2. Excitation due to surface-wave–seafloor interactions

When a source of excitation in the water or on the seafloor has a wavenumber and frequency that falls on the dispersion surface for the acoustic-gravity–Scholte wave field in the shallow-water–seafloor coupled system (§ 5), natural modes of the acoustic-gravity–Scholte wave field are excited, causing resonance. Here, the wave field is forced by the external source generated by the nonlinear interaction between surface waves and a non-uniform seafloor (Hasselmann Reference Hasselmann1963). The surface-wave field is assumed to include both shoreward-propagating waves and their reflections. The method developed in Ardhuin (Reference Ardhuin2018) (main document and Supplementary information) may be used to understand the excitation of acoustic-gravity–Scholte wave groups by surface waves propagating over a non-uniform seafloor. Here a somewhat different approach is investigated, as described below. A coordinate system is introduced where the axes $(X,z)$ are located at the undisturbed free surface, and

(2.1) \begin{equation} X = x,\ \ \ z = -H+h(x), \end{equation}

where $x$ and $z$ are the cross-shore and vertical coordinates, and $h(x)$ describes the bottom profile. Thus, if $h(x) = 0$ , $z = -H$ . It is assumed here that $h(x)$ is a mildly varying function of $x$ in the sense that a uniform-depth approximation may be used in deriving the boundary conditions as long as the correct depth is substituted at each $x$ in the final steps (Whitham Reference Whitham1973). Under linear finite depth theory, for a particular wavenumber–frequency combination $(\kappa ,\sigma )$ , the z-component $Z(z)$ of the assumed solution $X(x)Z(z)$ to the Laplace equation is

(2.2) \begin{equation} Z(z) = C_1\sinh \kappa z + C_2\cosh \kappa z, \end{equation}

where $C_1$ and $C_2$ are constants, and $\kappa$ denotes the surface-wave wavenumber. Letting $\varGamma (x,z;t)$ represent the velocity potential for surface waves, for $X(x) \sim \exp [i(\kappa x - \sigma t)]$ , the free surface condition becomes

(2.3) \begin{equation} \frac {\partial \varGamma }{\partial z} - \frac {\sigma ^2}{g}\varGamma = 0, \end{equation}

where $\sigma$ is the surface-wave angular frequency, $t$ is time and $g$ is gravitational acceleration. Substitution of $Z(z)$ from (2.2) into the boundary condition at $z = 0$ implies

(2.4) \begin{equation} \kappa C_1 - \frac {\sigma ^2}{g} C_2 = 0. \end{equation}

At the boundary $z = -H + h$ , to first order, $\partial \varGamma /\partial z = 0$ . This condition implies that

(2.5) \begin{equation} \kappa C_1\cosh \kappa (-H+h) + \kappa C_2\sinh k(-H+h) = 0. \end{equation}

Equations (2.4) and (2.5) lead to the surface-wave dispersion relation subject to the mildly variable depth approximation,

(2.6) \begin{equation} \sigma ^2 = g\kappa \tanh \kappa H\left (1- h/H\right )\!. \end{equation}

The surface-wave velocity potential $\varGamma (x,z; t)$ then becomes

(2.7) \begin{equation} \varGamma (x, z; t) = C_2 \left [ \frac {\sinh \kappa H(1-h/H)}{\cosh \kappa H(1-h/H)} \sinh \kappa z + \cosh \kappa z\right ]\exp \left (i(\kappa x - \sigma t)\right )\!. \end{equation}

Following some algebra,

(2.8) \begin{equation} \varGamma (x, z; t) = \frac {C_2 \cosh \kappa \left (z + H(1-h/H)\right )}{\cosh \kappa H(1-h/H)}\exp \left (i(\kappa x - \sigma t)\right )\!, \end{equation}

which can be expressed as

(2.9) \begin{equation} \varGamma (x, z; t) = C_2 m_d(\kappa z,\kappa h)\exp \left (i(\kappa x - \sigma t)\right )\!, \end{equation}

where it is noted that $h$ is a function of $x$ .Here

(2.10) \begin{equation} m_d(\kappa z,\kappa h(x)) = \frac {\cosh \kappa \left (z + H(1-h(x)/H)\right )}{\cosh \kappa H(1-h(x)/H)}. \end{equation}

With the coefficient $C_2$ determined as shown below, $m_d(\kappa z, \kappa h(x))$ in (2.9) describes the modulation of the surface-wave field by variable seafloor topography (cf. equations (9) and (12) in Ardhuin Reference Ardhuin2018). Further, because the present formulation is in terms of wavenumber–frequency spectra, the spatial modulation of amplitudes of individual wave components is mapped into the wavenumber domain.

The application of the linearized dynamic free-surface condition at $z = 0$ ,

(2.11) \begin{equation} g\zeta = \frac {\partial \varGamma }{\partial t} \end{equation}

leads to, from (2.8), $C_2 = {i g {\rm d}Z(\kappa , \sigma )}/{\sigma }$ . The dynamic pressure at the seafloor $z = -H+h$ can be expressed as

(2.12) \begin{equation} p(x, -H+h; t) = \frac {\rho g \zeta (x,t)}{\cosh \kappa H(1-h/H)}, \end{equation}

where $\rho$ is the density of seawater. It can be argued that

(2.13) \begin{equation} \cosh \kappa H(1-h/H) = \cosh (\kappa H - \kappa h) = \cosh kH\cosh \kappa h - \sinh \kappa H \sinh \kappa h, \end{equation}

and for small $h$ , $\cosh \kappa h \approx 1$ and $\sinh \kappa h \approx \kappa h$ , so that

(2.14) \begin{equation} \cosh \kappa (H-h/H) \approx \cosh \kappa H - \kappa h \sinh \kappa H = \cosh \kappa H\left ( 1 - \kappa h\tanh \kappa H\right )\!. \end{equation}

Hence,

(2.15) \begin{equation} \frac {1}{\cosh \kappa H(1-h/H)} \approx \frac {1}{\cosh \kappa H}\left (1-\kappa h\tanh \kappa H\right )^{-1} \approx \frac {(1+\kappa h\tanh \kappa H)}{\cosh \kappa H}. \end{equation}

The pressure at the seafloor then can be approximated as

(2.16) \begin{equation} p(x,-H+h;t) = \rho g\left [\frac {\zeta (x,t)}{\cosh \kappa H} + \frac {\kappa \tan \kappa H}{\cosh \kappa H}\zeta (x,t)h(x)\right ], \end{equation}

where it is made explicit that $h$ is a function of $x$ . The second term in the square brackets contains a product of two $x$ -dependent quantities that can be expressed in the wavenumber domain by means of a Fourier–Stieltjes integral as (Phillips Reference Phillips1960)

(2.17) \begin{equation} \int \limits _{\boldsymbol{k}} {\rm d}Z_h({\boldsymbol{k}},t) = \iint \limits _{{\boldsymbol{\kappa }}',{\boldsymbol{k}}_s}{\rm d}Z({\boldsymbol{\kappa }}',t){\rm d}H({\boldsymbol{k}}_s)\exp \left [i\left ({\boldsymbol{\kappa }}'+{\boldsymbol{k}}_s\right )\boldsymbol{\cdot }{\boldsymbol{x}}\right ], \end{equation}

where bold indicates a vector, ${\boldsymbol{\kappa }}'$ represents the wavenumber vectors for the surface elevation and ${\boldsymbol{k}}_s$ denotes the wavenumber vectors for the seafloor. The right-hand side of (2.17) represents infinite combinations of ${\boldsymbol{\kappa }}'$ and ${\boldsymbol{k}}_s$ arising from the interaction of the surface wave and the seafloor as approximated above. The surface-wave dispersion relation (2.6) implies that the wavenumber is a function of $x$ for a given frequency, and thus a continuous spectrum of wavenumbers is required to satisfy (2.6), with the seafloor contributing its own wavenumbers. Here, the focus is on a subset of $({\boldsymbol{\kappa }}',{\boldsymbol{k}}_s)$ combinations for which for a given frequency $\sigma$ the vector sum ${\boldsymbol{\kappa }}'+{\boldsymbol{k}}_s$ lies on a dispersion surface for acoustic-gravity waves in the water-column–seafloor system (see § 5). A further restriction is introduced here by choosing ${\boldsymbol{\kappa }}'$ to be ${\boldsymbol{\kappa }}_0$ where ${\boldsymbol{\kappa }}_0$ satisfies the surface-wave dispersion relation at a known depth for a mildly varying seafloor, and the wave–seafloor interaction is observed shoreward of this location. Thus, here the pressure $p(x,-H+h;t)$ is approximated as

(2.18) \begin{align} &p(x,-H+h;t) \approx \frac {\rho g\zeta (x;t) }{\cosh \kappa _0H} + p_\Delta (x,t),\ \textrm {where}\nonumber \\ &p_\Delta (x,t) = \frac {\kappa _0 \rho g\tanh {\kappa _0 H}}{\cosh \kappa _0 H}\zeta (x,t) h(x),\ \ \sigma ^2 = g\kappa _0\tanh \kappa _0 H. \end{align}

In terms of wavenumber spectra,

(2.19) \begin{equation} p_\Delta (x,t) = \int \limits _{{\boldsymbol{k}}}{\rm d}P_{\Delta }\textrm {e}^{i{\boldsymbol{k}}\boldsymbol{\cdot }{\boldsymbol{x}}} = \rho g\int \limits _{{\boldsymbol{k}}_s}\frac {{\boldsymbol{\kappa }}_0\rho g\tanh \kappa _0H}{\cosh \kappa _0H}{\rm d}Z({\boldsymbol{\kappa }}_0, t){\rm d}H({\boldsymbol{k}}_s)\exp \left [i\left ({\boldsymbol{\kappa }}_0 +{\boldsymbol{k}}_s\right )\boldsymbol{\cdot }{\boldsymbol{x}}\right ]. \end{equation}

Although a more general seafloor profile $h(x)$ such as expanded above and as described in (5.18) and (5.19) (§ 5) could be used, here a depth profile is employed where random features are superimposed on a mildly sloping seafloor, which can be represented as

