2.1. The first-order analysis

The first-order problem consists of two wave components: one propagating forward and the other either backward (subharmonic resonance) or forward (superharmonic resonance), where the corresponding first-order BVP is homogeneous. Without loss of generality, the first-order solutions for the velocity potential and free-surface elevation can be expressed as

(2.8) \begin{equation} \phi _1=\psi ^\alpha A^\alpha S^{k_\alpha ,-\omega }+\psi ^\beta A^\beta S^{k_\beta ,-2\omega }+\text{c.c.}, \end{equation}

and

(2.9) \begin{equation} \eta _1=A^\alpha S^{k_\alpha ,-\omega }+A^\beta S^{k_\beta ,-2\omega }+\text{c.c.}, \end{equation}

where c.c. denotes the conjugate component. The superscripts $\alpha$ and $\beta$ refer to the incident wave and the scattered wave, respectively. Here $\psi ^\alpha$ and $\psi ^\beta$ are the corresponding vertical profiles,

(2.10) \begin{equation} \psi ^\alpha =-\frac {i g}{2\omega }\,\frac {\cosh k_\alpha (z+h)}{\cosh k_\alpha h}, \quad \psi ^\beta =-\frac {i g}{4\omega }\,\frac {\cosh k_\beta (z+h)}{\cosh k_\beta h}. \end{equation}

Here $A^\alpha =A^\alpha (\xi _1,\tau _1,\xi _2,\tau _2 )$ and $A^\beta =A^\beta (\xi _1,\tau _1,\xi _2,\tau _2 )$ are complex wave amplitudes modulated by the slow variables. Here, $S^{k,\omega }=\exp{(i kx+ i \omega t)}$ denotes the phase function, and thus we obtain

(2.11) \begin{equation} S^{k_\alpha ,-\omega }=\exp{( ik_\alpha x - i\omega t )}, \quad S^{k_\beta ,-2\omega }=\exp{( i k_\beta x-2 i\omega t )}, \end{equation}

where $\omega$ is the angular frequency, and $k_\alpha$ and $k_\beta$ can be derived from the following dispersion equations, respectively:

(2.12) \begin{equation} \omega ^2=gk_\alpha \ \tanh k_\alpha h,\quad 4\omega ^2=gk_\beta \ \tanh k_\beta h. \end{equation}

To capture the behaviour in the vicinity of resonance, the wavenumbers of the waves and the topography should satisfy the standard condition proposed by Liu & Yue (Reference Liu and Yue1998),

(2.13) \begin{equation} k_d=\pm (2k_\alpha -k_\beta ), \end{equation}

where the positive and negative signs correspond to the conditions for subharmonic and superharmonic resonance, respectively. Therefore, the cosine topography can be expressed in terms of the wavenumbers $k_\alpha$ and $k_\beta$ :

(2.14) \begin{equation} \sigma =D\cos (2k_\alpha -k_\beta ) x. \end{equation}

2.2. The second-order analysis

In this section, we will derive the second-order solutions for velocity potential and surface elevation. At $\mathcal{O}(\varepsilon ^2)$ , the Laplace equation becomes an inhomogeneous one:

(2.15) \begin{equation} \begin{aligned} \frac {\partial ^2 \phi _2}{\partial x^2}+\frac {\partial ^2 \phi _2}{\partial z^2} &=-\frac {gk_\alpha }{\omega }\, \frac {\cosh k_\alpha (z+h)}{\cosh k_\alpha h} \ \frac {\partial A^\alpha }{\partial \xi _1} \, S^{k_\alpha ,-\omega } \\ &\quad -\frac {gk_\beta }{2\omega }\, \frac {\cosh k_\beta (z+h)}{\cosh k_\beta h} \ \frac {\partial A^\beta }{\partial \xi _1} \, S^{k_\beta ,-2\omega } +\text{c.c.},\ (-h\lt z\lt 0) .\end{aligned} \end{equation}

On the free surface $(z=0)$ , we have

(2.16) \begin{align} \frac {\partial ^2 \phi _2}{\partial t^2}+g \frac {\partial \phi _2}{\partial z} &= g\left (\frac {\partial A^\alpha }{\partial \tau _1} S^{k_\alpha ,-\omega } +\frac {\partial A^\beta }{\partial \tau _1} S^{k_\beta ,-2\omega }\right ) \nonumber \\ &\quad -3i\,\frac {\omega ^4-g^2 k_\alpha ^2}{4\omega }\,\left(A^\alpha \right)^2 S^{2k_\alpha ,-2\omega } -3i\,\frac {16\omega ^4-g^2 k_\beta ^2}{8\omega }\,\left(A^\beta \right)^2 S^{2k_\beta ,-4\omega } \nonumber \\ &\quad -i\,\frac {42\omega ^4-g^2 \left(2k_\alpha ^2+6k_\alpha k_\beta +k_\beta ^2\right)}{8\omega }\,A^\alpha A^\beta S^{k_\alpha +k_\beta ,-3\omega } \nonumber \\ &\quad -i\,\frac {6\omega ^4+g^2 \left(2k_\alpha ^2-2k_\alpha k_\beta -k_\beta ^2\right)}{8\omega }\,\overline {A^\alpha }\,A^\beta S^{k_\beta -k_\alpha ,-\omega } +\text{c.c.} \end{align}

