta _2 = \frac {1}{\omega _2}\left(\frac {1}{2} - \frac {\lambda ^2_2}{\omega ^2_2}\right). \end{gathered} \end{align}

Multiplying the terms in (5.1) by their associated amplitude’s complex conjugate, and adding the product of the amplitudes with the complex conjugate of the terms, results in

(5.3) \begin{align} \begin{gathered} \frac {\mathrm {d}|A_1|^2}{\mathrm {d} T} = \alpha _1A_1A_2S^{\ast } + \alpha _1A^{\ast }_1A^{\ast }_2S,\\ \frac {\mathrm {d}|A_2|^2}{\mathrm {d} T} = \alpha _2A_1A_2S^{\ast } + \alpha _2A^{\ast }_1A^{\ast }_2S,\\ \frac {\mathrm {d}|S|^2}{\mathrm {d} T} = -\alpha _3A_1A_2S^{\ast } - \alpha _3A^{\ast }_1A^{\ast }_2S. \end{gathered} \end{align}

Through straightforward algebra, the following invariants (constants of motion) can be derived from (5.3):

(5.4) \begin{align} \begin{gathered} E_0 \equiv \alpha _1\alpha _2|S|^2 + \frac {\alpha _3\alpha _2}{2}|A_1|^2 + \frac {\alpha _3\alpha _1}{2}|A_2|^2,\\ I_0 \equiv \left(\alpha _2 |A_1|^2 - \alpha _1 |A_2|^2 \right), \quad I_j \equiv \left(\alpha _3 |A_j|^2 + \alpha _j |S|^2 \right), \quad j=1,2,\\ Q \equiv |A_1||A_2||S|\sin (\arg {S} - \arg {A_1} - \arg {A_2}) + (\beta _1/\alpha _1)|A_1|^2 + (\beta _2/\alpha _2)|A_2|^2, \end{gathered} \end{align}

where $E_0$ represents the initial energy of the system (Craik Reference Craik1988), $I_j$ for $j = 0, 1, 2$ are the Manley–Rowe relations (Manley & Rowe Reference Manley and Rowe1956) and $Q$ is a Hamiltonian of the system (Martin & Segur Reference Martin and Segur2016, pp. 74–75).

The conservation of energy equation is useful for reducing system (5.1) to a single ordinary differential equation for the gravity-wave amplitude S. By differentiating (5.1c) with respect to time and utilising (5.1a), (5.1b) alongside (5.4), we obtain

(5.5) \begin{align} \ddot {S} = -{\mathrm {i}}(\beta _1 + \beta _2)\dot {S} - 2S\left(E_0 - \alpha _1\alpha _2|S|^2\right). \end{align}

We solve equation (5.5) with the initial conditions $S(0) = 0.5$ and $\dot {S}(0) = 0$ . The solutions exhibit periodic behaviour, as shown in figure 1, across different resonant triads (distinguished by varying $k_3$ ) and for two sets of acoustic axial mode numbers: $n=(0,0)$ , i.e. $\lambda _1 = \lambda _2 = \pi /2$ in the upper panels, and $n=(0,1)$ , i.e. $(\lambda _1, \lambda _2) = (\pi /2, 3\pi /2)$ in the lower panels. Following Craik (Reference Craik1988, pp. 129–130) and Martin & Segur (Reference Martin and Segur2016), it can be shown that the solutions in figure 1 exhibit a non-explosive periodic behaviour (see Appendix A).

Notably, for the case $n=(0,1)$ , the last two panels highlight a pronounced decrease followed by an increase in the amplitude of $S$ within the considered time range. This rapid variation in the gravity-wave amplitude is largely absent in the other panels, where the system’s orbits remain consistent.

Figure 1.

Phase portraits for $n=(0,0)$ (top panels) and $n=(0,1)$ (bottom panels) for initial conditions $S(0) = 0.5$ , $\dot {S}(0) = 0$ : (a) $k_3 = 1$ ; (b) $k_3 = 1.5$ ; (c) $k_3 = 2$ . Red dot indicates the starting point $(T = 0)$ ; green dot the end point $(T = 20)$ in the phase portrait. For each case, the periodic time behaviour is shown above the phase portraits over an extended time frame $T=200$ .

Analytical solutions to (5.5) can be derived in closed form, particularly when $\beta _1 + \beta _2 \ll 1$ . Under this condition, the imaginary term in (5.5) vanishes, allowing $S(T)$ to be treated as a purely real function. It satisfies a Duffing-type equation:

(5.6) \begin{align} \ddot {S} + 2E_0 S - 2\alpha _1\alpha _2S^3 = 0, \quad \end{align}

with initial conditions $S(0) = 1/2$ ; $\dot {S}(0) = 0$ . Following Salas & Castillo (Reference Salas and Castillo2014), this differential problem (5.6) admits the closed-form solution

(5.7) \begin{align} \begin{aligned} & S(T) = \frac {1}{2}\mathrm {cn}\left[ \left(2E_0 - \frac {\alpha _1\alpha _2}{2}\right)^{1/2}t, \left(\frac {-\alpha _1\alpha _2}{8E_0 - 4\alpha _1\alpha _2}\right)^{1/2} \right], \end{aligned} \end{align}

where $\mathrm {cn}[\cdot ]$ denotes the elliptic Jacobi function (Abramowitz & Stegun Reference Abramowitz and Stegun1965). A comparison between numerical and analytical solutions (5.7) reveals that the system exhibits a periodic trend, with good agreement between the two approaches for smaller values of $k_3$ (figure 2). As $k_3$ increases, differences in the phase of the oscillations become apparent, due to the deviation of $\beta _1 + \beta _2$ from zero. This leads to a growing discrepancy between the analytical and numerical solutions. However, for relatively small values of $k_3$ , the deviation between the analytical and the numerical solutions becomes small.

Figure 2.

Numerical and analytical solutions (5.7) for $n = (0, 0)$ .

Figure 3.

Difference function $D_{S}$ (equation (6.2)) for $\sigma = 1$ , $n=(0,0)$ and several times $T$ .

Figure 4.

Difference function $D_S$ as a function of slow time scale $T$ , acoustic width $\sigma$ , mode numbers $n = (0,0)$ , resonant gravity wavenumber $k_3 = 1$ and different initial acoustic amplitudes: (a) $A = 0.25$ ; (b) $A = 0.5$ ; (c) $A = 0.75$ ; (d) $A = 1$ .

Figure 5.

Difference function $D_{S}$ for $\sigma = 10$ , $n=(0,0)$ and several times $T$ .

Figure 6.

Difference function $D_{S}$ as a function of slow time scale $T$ , with $k_3 = 1$ and acoustic amplitudes $A=0.25$ (top panels), $A=0.5$ (middle panels), $A=0.75$ (bottom panels); and $\sigma =1$ (left-hand panels) and $\sigma =10$ (right-hand panels). Black: $n = (0,0)$ . Blue: $n = (0,1)$ . Red: $n=(1,1)$ .