(2.20) \begin{equation} h(x) = bx + cx^2 + \ldots + r(x), \end{equation}

where $r(x)$ is an amplitude-limited random function of $x$ and may be treated separately (§ 5), and hence is not considered further here. Assuming $c \ll b$ , the seafloor profile is simply $h(x) \approx bx$ . The slope $b \gt 0$ for $-\infty \lt x \leqslant 0$ , and $b \lt 0$ for the image space $0 \lt x \lt \infty$ . This function is Laplace transformable, and its Fourier transform $F_s(\gamma )$ can be composed using

(2.21) \begin{equation} F_s(\gamma ) = \int \limits _{-\infty }^0b(-x')\textrm {e}^{-\gamma x'} {\rm d}x' + \int \limits _0^\infty b x'\textrm {e}^{-\gamma x'} {\rm d}x', \end{equation}

where $x'$ is a dummy variable for the horizontal coordinate, $\gamma = \epsilon + i k_s$ with $0\lt \epsilon \ll 1$ , and $k_s$ is the seafloor wavenumber. It is seen that, as $\epsilon \rightarrow 0$ ,

(2.22) \begin{equation} F_s(\gamma ) = F_s(k_s) = \frac {2b}{k^2_s}. \end{equation}

To force $F_s(k_s)$ to have finite values as $k_s \rightarrow 0$ it is written as (Hasselmann Reference Hasselmann1976)

(2.23) \begin{equation} F_s(k_s) = \frac {2b(1-\cos k_s x)}{k^2_s}. \end{equation}

With two applications of L’Hôspital’s rule, the limit of $F_s(k_s)$ as $k_s \rightarrow 0$ is

(2.24) \begin{equation} \lim _{k_s\rightarrow 0}\frac {2(1-\cos k_s x)}{k^2_s} = bx^2. \end{equation}

As $x \rightarrow \infty$ with $k_s\rightarrow 0$ , the limiting value $bx^2$ gets increasingly larger, forming a peak with a width decreasing as $1/x$ . The Fourier transform $F_s(k_s)$ can then be expressed as $b x\delta (k_s)$ , where $\delta (\boldsymbol{\cdot })$ is the Dirac delta function (Hasselmann Reference Hasselmann1976). Here, $F_s(k_s)$ is averaged (indicated with an overbar) over the shelf length $L_s$ assuming that $L_w \ll L_s$ where $L_w$ is the surface-wave length. Thus, for $k_s \rightarrow 0$ ,

(2.25) \begin{equation} \overline {F_s(k_s)} = \frac {2}{L_s}\int \limits _0^{L_s} b x \delta (k_s) {\rm d}x = b L_s \delta (k_s),\ \ k_s \rightarrow 0. \end{equation}

Whereas for $k_s \gt 0$ ,

(2.26) \begin{align} \overline {F_s(k_s)} & = \frac {2}{L_s}\int \limits _0^{L_s} \frac {2(1-\cos k_s x)}{k^2_s} {\rm d}x, = \frac {2b}{k^2_s}\left (1-\frac {\sin k_sL_s}{k_sL_s}\right )\!, \ \ k_s \gt 0. \end{align}

Hence,

(2.27) \begin{equation} \overline {F_s(k_s)} = \left [ b L_s \delta (k_s) + \frac {2b}{k^2_s}\left (1-\frac {\sin k_sL_s}{k_sL_s}\right )\right ]. \end{equation}

The seafloor profile now can be represented as

(2.28) \begin{equation} h(x) = \frac {1}{2\pi }\int \limits _{k_s}\overline {{\rm d}F_s(k_s)}\exp (i {\boldsymbol{k}_s}\boldsymbol{\cdot }{\boldsymbol{x}}),\ \overline {{\rm d}F_s({k_s})} = \overline {F_s(k_s)}{\rm d}{k}_s, \end{equation}

where $k_s$ is the magnitude of ${\boldsymbol{k}}_s$ .Here $\overline {{\rm d}F_s(k)}$ is the Fourier–Stieltjes representation ${\rm d}H$ in (2.19).Here

(2.29) \begin{equation} {\rm d}P_{\Delta }({\boldsymbol{k}}, t) = \rho g \left [\left (\kappa _0\frac {{\rm d}Z({\boldsymbol{\kappa }}_0, t)\tanh \kappa _0 H}{\cosh \kappa _0 H}\right ) {\overline {{\rm d}F_s({\boldsymbol{k}_s}})}\right ], \ {\boldsymbol{k}} = {\boldsymbol{\kappa }} + {\boldsymbol{k}_s}. \end{equation}

The surface-wave wavenumber ( $\boldsymbol{\kappa }$ ) in the following is the same as that in (2.18), with the subscript $0$ dropped. The seafloor pressure Fourier coefficient as shown in (2.29) indicates that the resulting pressure results from an interaction between the surface wave and the seafloor and depends on the surface wavenumber $\boldsymbol{\kappa }$ and the seafloor wavenumber $\boldsymbol{k}_s$ and has a wavenumber $\boldsymbol{k}$ as determined by the vector sum of $\boldsymbol{\kappa }$ and $\boldsymbol{k}_s$ . This pressure leads to an acoustic-gravity–Scholte wave group at a wavenumber–frequency combination $({\boldsymbol{k}},\omega )$ , where $\omega = \sigma$ . For propagation along the $x$ -axis, the temporal evolution of the wavenumber spectrum $S_{P_\Delta }({\boldsymbol{k}}, t)$ for ${\rm d}P_\Delta ({\boldsymbol{k}},t)$ can be found as

(2.30) \begin{equation} {\rm d}P_{\Delta } {\rm d}P_{\Delta } = \rho ^2 g^2\left (\frac {\kappa \tanh \kappa H}{\cosh \kappa H}\right )^2 {\rm d}Z(\kappa , t){\rm d}Z^\ast (\kappa , t)\overline {{\rm d}F_s(k_s)} \overline {{\rm d}F^\ast _s(k_s)}. \end{equation}

When $\boldsymbol{\kappa }$ is determined as described above (cf. (2.18),

(2.31) \begin{equation} S_{P_\Delta }({\boldsymbol{k}}, t) = \frac {{\rm d}P_{\Delta } {\rm d}P^\ast _\Delta }{{\rm d}\boldsymbol{k}}, \end{equation}

which can be expressed as

(2.32) \begin{equation} S_{P_\Delta }({\boldsymbol{k}}, t) =\rho ^2 g^2\iint \limits _{{\boldsymbol{\kappa } {\boldsymbol{k}}_s}}\left (\frac {\tanh \kappa H}{\cosh \kappa H}\right )^2\kappa ^2S_e({\boldsymbol{\kappa }},t)S_{sf}({\boldsymbol{k}_s})\delta ({\boldsymbol{k}} - {\boldsymbol{\kappa }}-{\boldsymbol{k}}_s) {\rm d}{\boldsymbol{\kappa }}{\rm d}{\boldsymbol{k}}_s, \end{equation}

where $S_e$ is the wavenumber spectrum of the surface waves and $S_{sf}$ is the wavenumber spectrum of the seafloor. Here $S_{P_\Delta }({\boldsymbol{k}}, t)$ provides the forcing for the acoustic-gravity wave field generated by the interaction of the surface waves with the sloping bottom. Equation (2.32) can be expressed in terms of wavenumber–frequency spectra via a Fourier–Stieltjes expansion into wavenumber space. Resonance occurs for propagation in the $x$ direction when for a particular frequency $\omega$ , the wavenumber $k$ is such that $(k,\omega )$ lie on a dispersion surface for an acoustic-gravity–Scholte wave group excited by the pressure-spectrum in equation (2.32). The present approach for modelling the seafloor slope is convenient and is general enough to allow extensions to more complicated seafloor topographies.

3. Direct second-order pressure generation

Except as the water depth approaches zero, three surface waves with frequencies $\sigma _1$ , $\sigma _2$ and $\sigma _3$ where $\sigma _1 + \sigma _2 = \sigma _3$ do not have corresponding wavenumbers that satisfy the resonance condition ${\boldsymbol{\kappa }}_1 + {\boldsymbol{\kappa }}_2 = {\boldsymbol{\kappa }}_3$ (Phillips Reference Phillips1960; Hasselmann Reference Hasselmann1962). However, a three-wave interaction such as

(3.1) \begin{align} &{\boldsymbol{\kappa }}_1 + {\boldsymbol{\kappa }}_2 = {\boldsymbol{k}},\nonumber \\ &\sigma _1 + \sigma _2 = \omega \end{align}

can produce a combination $({\boldsymbol{k}},\omega )$ that lies on one (or more) of the dispersion surfaces for an acoustic-gravity–Scholte wave field and hence can excite an energetic acoustic-gravity wave field.

The nonlinear terms $\boldsymbol{\nabla} _h\varGamma \boldsymbol{\cdot }\boldsymbol{\nabla} _h\zeta$ , $(\partial \varGamma /\partial z)^2$ and $\zeta (\partial ^2\varGamma /\partial z^2)$ also contribute to the second-order pressures on the seafloor ( $\boldsymbol{\nabla} _h$ denotes the gradient operator with just the horizontal components included, to distinguish it from the gradient operator $\boldsymbol{\nabla}$ as used elsewhere) (Ardhuin & Herbers Reference Ardhuin and Herbers2013). The three terms are considered further in the following.