At $z=-h$ , the boundary condition is

(2.17) \begin{equation} \begin{aligned} \frac {\partial \phi _2}{\partial z} &=\frac {igk_\alpha D}{4\omega \cosh k_\alpha h}\,\big [\left(k_\beta -k_\alpha \right) S^{k_\beta -k_\alpha ,-\omega } +\left(3k_\alpha -k_\beta \right) S^{3k_\alpha -k_\beta ,-\omega }\big ]A^\alpha \\ &\quad +\frac {igk_\beta D}{4\omega \cosh k_\beta h}\,\big [k_\alpha S^{2k_\alpha ,-2\omega } +\left(k_\beta -k_\alpha \right) S^{2k_\beta -2k_\alpha ,-2\omega }\big ]A^\beta +\text{c.c.} \end{aligned} \end{equation}

By solving (2.15)–(2.17), $\phi _2$ can be expressed as the sum of three solutions that are forced by (2.15), (2.16) and (2.17), respectively, with $\phi _2=\phi _2^{(1)}+\phi _2^{(2)}+\phi _2^{(3)}$ ,

(2.18) \begin{align} \phi _2^{(1)} & =-\frac {g}{2\omega } \frac {(z+h)\ \sinh k_\alpha (z+h)}{\cosh k_\alpha h} \frac {\partial A^\alpha }{\partial \xi _1} S^{k_\alpha ,-\omega } \nonumber \\ & \quad -\frac {g}{4\omega } \frac {(z+h)\ \sinh k_\beta (z+h)}{\cosh k_\beta h} \frac {\partial A^\beta }{\partial \xi _1} S^{k_\beta ,-2\omega } +\text{c.c.}, \end{align}

(2.19) \begin{equation} \begin{aligned} \phi _2^{(2)} &=-\frac {3i\left(\omega ^4-g^2 k_\alpha ^2\right)\ \big(A^\alpha \big)^2}{4\omega }\ F(2k_\alpha ,-2\omega ) -\frac {3i \left(16\omega ^4-g^2 k_\beta ^2 \right)}{8\omega }\ \left(A^\beta \right)^2\\ & \quad F(2k_\beta ,-4\omega )-\frac {i\left[6\omega ^4+g^2\left(2k_\alpha ^2-2k_\alpha k_\beta -k_\beta ^2\right)\right]}{8\omega }\ A^\beta \ \overline {A^\alpha }\ F(k_\beta -k_\alpha ,-\omega ) \\&\quad -\frac {i\left[42\omega ^4-g^2\left(2k_\alpha ^2+6k_\alpha k_\beta +k_\beta ^2\right)\right]}{8\omega }\ A^\alpha A^\beta \ F(k_\alpha +k_\beta ,-3\omega ) +\text{c.c.}, \end{aligned} \end{equation}

and

(2.20) \begin{equation} \begin{aligned} \phi _2^{(3)} &=\frac {igk_\alpha D}{4\omega \cosh k_\alpha h}\ \left[\left(k_\beta -k_\alpha \right)\ G\left(k_\beta -k_\alpha ,-\omega \right) +\left(3k_\alpha -k_\beta \right)\ G\left(3k_\alpha -k_\beta ,-\omega \right)\right] A^\alpha \\&\quad +\frac {igk_\beta D}{4\omega \cosh k_\beta h}\ \left [k_\alpha \ G\left(2k_\alpha ,-2\omega \right) +\left(k_\beta -k_\alpha \right)\ G\left(2k_\beta -2k_\alpha ,-2\omega \right)\right ] A^\beta +\text{c.c.}, \end{aligned} \end{equation}

in which

(2.21) \begin{equation} F(k,\omega )=\frac {\cosh k(z+h)\ \mathrm{sech}\,kh}{gk\tanh kh-\omega ^2}\ S^{k,\omega }, \quad G(k,\omega )=\frac {gk\cosh kz+\omega ^2\sinh kz}{k\omega ^2\cosh kh-gk^2\sinh kh}\ S^{k,\omega }. \end{equation}

Then $\eta _2$ can be derived from the dynamic surface boundary condition,

(2.22) \begin{align} & \eta _2 = \frac {4\left(\omega ^4 - g^2 k_\alpha ^2\right)\,\left|A^\alpha \right|^2 + \left(16\omega ^4 - g^2 k_\beta ^2\right)\,\left|A^\beta \right|^2}{16g\omega ^2} \nonumber \\[4pt] &\quad + \Bigg \{ - \frac {\left(g^2 k_\beta ^2 - 48\omega ^4\right)\tanh 2k_\beta h + 16g\omega ^2 k_\beta } {32\omega ^2 \left(g k_\beta \tanh 2k_\beta h - 8\omega ^2 \right)}\, k_\beta \, \left(A^\beta \right)^2 S^{2k_\beta ,-4\omega } \nonumber \\[4pt] &\quad - \Bigg [ \frac {\left(g^2 k_\alpha ^2 - 3\omega ^4\right)\tanh 2k_\alpha h + 4g\omega ^2 k_\alpha } {8\omega ^2 \left(g k_\alpha \tanh 2k_\alpha h - 2\omega ^2\right)} \left(A^\alpha \right)^2 - \frac {2Dg k_\beta \, \operatorname {sech} 2k_\alpha h\, \operatorname {sech} k_\beta h\, A^\beta } {8\left(g k_\alpha \tanh 2k_\alpha h - 2\omega ^2\right)} \Bigg ] k_\alpha \, S^{2k_\alpha ,-2\omega } \nonumber \\[4pt] &\quad - \Bigg [ \frac {\left(g^2 k_\alpha k_\beta - 6\omega ^4\right)\tanh \left(k_\beta - k_\alpha \right)h + g\omega ^2 \left(2k_\alpha + k_\beta \right)} {8\omega ^2 \left(g (k_\beta - k_\alpha )\tanh (k_\beta - k_\alpha )h - \omega ^2\right)} A^\beta \, \overline {A^\alpha } \nonumber \\[4pt] &\quad \qquad - \frac {2Dg k_\alpha \, \operatorname {sech} k_\alpha h\, \operatorname {sech} \left(k_\beta - k_\alpha \right)h\, A^\alpha } {8 \left(g (k_\beta - k_\alpha )\tanh (k_\beta - k_\alpha )h - \omega ^2 \right)} \Bigg ] \left(k_\beta - k_\alpha \right)\, S^{k_\beta - k_\alpha ,-\omega } \nonumber \\[4pt] &\quad - \frac {\left(g^2 k_\alpha k_\beta - 14\omega ^4\right)\tanh \left(k_\alpha + k_\beta \right)h + 3g\omega ^2 \left(2k_\alpha + k_\beta \right)} {8\omega ^2 \left[g \left(k_\alpha + k_\beta \right)\tanh \left(k_\alpha + k_\beta \right)h - 9\omega ^2\right] \left(k_\alpha + k_\beta \right)^{-1}}\, \, A^\alpha A^\beta \, S^{k_\alpha + k_\beta ,-3\omega } + \text{c.c.} \Bigg \} \nonumber \\[4pt] & \qquad + \text{other terms.} \end{align}

It is worth noting that, on the right-hand side of (2.22), only the terms contributing to the generation of resonance in the third-order problem are explicitly retained, while the remaining non-resonant terms are grouped as ‘other terms’.

As demonstrated by Fang et al. (Reference Fang, Tang and Lin2024a ), for the singular modes $S^{k_\alpha ,-\omega }$ and $S^{k_\beta ,-2\omega }$ , the following solvability conditions should be satisfied:

(2.23) \begin{equation} \int _{-h}^{0} \left ( -\frac {g k_\gamma }{\delta e_\gamma \,\omega }\, \frac {\cosh k_\gamma (z+h)}{\cosh k_\gamma h}\, \frac {\partial A^\gamma }{\partial \xi _1}\, \psi ^\gamma \right) \, {\rm d}z = \frac {\partial A^\gamma }{\partial \tau _1}\, \psi ^\gamma \big |_{z=0}, \quad (\gamma =\alpha , \beta ) \end{equation}

where $\delta e_\alpha =1$ and $\delta e_\beta =2$ . Equation (2.23) can be simplified as

(2.24) \begin{equation} \frac {\partial A^\gamma }{\partial \tau _1} + C_g^\gamma \, \frac {\partial A^\gamma }{\partial \xi _1} = 0, \end{equation}

in which $C_g^\gamma$ represent the group velocities,

(2.25) \begin{equation} C_g^\alpha =\frac {\omega }{2k_\alpha }+h\omega \,\mathrm{csch}\,2k_\alpha h, \qquad C_g^\beta =\frac {\omega }{k_\beta }+2h\omega \,\mathrm{csch}\,2k_\beta h. \end{equation}

Note that, unlike Class-I Bragg resonance (Mei Reference Mei1985; Kirby Reference Kirby1988), the second-order solvability conditions have not included the effects of topography, since no singular terms appear in the bottom boundary condition, (2.17). For the third-order problem, the influence of topography can be incorporated into the solvability conditions, as will be shown.