Using the surface elevation $\zeta$ based on linear theory, the Fourier–Stieltjes expansion for the pressure term $p_{d1}({\boldsymbol{x}},t)$ corresponding to $\boldsymbol{\nabla} _h\varGamma \boldsymbol{\cdot }\boldsymbol{\nabla} _h \zeta$ at $\zeta = 0$ can be expressed as

(3.2) \begin{align} p_{d1}({\boldsymbol{x}},t) &= -\rho g^2\iint \limits _{{\boldsymbol{\kappa },\kappa '}}\iint \limits _{\sigma ,\sigma '}\frac {{\boldsymbol{\kappa }} \boldsymbol{\cdot }{\boldsymbol{\kappa }'}}{\sigma \sigma '}\frac {\cosh \kappa (z+H-h)}{\cosh \kappa (H-h)}\nonumber \\ &\quad \times {\rm d}Z({\boldsymbol{\kappa }},\sigma ){\rm d}Z'({\boldsymbol{\kappa }'},\sigma ')\exp \left [\left ({\boldsymbol{\kappa }} + {\boldsymbol{\kappa }'}\right )\boldsymbol{\cdot }{\boldsymbol{x}} - \left (\sigma + \sigma '\right )t\right ]. \end{align}

Similarly, the pressure terms $p_{d2}({\boldsymbol{x}},t)$ and $p_{d3}({\boldsymbol{x}},t)$ corresponding to $(\partial \varGamma /\partial z)^2$ , and $\zeta (\partial ^2\varGamma /\partial z^2)$ can be expressed in terms of their Fourier–Stieltjes expansions as

(3.3) \begin{align} p_{d2}({\boldsymbol{x}},t)&= -\rho g^2 \iint \limits _{{\boldsymbol{\kappa }} {\boldsymbol{\kappa }'}}\iint \limits _{\sigma \sigma '}\frac {\kappa \kappa '}{\sigma \sigma '}\left (\frac {\sinh \kappa (z+H-h)}{\cosh \kappa (H-h)}\right )\left (\frac {\sinh \kappa '(z+H-h)}{\cosh \kappa '(H-h)}\right )\nonumber \\ &\quad\times \exp \left [\left ({\boldsymbol{\kappa }} + {\boldsymbol{\kappa }'}\right )\boldsymbol{\cdot }{\boldsymbol{x}} - \left (\sigma + \sigma '\right )t\right ], \end{align}

and

(3.4) \begin{align} p_{d3}({\boldsymbol{x}},t)=-\rho g^2\iint \limits _{{\boldsymbol{\kappa }} {\boldsymbol{\kappa }'}}\iint \limits _{\sigma \sigma '}\frac {{\boldsymbol{\kappa }}\boldsymbol{\cdot }{\boldsymbol{\kappa }'}}{\sigma \sigma '}{\rm d}Z({\boldsymbol{\kappa }},\sigma ){\rm d}Z'({\boldsymbol{\kappa }'},\sigma ') \exp \left [\left ({\boldsymbol{\kappa }} + {\boldsymbol{\kappa }'}\right )\boldsymbol{\cdot }{\boldsymbol{x}} - \left (\sigma + \sigma '\right )t\right ]. \end{align}

For all three pressure components, for the unidirectional case when $\kappa$ and $\kappa '$ are both shoreward propagating,

(3.5) \begin{align} &\kappa + \kappa ' = k,\nonumber \\ &\sigma + \sigma ' = \omega . \end{align}

Here $k \gt \kappa , \kappa '$ and $\omega \gt \sigma , \sigma '$ for the wavenumbers and frequencies commonly observed in surface gravity wave spectra in the $(0.01, 2.00)$ Hz range. No wavenumbers $\kappa$ for such waves are small enough to form a combination $(\kappa ,\sigma )$ that lies on one of the first three acoustic-gravity dispersion surfaces (wave scattering can be significant for higher modes (Hasselmann Reference Hasselmann1963; Jensen et al. Reference Jensen, Kuperman, Porter and Schmidt2011)), and thus the second-order terms do not lead to resonant interactions and hence do not produce longer-lasting interactions such as the surface-wave–seafloor interactions examined in § 2. An interaction between oppositely propagating incident ( $I$ ) and reflected ( $R$ ) surface waves would produce an acoustic-gravity wave field with ${\boldsymbol{k}} = 0$ , and $\omega = 2\sigma$ . The dispersion relation defining this wave field is given by (Jensen et al. Reference Jensen, Kuperman, Porter and Schmidt2011; Korde Reference Korde2024)

(3.6) \begin{equation} \alpha \tan \alpha H = \frac {\lambda ' + 2\mu '}{\lambda }\alpha ',\ \ \alpha = \frac {\omega }{c_1},\ \ \alpha ' = \frac {\omega }{c_p}, \end{equation}

where $\lambda$ and $\lambda '$ are Lamé constants associated with dilatation of the fluid particles and of the seafloor, respectively, $\mu '$ is the Lamé constant associated with distortion of the seafloor, $c_1$ denotes the acoustic velocity in salt water and $c_p$ is the dilatational wave velocity in the seafloor.

In the limiting case of a rigid seafloor, $\lambda ', \mu ' \rightarrow \infty$ , and

(3.7) \begin{align} &\tan \alpha H \rightarrow \infty ,\ \Rightarrow \cos \alpha H\rightarrow 0, \nonumber \\ &\textrm {or},\ \frac {\omega _n H}{c_1} \rightarrow (2n+1)\frac {\pi }{2},\ n= 0,\ 1,\ 2,\ \ldots , \end{align}

because $k = 0$ (Longuet-Higgins Reference Longuet-Higgins1950; Kadri & Akylas Reference Kadri and Akylas2016; Korde Reference Korde2024). The resonance frequencies for the elastic seafloors would be in the neighbourhood of these $\omega _n$ values. The fundamental-mode ( $n=0)$ resonance frequency for $H \rightarrow 13$ m is approximately 28 Hz, which is far out of the frequency range of surface-gravity wave spectra.

For the shallow water depths ( $\sim O(10 \,\textrm {m})$ ) resonance can be expected only for the higher modes $n \geqslant 3$ , for which scattering effects may be large and the resulting acoustic-gravity wave field may not be sufficiently energetic.

For these reasons, the second-order pressure generation mechanisms described above are not considered further, and the focus is on wave interactions in which the seafloor plays a role. Interactions between incident and reflected waves of nearly equal frequencies and the seafloor are considered in § 4.

4. Wave–wave–seafloor interactions

Two types of three-wave interaction in shallow water are considered here: (i) one involving a shoreward-propagating wave, a reflected wave and the seafloor; (ii) the other involving the difference frequency wave resulting from quadratic interaction of two shoreward waves and the seafloor. Wave–wave–seafloor interactions generating another surface wave have been investigated (Arduin, Drake & Herbers Reference Arduin, Drake and Herbers2002). In contrast, the object here is to study such interactions when they lead to acoustic-gravity waves. An interaction between incident and reflected waves of unequal, but comparable frequencies and a sloping seafloor can lead to a resonant combination $({\boldsymbol{k}},\omega )$ such that

(4.1) \begin{align} {\boldsymbol{\kappa }}_I + {\boldsymbol{\kappa }}_R + {\boldsymbol{k}}_s &= {\boldsymbol{k}},\nonumber \\ \sigma _I + \sigma _R + 0&= \omega . \end{align}

For unidirectional surface waves and seafloor contours,

(4.2) \begin{align} \kappa _I - \kappa _R - k_s &= k,\nonumber \\ \sigma _I + \sigma _R &=\omega , \end{align}

where $(k,\omega )$ is a point on the acoustic-gravity–Scholte wave dispersion curve for the depth $H$ and the known $\lambda$ , $\lambda '$ and $\mu '$ .

Other interactions of interest here involve difference-frequency waves resulting from quadratic interactions between two shoreward propagating waves with closely spaced frequencies. Nonlinear interaction of these low-frequency, low-wavenumber waves with the seafloor could produce energetic acoustic-gravity and Scholte waves. When the wave components are normal to the seafloor contours,

(4.3) \begin{align} \kappa _1 - \kappa _2 - k_s &= k,\nonumber \\ \sigma _1 - \sigma _2 &= \omega . \end{align}

Letting $\kappa$ represent $\kappa _I$ or $\kappa _1$ and $\kappa '$ denote $\kappa _R$ or $\kappa _2$ , the treatment below can be applied to both types of interaction ((4.2) and (4.3)). Specifically, the free-surface boundary condition is linearized and satisfied at the undisturbed free surface $z = 0$ . It is proposed that the wave–wave–seafloor interactions of interest can be understood as a second-order interaction between two surface waves as represented by the quantity $({\boldsymbol {\nabla }} \varGamma )^2$ (where $\boldsymbol{\nabla}$ represents the full gradient) and its second-order interaction with the seafloor. In shallow water over a flat seafloor, the pressure resulting from the $({\boldsymbol{\nabla }} \varGamma )^2$ term is cancelled by the pressure from the $u^2$ term at the seafloor (Ardhuin & Herbers Reference Ardhuin and Herbers2013). When the seafloor profile is mildly non-uniform the $\partial \varGamma /\partial z$ term is non-zero at second order, which also results in second-order variation in the horizontal velocity (because ${\boldsymbol{\nabla } \boldsymbol{\cdot }{\boldsymbol{u}}} = 0$ ). The goal here is to quantify the pressure resulting from second-order interactions between the $(\boldsymbol{\nabla }\varGamma )^2$ term and the seafloor profile.

The pressure spectrum due to the $(\boldsymbol{\nabla }\varGamma )^2$ at $z = 0$ is evaluated for two interacting surface waves using a modified form of the spectrum in Hasselmann (Reference Hasselmann1963) that includes a depth correction factor $d_c = \tanh \kappa H \tanh \kappa ' H$ attached to the wavenumber product representing the vertical derivatives,

(4.4) \begin{align} S'_e({\boldsymbol{k}}',\omega ) &= g^4\iint \limits _{{\boldsymbol{\kappa }}, \sigma }\iint \limits _{{\boldsymbol{\kappa }}',\sigma '}S_{\zeta }({\boldsymbol{\kappa }},\sigma )S_{\zeta '}({\boldsymbol{\kappa }}',\sigma ')\frac {{\bigl(\kappa \kappa ' d_c - {\boldsymbol{\kappa }}{\boldsymbol{\kappa }}'\bigr)}^2}{(\sigma \sigma ')^2}\delta \bigl({\boldsymbol{k}}' - {\boldsymbol{\kappa }}-{\boldsymbol{\kappa }}'\bigr)\nonumber \\ &\quad\times \delta \bigl(\omega -\sigma \mp \sigma '\bigr){\rm d}{\boldsymbol{\kappa }} \,{\rm d}{\boldsymbol{\kappa }}'{\rm d}\sigma {\rm d}\sigma '. \end{align}

Equation (4.4) is consistent with (A.2) in (Ardhuin & Herbers Reference Ardhuin and Herbers2013) when applied at $z = 0$ .

The wavenumber spectrum in (4.4) is based on the second-order pressure as defined by $\rho (\boldsymbol{\nabla }\varGamma )^2$ , where the gradient operator includes the vertical component (the terms $\kappa \kappa '$ appearing alongside the terms $\boldsymbol{\kappa } \boldsymbol{\cdot }\kappa '$ in (2.12) in Hasselmann (Reference Hasselmann1963), which was adapted into equation (5.5) in Korde (Reference Korde2024)),

(4.5) \begin{align} {\boldsymbol{\kappa }} + {\boldsymbol{\kappa }}' &= {\boldsymbol{k}}',\nonumber \\ \sigma \pm \sigma ' &= \omega . \end{align}

The pressure variation due to $(\boldsymbol{\nabla }\varGamma )^2$ acts at $z = 0$ . The effect of this pressure variation can be replaced by an equivalent free-surface elevation $\zeta _{eq}$ given by

(4.6) \begin{equation} \zeta _{eq} = \frac {(\boldsymbol{\nabla }\varGamma )^2}{2g}. \end{equation}

The interaction of the equivalent surface elevation $\zeta _{eq}$ with the seafloor can be evaluated using the method of § 2 for propagation in the $x$ direction, starting with

(4.7) \begin{equation} p_{\Delta q}(x,-H+h;t) = \rho g \frac {k'\tan k'H}{\cosh k'H}\zeta _{eq}(x,t)h(x). \end{equation}

Subsequent steps similar to those in § 2 lead to

(4.8) \begin{equation} S_{Ps}({\boldsymbol{k}}, t) = \rho ^2 g^2\iint \limits _{{\boldsymbol{k}}' {\boldsymbol{k}_s}} {\boldsymbol{k}}'\boldsymbol{\cdot }{\boldsymbol{k}}'S'_{eq}({\boldsymbol{k}}', t) S_{sf}({\boldsymbol{k}}_s) \delta ({\boldsymbol{k}}-{\boldsymbol{k}}' - {\boldsymbol{k}}_s) {\rm d}{\boldsymbol{k}}'{\rm d}{\boldsymbol{k}_s}, \end{equation}

where, using (4.6),

(4.9) \begin{equation} S_{eq}({\boldsymbol{k}}',t) = \frac {S'_e({\boldsymbol{k}}',t)}{(2g)^2}. \end{equation}

Equation (4.9) can be expressed in terms of wavenumber–frequency spectra via a Fourier–Stieltjes expansion into wave frequency space. This pressure variation at the seafloor (representing the two types of wave–wave–seafloor interactions above) contributes one source of excitation to the acoustic-gravity–Scholte wave field in the two media system comprised of the water column and the seafloor (see § 5). The temporal evolution of the acoustic-gravity–Scholte wave field can then be derived using the approach summarized in § 6.

5. Acoustic-gravity–Scholte wave field in shallow water over non-uniform depth

Amplitude variations for acoustic-gravity waves in shoaling depths were computed using energy flux arguments (Eyov et al. Reference Eyov, Klar, Kadri and Stiassnie2013), and a perturbation approach within a depth-averaged formulation (Renzi Reference Renzi2017) was used to evaluate the amplitudes and frequencies of acoustic-gravity waves over a mildly sloping seafloor. Here, an approach based on the coupled mechanics of acoustic-gravity and Scholte waves is employed, and the different wave fields are assumed to be stationary random processes. Wavenumber–frequency spectra are derived for a wave field comprised of compression waves in the shallow ocean layer and Scholte waves on a linearly sloping seafloor with random features superimposed. A perturbation approach is used to accommodate the depth non-uniformity.

The domain consists of a finite ocean layer and a semi-infinite seafloor layer. Surface-wave forcing by the acoustic-gravity–Scholte wave field represented by a gravity term (Bondi Reference Bondi1947; Abdolali & Kirby Reference Abdolali and Kirby2017) is neglected because it is relatively small for the frequency range considered here (see § 6 and Appendix A) given the focus on the nearshore acoustic-gravity–Scholte wave groups. However, inclusion of the gravity term in the equations of motion impacts the phase velocities of generated wave fields for long wavelengths in the presence of water compressibility and seafloor elasticity (Abdolali et al. Reference Abdolali, Kadri and Kirby2019), and the treatment below is limited to frequencies $f \geqslant 0.01$ Hz. However, inclusion of the gravity term in the derivation of the dispersion relations below would enable an extension of the results here to include similar interactions with longer-period, lower-frequency waves.

Ocean floors generally have properties varying with depth below the seabed, and often a multilayer approximation for the semi-infinite seafloor (each with different elastic constants and densities (Dziewonski & Anderson Reference Dziewonski and Anderson1981)) is used. A multilayer model with one layer modelling the ocean and six layers representing the seafloor was investigated in $\sim$ 100 m water depths (Kibblewhite & Wu Reference Kibblewhite and Wu1993). Although a multilayered seafloor representation would be preferred in a study that includes seismic surface waves and body waves (e.g. P and S waves) generated by sea–surface-wave interactions, here the focus is on seafloor surface waves that decay rapidly with depth. Thus, here the seafloor is represented as a single layer with properties averaged over the top $\sim 1000$ m. The origin is located at the seafloor with the $x$ axis along the propagation direction approaching from $x\rightarrow -\infty$ , with the $y$ axis pointing vertically up (Korde Reference Korde2024). The shore is at a distance $L_s$ along the propagation direction. Part of the wave field is assumed to be reflected back from the shore, with the rest manifesting as a Rayleigh wave propagating along the land surface. The effect of shore reflection is modelled using an image domain from $L_s$ to $\infty$ . Assuming $L_s$ to be long, the entire domain is taken to be $-\infty \lt x \lt \infty$ and $0 \leqslant y \leqslant H-h$ , where $H$ denotes the water depth measured from $y = 0$ at the bottom of the continental shelf, and $y = h(x)$ represents the seafloor profile superimposed over the plane $y = 0$ . Part of the goal is to derive the dispersion relation relating $\boldsymbol{k}$ , $\alpha$ and $\omega$ for the coupled compression-wave–seafloor-wave system, where $\boldsymbol{k}$ denotes the horizontal wavenumber, $\alpha$ denotes the vertical wavenumber and $\omega$ is the angular frequency of the wave field. Free acoustic-gravity–Scholte wave groups (such as created by impulsive plate movement or an underwater explosion) satisfy the dispersion relation, and resonance occurs when wavenumber–frequency combinations for acoustic-gravity–Scholte waves forced by interacting surface waves or surface-wave–seafloor interactions satisfy the dispersion relation. The dispersion relation defines multiple compression-wave–seafloor-wave modes, with each mode defining a unique dispersion surface.

The underwater wave field must satisfy the interface conditions at the water–air interface $y = H-h$ and at the water–seafloor interface $y = h$ , $h \neq 0$ , in addition to the radiation condition that the waves are incoming at $x \rightarrow -\infty$ and outgoing at $x \rightarrow \infty$ . The water layer can support only dilatational waves (here referred to as compression waves or acoustic-gravity waves) for which $\phi (x,t)$ denotes the scalar displacement potential, which defines the displacement vector as

(5.1) \begin{equation} {\boldsymbol{u}} = {\boldsymbol{\nabla }} \phi ,\ \ \textrm {or}\ u = \frac {\partial \phi }{\partial x},\ \ v = \frac {\partial \phi }{\partial y}. \end{equation}

The seafloor can support dilatational and distortional (i.e. shear mode) waves, requiring a scalar potential $\phi '(x,t)$ and a vector potential $\boldsymbol{\varPsi }$ . For propagation in the $x$ direction, only the vertical shear mode exists, so that ${\boldsymbol{\varPsi }} = [0, \ \varPsi _y]^T$ . The seafloor displacements $u'$ and $v'$ ( $'$ denoting seafloor) can be expressed as

(5.2) \begin{equation} {\boldsymbol{u}'} = {\boldsymbol{\nabla }\phi '} + {\boldsymbol{\nabla }}\times {\boldsymbol{\varPsi }}. \end{equation}

Thus,

(5.3) \begin{equation} u' = \frac {\partial \phi '}{\partial x} + \frac {\partial \varPsi _y}{\partial y}, \ \textrm {and}\ v' = \frac {\partial \phi '}{\partial y} - \frac {\partial \varPsi _y}{\partial x}. \end{equation}

The boundary condition at $y = h(x)$ , or $f(x, y) =y - h(x) =0$ is that the fluid and solid particles on either side of the seafloor remain in contact with the seafloor and hence their components normal to the seafloor are equal. Thus (Whitham Reference Whitham1973),

(5.4) \begin{align} \frac {{\rm d}f}{{\rm d}t} = \frac {\partial f}{\partial t} + u\frac {\partial f}{\partial x} + v\frac {\partial f}{\partial y} = 0,\nonumber \\ \frac {{\rm d}f}{{\rm d}t} = \frac {\partial f}{\partial t} + u'\frac {\partial f}{\partial x} + v'\frac {\partial f}{\partial y} = 0. \end{align}

Since the seafloor shape is independent of time at the time scales of this problem, (5.4) leads to

(5.5) \begin{equation} v - u\frac {{\rm d}h}{{\rm d}x} = 0,\ \textrm {and}\ v' - u'\frac {{\rm d}h}{{\rm d}x} = 0. \end{equation}

Subtracting the second equation from the first,

(5.6) \begin{equation} (v-v') - (u-u')\frac {{\rm d}h}{{\rm d}x} = 0. \end{equation}

The boundary condition at the free-surface $y = H-h$ is kinetic and requires that the normal stress $\tau _{yy}$ opposing fluid oscillation at the free surface equal the pressure $p$ on the free surface. That is

(5.7) \begin{equation} \tau _{yy} = \lambda \left (\frac {\partial u}{\partial x} + \frac {\partial v}{\partial y}\right ) = p, \end{equation}

where $\lambda$ is a Lamé constant associated with dilatation of the fluid particles and $p$ denotes the pressure field at the free surface due to wave–wave and wave–seafloor interactions. For free waves, $p = 0$ . At the fluid–solid interface at $y = h$ , the normal stress on the water side must equal the normal stress on the seafloor side. The stress components normal to and tangential to the inclined seafloor at $(x, h)$ are given by the terms with overbars,

(5.8) \begin{align} &{\overline \tau }_{yy} = \frac {\tau _{xx}+\tau _{yy}}{2} - \frac {\tau _{xx} - \tau _{yy}}{2}\cos 2\theta ' - \tau _{xy}\sin 2\theta ',\nonumber \\ &{\overline \tau }_{xy} = \frac {\tau _{yy} - \tau _{xx}}{2} + \tau _{xy}\cos 2\theta '. \end{align}

Here, $\theta '$ denotes the local angle of inclination of the seafloor such that $\tan \theta '$ is the local slope of the seafloor. Assuming $\theta '$ to be small, to first order,

(5.9) \begin{equation} \tan \theta ' \approx \sin \theta '\approx \theta ', \ \ \textrm {and}\ \ \cos \theta ' \approx 1. \end{equation}

The horizontal stress component on a flat seafloor on the water side is $\tau _{xx} = 0$ , because the horizontal oscillation is not opposed in the absence of viscous friction, which is neglected here. The shear stress is zero because the fluid cannot support shear oscillations. The stress components on the inclined seafloor $y = h(x)$ are (Auld (Reference Auld1990), chapters 1 and 2)

(5.10) \begin{align} &{\overline \tau }_{yy} = \tau _{yy},\ \textrm {and}\nonumber \\ &{\overline \tau }_{xy} = \frac {\tau _{yy}}{2}\sin 2\theta \approx \frac {\tau _{yy}}{2} 2\theta ' \approx \tau _{yy}\frac {{\rm d}h}{{\rm d}x}. \end{align}

On the solid side, the shear stress is non-zero, and

(5.11) \begin{align} &{\overline \tau '}_{yy} = {\tau '}_{yy} - 2{\tau '}_{xy}\frac {{\rm d}h}{{\rm d}x},\nonumber \\ &{\overline \tau '}_{xy} = {\tau '}_{xy} + {\tau '}_{yy}\frac {{\rm d}h}{{\rm d}x}. \end{align}

The kinetic conditions at $y = h$ are

(5.12) \begin{align} &{\overline \tau }_{yy} = {\overline \tau '}_{yy},\ \textrm {and}\nonumber \\ &{\overline \tau }_{xy} = {\overline \tau '}_{xy}. \end{align}

It is easier to work in the original coordinate system as long as the relationships in (5.11) and (5.12) are correctly applied. The stress components on the solid side of the seafloor are related to the particle displacements in the solid as

(5.13) \begin{align} &{\tau '}_{yy} = \lambda '\left (\frac {\partial u'}{\partial x} + \frac {\partial v'}{\partial y}\right ) + 2\mu '\frac {\partial v'}{\partial y},\ \textrm {and}\nonumber \\ &{\tau '}_{xy} = \mu '\left (\frac {\partial u'}{\partial y} + \frac {\partial v'}{\partial x}\right )\!. \end{align}

It follows that, at $y = h$ ,

(5.14) \begin{align} \lambda \left (\frac {\partial u}{\partial x} + \frac {\partial v}{\partial y}\right ) &= \lambda '\left (\frac {\partial u'}{\partial x} + \frac {\partial v'}{\partial y}\right ) + 2\mu '\frac {\partial v}{\partial y} - 2\mu '\left (\frac {\partial u'}{\partial y} + \frac {\partial v'}{\partial x}\right )\!,\ \textrm {and}\nonumber \\ \lambda \left (\frac {\partial u}{\partial x} + \frac {\partial v}{\partial y}\right )\frac {{\rm d}h}{{\rm d}x} &= \mu '\left (\frac {\partial u'}{\partial y} + \frac {\partial v'}{\partial x}\right ) + \left [\lambda '\left (\frac {\partial u'}{\partial x} + \frac {\partial v'}{\partial y}\right ) + 2\mu '\frac {\partial v'}{\partial y}\right ]\frac {{\rm d}h}{{\rm d}x}. \end{align}

Although the seafloor bathymetry $h(x)$ may be known, it could be represented as an approximately known depth variation function with a random term added. Further, because $\partial h/\partial x$ is dimensionless, (5.6) has the dimension of the deflections $u, v, u'$ and $v'$ . Hence, even though a perturbation series is needed for investigating nonlinear interactions, the expansions may be applied to the original equations without affecting the dimensions of the terms appearing in the expansions for each order. Here, $\epsilon = h_a k_a/2\pi$ is used as a perturbation parameter, with $h_a$ denoting the nominal depth of the seafloor, and $k_a$ a nominal wavenumber in the acoustic-gravity wave field. For the $i$ th dispersion mode, $^i\phi$ , $^i\phi '$ and $^iH_y$ are expanded as

(5.15) \begin{align} ^i\phi &= { }^i{\phi }_1 + { }^i{\phi }_2 + { }^i{\phi }_3 + \ldots ,\nonumber \\ ^i\phi ' &= { }^i{\phi '}_1 + { }^i{\phi '}_2 + { }^i{\phi '}_3 + \ldots ,\nonumber \\ ^iH_{y} &= { }^i{H}_{y1} + { }^i{H}_{y2} + { }^i{H}_{y3} + \ldots , \end{align}

with ${ }^i{\phi }_1 = { }^i{\phi }_1$ , each subsequent order in the expansion can be expressed as $^i\phi _2 = \epsilon { }^i{\phi }_1$ , $^i\phi _3 = \epsilon ^2{ }^i{\phi }_1$ , and so forth (Hasselmann Reference Hasselmann1966). The acoustic-gravity–Scholte wave field resulting from a stationary random surface-wave field also is a stationary random process, and thus the potentials are expanded using Fourier–Stieltjes integrals as

(5.16) \begin{align} &^i\phi _{\!j} = \iint \limits _{ }^i{k_{\!j}, ^i\omega _{\!j}}\bigl(^i{\rm d}A_{j1}\cos { }^i{\alpha }_{\!j}y + { }^i{{\rm d}A}_{j2}\sin { }^i{\alpha }_{\!j}y\bigr)\exp \bigl(^ik_{\!j} x-{ }^i{\omega }_{\!j} t\bigr),\ i, j = 1, 2, 3, \ldots ,\nonumber \\ &^i{\phi '}_{\!j} = \iint \limits _ { }^i{k_{\!j}, ^i\omega _{\!j}}(^i {{\rm d}A'}_{\!j}\exp \bigl({ }^i{\alpha '}_{\!j}y)\exp (^ik_{\!j} x-{ }^i{\omega }_{\!j} t\bigr),\ i, j = 1, 2, 3,\ldots ,\nonumber \\ &^iH_{yj} = \iint \limits _{ }^i{k_{\!j}, { }^i{\omega }_{\!j}}({ }^i{{\rm d}B'}_{\!j}\exp \bigl({ }^i{\beta '}_{\!j} y\bigr)\exp \bigl(^ik_{\!j} x - { }^i{\omega }_{\!j} t\bigr),\ i, j = 1, 2, 3, \ldots . \end{align}

Here,

(5.17) \begin{align} &^i\alpha _{\!j} = \sqrt {\left(^i\omega ^2_{\!j}/c^2_1 - { }^i{k}^2_{\!j}\right)},\nonumber \\ &^i {\alpha '}_{\!j} = \sqrt {\left(^ik^2_{\!j} - { }^i{\omega }^2_{\!j}/c^2_p\right)},\nonumber \\ &^i{\beta '}_{\!j} = \sqrt {\left(^ik^2_{\!j} - { }^i{\omega }^2_{\!j}/c^2_s\right)}. \end{align}

A real-valued ${ }^i{\alpha }_{\!j}$ represents an oscillatory wave solution in the water (for mode $i$ and expansion order $j$ ), whereas real-valued ${ }^i{\alpha '}_{\!j}$ and ${ }^i{\beta '}_{\!j}$ represent an exponentially decreasing solution for the vertical oscillations within the solid (Korde Reference Korde2024). The seafloor $y = h(x)$ can be expanded as

(5.18) \begin{equation} h(x) = H_1 + H_2 + H_3 + \ldots . \end{equation}

with

(5.19) \begin{equation} H_{\!j} = \int \limits _{k_{sj}}{\rm d}H_{\!j}\exp (-ik_{sj}x),\ j = 1, 2, 3, \ldots . \end{equation}

If the seafloor shape $y = h(x)$ is such that $h(x)$ can be treated as small, then the trigonometric and exponential functions containing $h$ (via $y$ ) in the expansions of equation (5.16) can be approximated to third order as

(5.20) \begin{align} { }^i{{\rm d}A}_{1j}\cos { }^i{\alpha }_{\!j} h + { }^i{{\rm d}A}_{2j}\sin { }^i{\alpha }_{\!j} h &\approx { }^i{{\rm d}A}_{1j}\left (1 + \frac {\bigl({ }^i{\alpha }_{\!j} h\bigr)^2}{2!}+ \ldots \right )\nonumber\\&\quad + { }^i{{\rm d}A}_{2j}\left (\bigl({ }^i{\alpha }_{\!j}h\bigr) + \frac {\bigl({ }^i{\alpha }_{\!j}h\bigr)^3}{3!}+ \ldots \right )\!, \end{align}

and

(5.21) \begin{align} &{ }^i{{\rm d}A'}_{\!j}\exp {\bigl({ }^i{\alpha '}_{\!j}h\bigr)} \approx { }^i{{\rm d}A'}_{\!j}\left (1 + { }^i{\alpha '}_{\!j} h + \frac {\bigl({ }^i{\alpha '}_{\!j} h\bigr)^2 }{2!} + \frac {\bigl({ }^i{\alpha '}_{\!j}h\bigr)^3}{3!} + \ldots \right )\!,\nonumber \\ &{ }^i{{\rm d}B'}_{\!j}\exp {({ }^i{\beta '}_{\!j}h)} \approx { }^i{{\rm d}B'}_{\!j}\left (1 + { }^i{\beta '}_{\!j} h + \frac {\bigl({ }^i{\beta '}_{\!j} h\bigr)^2 }{2!} + \frac {\bigl({ }^i{\beta '}_{\!j}h\bigr)^3}{3!} + \ldots \right )\!. \end{align}

The Fourier–Stieltjes expansions in (5.16) with the approximations in (5.20) and (5.21) are substituted into the four boundary conditions for the ocean–seafloor system. Carrying out the analysis up to third order, the following equations result:

order one,

(5.22) \begin{align} -\lambda ({ }^i{k}^2_1 + { }^i{\alpha }^2_1)\left ({ }^i{{\rm d}A}_{11}\cos { }^i{\alpha }_1 H + { }^i{{\rm d}A}_{21}\sin { }^i{\alpha }_1 H\right ) &= 0,\nonumber \\ { }^i{\alpha }_1^i{\rm d}A_2 - { }^i{\alpha '}_1 { }^i{{\rm d}A'}_1 + {i^ik}_1 { }^i{{\rm d}B'}_1 &= 0,\nonumber \\ -\lambda (^ik^2_1 + { }^i{\alpha ^2}_1) ^i{\rm d}A_1 + \left (\lambda '({ }^i{k}^2_1 - { }^i{\alpha '}^2_1) - 2\mu ' { }^i{{\alpha '}^2}_1\right ) { }^i{{\rm d}A'}_1 + 2i\mu '\ ^ik_1{ }^i{\beta '}_1{ }^i{{\rm d}B'}_1 &= 0,\nonumber \\ -2i\mu '\ ^ik_1{ }^i{\alpha '}_1 { }^i{{\rm d}A'}_1 + \mu '(^ik^2_1 + { }^i{{\beta '}^2}_1){ }^i{{\rm d}B'}_1 &= 0; \end{align}

order two,

(5.23) \begin{align} &-\lambda ({ }^i{k}^2_2 + { }^i{\alpha }^2_2)\left ({ }^i{{\rm d}A}_{12}\cos { }^i{\alpha }_2 H + { }^i{{\rm d}A}_{22}\sin { }^i{\alpha }_2H\right ) = -i\cot { }^i{\alpha }_1H { }^i{\alpha }_1{ }^i{{\rm d}A}_{11} {\rm d}H_1,\nonumber \\ &\quad {}^i\alpha _2^i{\rm d}A_{22} -{ }^i{\alpha '}_2 { }^i{{\rm d}A'}_2 + i^ik_2{ }^i{{\rm d}B'}_2 = -i { }^i{\alpha }^2_1{ }^i{{\rm d}A'}_1 {\rm d}H_1 + { }^i{\alpha '}^2_1{ }^i{{\rm d}A'}_1 {\rm d}H_1 - k_{s1}{ }^i{{\rm d}B'}_1{\rm d}H_1 \nonumber \\ &\quad -{ }^i{k}_1 k_{s1}{ }^i{{\rm d}A'}_1{\rm d}H_1 -\lambda ({ }^i{k^2}_2 + { }^i{\alpha ^2}_2) { }^i{{\rm d}A}_{11} + \left (\lambda '(^ik^2_2 - ^i{\alpha '}^2_2) - 2\mu '{ }^i{{\alpha '}^2}_2\right ) { }^i{{\rm d}A'}_2\nonumber \\ &\quad + 2i{\mu '}^ik_2{ }^i{\beta '}_2{ }^i{{\rm d}B'}_2 = +\lambda '({ }^i{k^2}_1 - { }^i{\alpha ^2}_1){ }^i{\alpha }_1{ }^i{{\rm d}A}_{21} {\rm d}H_1 + {i^ik}_1({ }^i{k}_1+k_{s1}) { }^i{\alpha }_1{ }^i{{\rm d}A'}_1{\rm d}H_1 \nonumber \\ &\quad +i\lambda '{ }^i{\alpha '}^3_1{ }^i{{\rm d}A}_1{\rm d}H_1 + 2i\mu ' i(^ik_1+k_{s1}) { }^i{\beta '}^2_1{ }^i{ {\rm d}B'}_1{\rm d}H_1,\nonumber \\ &\quad -2\mu '{ }^i{\alpha '}_2{ }^i{k_2}{ }^i{{\rm d}A'}_2 + \mu '\ ({ }^i{k}^2_2-{ }^i{\beta '}^2_2){ }^i{{\rm d}B'}_2 = -2i\mu ' { }^i{\alpha '}^2(k_1 + k_{s1}){ }^i{{\rm d}A'}_1{\rm d}H_1 \nonumber \\ &\quad -2i\mu '{ }^i{\beta }^2{ }^i{{\rm d}B'}_1{\rm d}H_1 - 2i\mu ' { }^i{k}_1{ }^i{\beta '}({ }^i{k}_1+k_{s1}){ }^i{{\rm d}B'}_1{\rm d}H_1; \end{align}

order three,

(5.24) \begin{align} &-\lambda \left({ }^i{k}^2_3 + { }^i{\alpha }^2_3\right)\left ({ }^i{{\rm d}A}_{13}\cos { }^i{\alpha }_3H + { }^i{{\rm d}A}_{23}\sin { }^i{\alpha }_3 H\right ) = -i\cot { }^i{\alpha }_2 H { }^i{\alpha }_2{ }^i{{\rm d}A}_{12}{\rm d}H_1,\nonumber \\ &{ }^i{\alpha }_3 { }^i{{\rm d}A}_{23} + i{ }^i{k}_3{ }^i{{\rm d}B'}_3 - i { }^i{\alpha '}_3 { }^i{{\rm d}A'}_3 = { }^i{\alpha }^2_1 { }^i{{\rm d}A}_{21}{\rm d}H_2 + { }^i{\alpha }^2_2 { }^i{{\rm d}A}_{22}{\rm d}H_1 + { }^i{\alpha '}^2_1 { }^i{{\rm d}A'}^2_1{\rm d}H_2\nonumber \\ &\quad+ { }^i{\alpha '}^2_2 { }^i{{\rm d}A'}_2{\rm d}H_1+{ }^i{k}_1 { }^i{\beta }_1{ }^i{{\rm d}B'}_1 {\rm d}H_2 + { }^i{k}_1 k_{s1}{ }^i{{\rm d}A}_{21}{\rm d}H_1 + {i^ik}_1{ }^i{\alpha }_1k_{s1}{ }^i{{\rm d}A}_{11}{\rm d}H_2\nonumber \\ &\quad -{ik}^2_{s1}{ }^i{\beta }_1{ }^i{{\rm d}B'}_1{ }^i{{\rm d}H}_1 { }^i{dH}_1+ { }^i{\beta '}_1k_{s2}{ }^i{{\rm d}B'}_1 {\rm d}H_2 + { }^i{\beta '}_2k_{s1}{ }^i{{\rm d}B'}_2{\rm d}H_1,\nonumber \\ &-\lambda \left({ }^i{k}^2_3+{ }^i{\alpha }^2_3\right){ }^i{{\rm d}A}_{13} + \left (\lambda '\left({ }^i{k}^2_3 - { }^i{\alpha '}^2_3\right) - 2\mu ' { }^i{\alpha '}^2_3\right ) { }^i{{\rm d}A'}_3 + 2i\mu ' { }^i{k}_3{ }^i{\beta '}_3{ }^i{{\rm d}B'}_3 \nonumber \\ &\quad = -i\lambda { }^i{k}^2_1 { }^i{\alpha }^3_1{ }^i{{\rm d}A}_{21}{\rm d}H_2-i\lambda { }^i{k}^2_2{ }^i{\alpha }^3_2 { }^i{{\rm d}A}_{22}{\rm d}H1 + i\lambda '{ }^i{k}_1({ }^i{k}_1+k_{s2}){ }^i{\alpha '}_1{ }^i{{\rm d}A'}_{11}{\rm d}H_{s2}\nonumber \\ &\quad +i\lambda ' { }^i{k}_2(^ik_2+k_{s2}) ^i{\alpha '}_2^i {{\rm d}A'}_{21}{\rm d}H_1-2i{\mu '}^i{{\alpha '}^2}_1^i{{\rm d}A'}_1{\rm d}H_2 - 2{\mu '}^i{{\alpha '}^2}_2^i{{\rm d}A'}_2 {\rm d}H_1\nonumber \\ &\quad +2i{\mu '}^i{{\beta '}^2}_1(^ik_1 + k_{s2}) ^i{\rm d}B'_1{\rm d}H_2 + 2i{\mu '}^ik_2^i{{\beta '}^2}_2 ^i{\rm d}B'_2{\rm d}H_1,\nonumber \\ &-2i\mu '{ }^i{\alpha '}_3{ }^i{k_3}{ }^i{{\rm d}A'}_3 + \mu '({ }^i{k^2}_3+{ }^i{\beta '}^2_3){ }^i{{\rm d}B'}_3 = 2{\mu '}^i{\alpha '}^2_2(^ik_{2s} +k_{s1})^i{{\rm d}A'}_2{\rm d}H_1\nonumber \\ &\quad +2{\mu '}^i{\alpha '}^2_1(^ik_1+k_{s2}) ^i{{\rm d}A'}_1{\rm d}H_2 -i{\mu '}^i{\beta '}^3 { }^i{{\rm d}B'}_1 {\rm d}H_2 - i\mu ' { }^i{\beta '}^3{ }^i{{\rm d}B'}_1{\rm d}H_1\nonumber \\ &\quad -i\mu ' { }^i{k}_2({ }^i{k}_2+k_{s1}){ }^i{\beta '}_2{{\rm d}B'}_2{\rm d}H_1 - i\mu ' { }^i{k}_1({ }^i{k}_1+k_{s2}){ }^i{\beta '}_1{{\rm d}B'}_1{\rm d}H_2 \nonumber \\ &\quad - i\lambda '{ }^i{\alpha '}^2k_{s1}{ }^i{{\rm d}A'}_1{\rm d}H^2_1-i { }^i{k}^2_2k_{s1}{ }^i{{\rm d}A'}_2{\rm d}H_1 - i\lambda '{ }^i{k}^2_1k_{s2}{ }^i{{\rm d}A'}_2{\rm d}H_2\nonumber \\ &\quad - 2i\mu ' { }^i{\alpha '}_1{ }^i{k}_1k_{s1}{ }^i{{\rm d}A'}_1{\rm d}H_1{\rm d}H_1-2\mu ' { }^i{k}_1 { }^i{\beta '}_1 k_{s1}{ }^i{{\rm d}B'}_1 {\rm d}H_1 {\rm d}H_1\nonumber \\ &\quad - 2i\mu ' { }^i{k}_2k_{s1}{ }^i{\beta '}_2 { }^i{{\rm d}B'}_2 {\rm d}H_1 -2i\mu '\ ({ }^i{k}_1+k_{s1}){ }^i{k}_1{ }^i{\beta '}_1{ }^i{{\rm d}B'}^2_1{\rm d}H_1\nonumber \\ &\quad - 2i\mu ' { }^i{k}_1{ }^i{\beta '}_1 k_{s2}{ }^i{{\rm d}B'}_1{\rm d}H_2. \end{align}

Equations (5.22) describe the conditions to be satisfied by the acoustic-gravity–Scholte wave field following an instantaneous excitation, such as that due to a heavy gust of wind. The interaction of this wave field with the seafloor non-uniformity then forces the second-order acoustic-gravity–Scholte wave field as described by (5.23). The interactions of the first- and second-order wave fields with the seafloor non-uniformity then produce a third-order wave field, and so forth. The entire wave field in the ocean–seafloor coupled system is described by the summation of the solutions to (5.22)–(5.24) and other higher-order solutions. The free wave field includes the order-one natural modes and the response to interactions between free acoustic-gravity waves and seafloor non-uniformity.

The left-hand sides of the three equations (5.22)–(5.24) are linear, and the excitation terms on the right-hand sides of (5.23) and (5.24) contain quantities known from solutions at the previous order. Equations (5.22)–(5.24) can be expressed in matrix form as

(5.25) \begin{align} &{\boldsymbol{D}}_1{\boldsymbol{{\rm d}A}}_1 = 0,\nonumber \\ &{\boldsymbol{D}}_2{\boldsymbol{{\rm d}A}}_2 = {\boldsymbol{{\rm d}F}}_2,\nonumber \\ &{\boldsymbol{D}}_3{\boldsymbol{{\rm d}A}}_3 = {\boldsymbol{{\rm d}F}}_3. \end{align}

In (5.25), ${\boldsymbol{{\rm d}F}}_2$ and ${\boldsymbol{{\rm d}F}}_3$ denote the terms on the right-hand sides of (5.23) and (5.24), respectively. These excitation terms represent how lower-order solutions force higher-order effects. In the absence of excitation at the free surface, the first of (5.25) has a non-trivial solution when the determinant of ${\boldsymbol{D}}_1$ vanishes, or

(5.26) \begin{equation} \Vert {\boldsymbol{D}}_1\Vert = 0. \end{equation}

The free-wave acoustic-gravity–Scholte wave field is any wave field that satisfies (5.26), which in full form can be written as

(5.27) \begin{equation} \begin{vmatrix} -\lambda \left(^ik^2_1+ { }^i{\alpha }^2_1\right) & -\lambda \left(^ik^2_1+ { }^i{\alpha }^2_1\right) & 0 & 0\\\cos { }^i{\alpha }_1H & \sin { }^i{\alpha }_1H & & \\ 0 & { }^i{\alpha }_1 & -{ }^i{\alpha }_1 & i{ }^i{k}_1\\ -\lambda \left (k^2 + {\alpha }^2\right ) & 0 & \left (\lambda '\left(^ik^2_1 - { }^i{\alpha '}^2_1\right) - 2\mu '{ }^i{\alpha '}^2_1\right ) & 2i{\mu '}^ik_1{ }^i{\beta '}_1\\ 0 & 0 & -2i\mu ' { }^i{k}_1{ }^i{\alpha '}_1 & \mu '\left(^ik^2_1 + { }^i{{\beta '}^2}_1\right) \end{vmatrix} = 0. \end{equation}

The condition in (5.27) leads to the dispersion relations for the $i$ th mode, and can be expressed in the form

(5.28) \begin{equation} \tan { }^i{\alpha }_1H = \frac {{ }^i{\alpha }_1\bigl({ }^i{A} { }^i{D}-{ }^i{B} { }^i{C}\bigr)}{\lambda \gamma ^2_1\bigl({ \alpha '}{ }^i{D} - i{ }^i{k_1}{ }^i{C}\bigr)}, \end{equation}

where

(5.29) \begin{align} &{ }^i{A} = \lambda '({ }^i{k}^2_1 - { }^i{\alpha '}^2_1) -2\mu ' { }^i{\alpha '}^2_1, \ \ { }^i{B} = 2i\mu ' { }^i{k}_1{ }^i{\beta '}_1,\nonumber \\ &{ }^i{C} = 2i{ }^i{ k}_1{ }^i{\alpha '}_1,\ \ { }^i{D} = { }^i{k}^2_1 + { }^i{\beta '}^2_1. \end{align}

Equations (5.28) and (5.29) define the dispersion relations for the fundamental and higher modes represented by $(\cos \alpha H, \sin \alpha H)$ , $(\cos 2\alpha H, \sin 2\alpha H)$ and so forth. These dispersion relations assume an undisturbed sea surface. Together with (5.17), they relate the horizontal and vertical wavenumber components to the angular frequency of the free wave acoustic-gravity–Scholte wave field supported by the coupled ocean-water column and seafloor system. The dispersion relations are discussed further in § 7.

The first term on the right-hand side of the first of (5.23) contains a product $^i{\rm d}A_{11} {\rm d}H_1$ which represents a second-order interaction between an acoustic-gravity wave and the seafloor. The case studied here is where an acoustic-gravity wave of mode one (i.e. $i=1$ ) interacts with the seafloor to excite a mode 2 resonance in the second-order acoustic-gravity wave field according to

(5.30) \begin{align} ^1{\boldsymbol{k}}_1 + {\boldsymbol{k}}_s &= {{ }^2{\boldsymbol{k}}}_2,\nonumber \\ \omega + 0 &= \omega . \end{align}

Modes 1 and 2 here refer to the first two modes of the dispersion relation. For waves propagating in a single direction normal to the seafloor contours,

(5.31) \begin{align} {{ }^1k}_1 - k_s &= {{ }^2k}_2,\nonumber \\ \omega + 0 &= \omega . \end{align}

Such a resonance is found to occur at $f = 0.03$ Hz for the present set of seafloor properties and water depth. Here ${\rm d}H_1$ for a sloping seafloor equals the Fourier transform derived in § 2:

(5.32) \begin{equation} {\rm d}H_1(k_s) = F_s(k_s). \end{equation}

The wavenumber–frequency spectrum for the excitation due to this interaction can be derived as

(5.33) \begin{align} S_{AH}(^2{\boldsymbol{k}}_2,\omega ) &= \left [i^1{\rm d}A_{11}(^1{\boldsymbol{k}}_1,\omega ) {\rm d}H_1({\boldsymbol{k}}_s)\right ]\left [ -i^1{\rm d}A^\ast _{11}(^1{\boldsymbol{k}}_1,\omega ) {\rm d}H^\ast _1({\boldsymbol{k}}_s)\right ],\nonumber \\ &=S_A(^1{\boldsymbol{k}}_1,\omega )S_f({\boldsymbol{k}_s}),\ \ S_f({\boldsymbol{k}_s})= {\overline F_s(k_s)}{\overline F^\ast _s(k_s)}. \end{align}

The calculation result for the $f= 0.11$ Hz case is discussed in § 8.

6. Forced wave generation

An energetic forced wave field is generated whenever one or more of the nonlinear wave interaction mechanisms above excites the water column–seafloor system into resonance. The analysis below largely is based on the method of Hasselmann (Reference Hasselmann1963) as applied with some extensions (Korde Reference Korde2024). A forced acoustic-gravity wave field would add a free-surface disturbance according to the term $g\alpha \varGamma$ (Bondi Reference Bondi1947; Abdolali & Kirby Reference Abdolali and Kirby2017), but is neglected because it has only a small effect within the frequency range of interest here (see Appendix A.1).

In the present situation, resonance may be excited by (i) nonlinear interaction between surface waves and the seafloor; (ii) difference-frequency waves arising from quadratic interaction between two closely spaced frequencies interacting with the seafloor; (iii) nonlinear interaction between surface waves, their reflections from the shore and the seafloor; (iv) acoustic-gravity waves interacting with the sloping seafloor. The forcing spectrum $S_E({\boldsymbol{k}},\omega )$ can be derived independently in each case, and they may act concurrently. In the case of direct linear forcing, if the pressure amplitudes are large enough to begin with, it is not as critical for the forcing wave field to excite an acoustic-gravity wave resonance as in the case of second-order wave–wave and wave–seafloor interactions for which energy transfer from the second-order effects to the acoustic-gravity wave field reaches a local maximum in the wavenumber–frequency domain at each resonance.

Here, the excitation is applied to the lowest order of the perturbation expansion. The forced-wave wavenumber spectrum $S_A({\boldsymbol{k}},t)$ is given by (Korde Reference Korde2024)

(6.1) \begin{equation} S_A({\boldsymbol{k}},t)\Big |_{DS} = \frac {\pi t { }^i{D}^2_1}{2\omega ^2_1} S_E(\omega _1,{\boldsymbol{k}})\Big |_{DS}. \end{equation}

Here $DS$ represents a dispersion surface, $S_E$ denotes the forcing spectrum causing resonance and $^iD_1$ for mode $i$ order-one for depth $H$ for excitation on the surface $\zeta = 0$ is

(6.2) \begin{equation} ^iD_1 = \frac {2 \left [\sin { }^i{\alpha }_1 H + f_1(1-\cos { }^i{\alpha }_1 H)\right ]}{\rho ^i\alpha _1 H(1+f^2_1)}. \end{equation}

Here, $f_1 = -\cot \alpha _1 H$ for mode $i$ . For a surface-wave frequency spectrum $S_e(\sigma )$ with the wave-elevation amplitude ${\rm d}Z(\sigma )$ and a reflection coefficient from shore $C_{SR}$ (assumed to be frequency-independent as a simplification), the excitation spectrum $S_E(\omega _1)$ can be written as

(6.3) \begin{equation} S_E(\omega _1) = C_{SR} S_e(\sigma ), \ \ \ \omega _1 = \sigma + \sigma , {\boldsymbol{k}} = {\boldsymbol \kappa } + {\boldsymbol{\kappa }}, \ \ \ k = \kappa - \kappa = 0. \end{equation}

The acoustic-gravity wave field in this case is purely vertical and there is no propagation in the horizontal direction. In shallow water, resonance is possible only for modes $i \geqslant 3$ , which undergo more scattering (Hasselmann Reference Hasselmann1963), and thus are not studied further here.

For acoustic-gravity wave interacting with the sloping seafloor with $S_{AH}({\boldsymbol{k}},\omega )$ , the Fourier coefficient is

(6.4) \begin{equation} {\rm d}S_{AH}({\boldsymbol{k}}, t) = \sqrt {S_{AH}({\boldsymbol{k}}, t){\rm d}{\boldsymbol{k}}}. \end{equation}

Thus

(6.5) \begin{equation} S_E({{{ }^2\boldsymbol{k}}}, t) = S_{AH}({{ }^1\boldsymbol{k}}, t) S_f({\boldsymbol{k}}_s),\ \ {{ }^2\boldsymbol{k}} = {{ }^1\boldsymbol{k}} + {\boldsymbol{k}}_s. \end{equation}

The excitation of acoustic-gravity and Scholte waves by the interaction of surface waves with the sloping seafloor needs a different approach. It follows from (2.30)–(2.32) that this interaction excites the acoustic-gravity–Scholte wave field from the seafloor. Further, the excitation is applied to the lowest-order system in (5.22). Hence, the forced system can be described as

(6.6) \begin{align} -\lambda ({ }^i{k}^2_1 + { }^i{\alpha }^2_1)\left ({ }^i{{\rm d}A}_{11}\cos ^i\alpha _1 H + { }^i{{\rm d}A}_{21}\sin { }^i{\alpha }_1H\right ) &= 0,\nonumber \\ { }^i{\alpha }_1^i{\rm d}A_2 - { }^i{\alpha '}_1 { }^i{{\rm d}A'}_1 + i^ik_1 { }^i{{\rm d}B'}_1 &= 0,\nonumber \\ -\lambda (^ik^2_1 + { }^i{\alpha ^2}_1) ^i{\rm d}A_1 + \left (\lambda '(^ik^2_1 - ^i{\alpha '}^2_1) - 2\mu '{ }^i{{\alpha '}^2}_1\right ) { }^i{{\rm d}A'}_1 + 2i{\mu '}^ik_1{ }^i{\beta '}_1{ }^i{{\rm d}B'}_1 &= {\rm d}P,\nonumber \\ -2i{\mu '}^ik_1{ }^i{\alpha '}_1 { }^i{{\rm d}A'}_1 + \mu '(^ik^2_1 + { }^i{{\beta '}^2}_1){ }^i{{\rm d}B'}_1 &= 0. \end{align}

For forced-wave solutions that lie on the dispersion surfaces for mode one, (5.22) can be used to express ${\rm d}A_{21}$ , ${\rm d}A'$ and ${\rm d}B'$ in terms of ${\rm d}A_{11}$ (Korde Reference Korde2024), Defining

(6.7) \begin{align} &A = \lambda '(k^2_1 - {\alpha '}^2_1) - 2 \mu '{\alpha '}^2_1, \ \ \ B = 2 i \mu ' k_1{\beta '}_1,\nonumber \\ &C = 2 i \mu 'k_1\beta '_1,\ \ \ D = \mu '(k^2_1 + {\beta '}^2_1), \end{align}

and

(6.8) \begin{align} & f_0 = -\lambda (k^2_1 + \alpha ^2_1),\ \ \ f_1 = -\cot \alpha _1 H, \nonumber \\ & f_2 = -\frac {\alpha _1}{\alpha ' - ik C/D} f_1, \ \ \ f_3 = -\frac {C}{D} f_2, \end{align}

the third of (6.6) can be written as

(6.9) \begin{equation} (f_0 + A f_2 + B f_3) {\rm d}A_{11} = {\rm d}P. \Rightarrow {\rm d}A_{11} = \left (\frac {1}{f_0 + A f_2 + B f_3}\right ) {\rm d}P. \end{equation}

Both ${\rm d}P$ and ${\rm d}A$ are the Fourier coefficients at each $(k,\omega )$ , and $f_0$ , $f_1$ and $f_2$ also are functions of $k$ and $\omega$ . Denoting the wavenumber as a vector to allow for directionality, $S_E({\boldsymbol{k}},t)$ can be expressed as

(6.10) \begin{equation} S_E({\boldsymbol{k}},t) = \frac {S_{P_\Delta }({\boldsymbol{k}}, t)}{(f_0 + Af_2 + Bf_3)^2}, \end{equation}

so that,

(6.11) \begin{equation} S_A({\boldsymbol{k}},t) = \frac {\pi t D^2_f}{2\omega ^2} S_E({\boldsymbol{k}}, t),\ \ \omega = \sigma . \end{equation}

Here

(6.12) \begin{equation} D_f = \frac {1}{(f_0 + Af_2 + Bf_3)}. \end{equation}

When the forcing mechanism acts on the surface $\zeta = 0$ , and hence, $S_E({\boldsymbol{k}},t) = S_{PS}({\boldsymbol{k}},t)$ (see (4.8)) so that

(6.13) \begin{equation} S_A({{ }^2\boldsymbol{k}},t)= \frac {\pi t {{ }^1D}^2_1}{2\omega ^2} S_E({\boldsymbol{k}},t). \end{equation}

The evolution equation for $S_A({\boldsymbol{k}}, t)$ (using $i = 1$ for example) can be written as (assuming stationarity of the surface-wave field) (Whitham Reference Whitham1973)

(6.14) \begin{equation} \frac {{\rm d}S_A({\boldsymbol{k}},t)}{{\rm d}t} = \frac {\pi D^2_{\!j}}{2\omega ^2_1} S_E(\omega _1,{\boldsymbol{k}}) - \frac {S_A}{t}. \end{equation}

Here $D_{\!j} = D_1$ or $D_f$ depending on the source of excitation. Interactions with bottom sediment, for example the formation of seafloor ripples, can cause dissipation (Arduin et al. Reference Arduin, Drake and Herbers2002). Using a damping rate of $c_t$ to include energy dissipation in the seafloor and (to a smaller extent) within the water layer,

(6.15) \begin{equation} \frac {{\rm d}S_A({\boldsymbol{k}},t)}{{\rm d}t} = \frac {\pi D^2_1}{2\omega ^2_1} S_E(\omega _1,{\boldsymbol{k}}) - \frac {S_A}{t} - c_tS_A. \end{equation}

If the interaction time is denoted as $t_e$ (Korde Reference Korde2024), the wavenumber spectrum at the present time $t \gt t_e$ can be found as

(6.16) \begin{equation} S_A({\boldsymbol{k}},t) = \int \limits _{0}^{t_e}\frac {\pi D^2_1}{2\omega ^2} S_E({\boldsymbol{k}},\tau ) {\rm d}\tau - \int \limits _{0}^{t}\left (\frac {1}{\tau } + c_t\right )S_A({\boldsymbol{k}},\tau ){\rm d}\tau . \end{equation}

If $t \leqslant t_e$ , then the upper limit on the first integral in (6.16) would change to $t$ . For a wave group propagating through the interaction region, $t\gt t_e$ means the group has moved out of the interaction region and is not being amplified further. With $\gamma = \omega /c_1 = \sqrt {k^2 + \alpha ^2}$ , the wavenumber spectrum $S_P({\boldsymbol{k}})$ for the pressure variation in the water column is (Korde Reference Korde2024)

(6.17) \begin{equation} S_P({\boldsymbol{k}},t) = \frac {{\rm d}P_1 {\rm d}P^\ast _1}{{\rm d}{\boldsymbol{k}}} = \left (\lambda \gamma ^2\right )^2 S_A({\boldsymbol{k}},t). \end{equation}

The wavenumber spectrum (convertible to a frequency spectrum) for the power density is then

(6.18) \begin{equation} S_W({\boldsymbol{k}},t) = \frac {S_P({\boldsymbol{k}},t)}{\rho c_1}. \end{equation}

The power density for the group ( ${\boldsymbol{k}},\omega )$ at the seafloor can be found using

(6.19) \begin{equation} \varPi ({\boldsymbol{k}},t) = \sqrt {S_W({\boldsymbol{k}},t) {\rm d}{\boldsymbol{k}}}, \end{equation}

where $\varPi$ is in units such as watts per square metre. The pressure amplitudes at the seafloor are given by the Fourier coefficients ${\rm d}P$ as

(6.20) \begin{equation} \vert {\rm d}P \vert = \sqrt {S_P({\boldsymbol{k}},t) {\rm d} \boldsymbol k},\ {\rm d}{\boldsymbol{k}} = {\rm d}k_1 {\rm d}k_2. \end{equation}

The wavenumber spectrum for the Scholte waves forming part of this group and the Scholte wave Fourier coefficients are given by

(6.21) \begin{align} S_R({\boldsymbol{k}},t) &= \left [\gamma ^2_p F_2({\boldsymbol{k}},\alpha ,\alpha ',\beta ) + \gamma ^2_s F_3({\boldsymbol{k}},\alpha ,\alpha ',\beta ')\right ] \frac {{\rm d}A_1 {\rm d}A^{\ast }_1}{d\vert {\boldsymbol{k}} \vert {\rm d}\omega }\nonumber \\ & = \omega ^2\left [\frac {F_2({\boldsymbol{k}},\alpha ,\alpha ',\beta ')}{c^2_p} + \frac {F_3({\boldsymbol{k}},\alpha ,\alpha ',\beta ')}{c^2_s}\right ]S_A({\boldsymbol{k}},t),\nonumber \\ \vert {\rm d}R \vert &= \sqrt {S_R({\boldsymbol{k}},t) {\rm d}{\boldsymbol{k}}}, \end{align}

where ${\rm d}R$ represents the Fourier coefficient for the Scholte-wave displacement amplitudes. Here, for mode one and order one,

(6.22) \begin{equation} F_2({\boldsymbol{k}},\alpha , \alpha ',\beta ') = f_2f^\ast _2, \ \ \ F_3({\boldsymbol{k}},\alpha ,\alpha ', \beta ') = f_3 f^\ast _3, \end{equation}

where $f_2$ and $f_3$ are as defined in (6.8).