2.3. The third-order analysis

For the third-order problem, the priority is given to the derivation of the solvability conditions of the singular modes $S^{k_\alpha ,-\omega }$ and $S^{k_\beta ,-2\omega }$ ; therefore, non-resonant modes are not considered here. The third-order BVP is given as follows:

(2.26) \begin{equation} \frac {\partial ^2 \phi _3}{\partial x^2}+\frac {\partial ^2 \phi _3}{\partial z^2} = T^\alpha S^{k_\alpha ,-\omega } + T^\beta S^{k_\beta ,-2\omega } + \text{c.c.},\quad (-h\lt z\lt 0), \end{equation}

(2.27) \begin{equation} \frac {\partial ^2 \phi _3}{\partial t^2}+g\,\frac {\partial \phi _3}{\partial z} = P^\alpha S^{k_\alpha ,-\omega } + P^\beta S^{k_\beta ,-2\omega } + \text{c.c.},\quad (z=0), \end{equation}

(2.28) \begin{equation} \frac {\partial \phi _3}{\partial z} = Q^\alpha S^{k_\alpha ,-\omega } + Q^\beta S^{k_\beta ,-2\omega } + \text{c.c.},\quad (z=-h), \end{equation}

in which the expressions of $T^\alpha , T^\beta , P^\alpha , P^\beta , Q^\alpha$ and $Q^\beta$ are outlined in Appendix A.

At the order of $\varepsilon ^3$ , applying the Green formula to the forcing terms on the right-hand side of (2.26)–(2.28) yields solvability conditions for singular components,

(2.29) \begin{equation} \int _{-h}^{0} T^\gamma \,\psi ^\gamma {\rm d}z = g^{-1} P^\gamma \,\psi ^\gamma \big |_{z=0} - Q^\gamma \,\psi ^\gamma \big |_{z=-h},\quad (\gamma =\alpha , \beta ) \end{equation}

which can lead to two coupled evolution equations for $A^\alpha$ and $A^\beta$ ,

(2.30) \begin{equation} \begin{aligned} &\frac {\partial A^\alpha }{\partial \tau _2} + C_g^\alpha \,\frac {\partial A^\alpha }{\partial \xi _2} + B_1^\alpha \,\frac {\partial ^2 A^\alpha }{\partial \tau _1^2} + B_2^\alpha \,\frac {\partial ^2 A^\alpha }{\partial \xi _1\,\partial \tau _1} + B_3^\alpha \,\frac {\partial ^2 A^\alpha }{\partial \xi _1^2} \\&\qquad + D_1^\alpha A^\alpha + D_2^\alpha \overline {A^\alpha }\,A^\beta + \left\{\sigma _1^\alpha \left|A^\alpha \right|^2 + \sigma _2^\alpha \left|A^\beta \right|^2\right\}\,A^\alpha = 0, \end{aligned} \end{equation}

(2.31) \begin{equation} \begin{aligned} &\frac {\partial A^\beta }{\partial \tau _2} + C_g^\beta \,\frac {\partial A^\beta }{\partial \xi _2} + B_1^\beta \,\frac {\partial ^2 A^\beta }{\partial \tau _1^2} + B_2^\beta \,\frac {\partial ^2 A^\beta }{\partial \xi _1\,\partial \tau _1} + B_3^\beta \,\frac {\partial ^2 A^\beta }{\partial \xi _1^2} \\ &\qquad + D_1^\beta A^\beta + D_2^\beta \left(A^\alpha \right)^2 + \left\{\sigma _1^\beta \left|A^\beta \right|^2 + \sigma _2^\beta \left|A^\alpha \right|^2\right\}\,A^\beta = 0, \end{aligned} \end{equation}

in which $|A^\pm |$ corresponds to the modulus of $A^\pm$ , given by $\left(\sqrt{A^\pm\overline{A^\pm}}\right)$ , and the coefficients are expressed as

(2.32) \begin{align} B_1^\alpha &=\frac {i}{2\omega },\quad B_1^\beta =\frac {i}{4\omega },\quad B_2^\alpha =-i h\, \tanh k_\alpha h,\quad B_2^\beta =-i h\, \tanh k_\beta h, \nonumber\\[2pt] B_3^\alpha &=-\frac {i h \omega }{2k_\alpha }\, \coth k_\alpha h, \! \quad B_3^\beta =-\frac {i h \omega }{k_\beta }\, \coth k_\beta h, \quad D_1^\alpha =\frac {i D^2 k_\alpha \omega \, \mathrm{csch}\,k_\alpha h \, \mathrm{sech}\,k_\alpha h} {8 F_1 (k_\alpha -k_\beta ,\omega ) F_1 (3k_\alpha -k_\beta ,\omega )}\, \widetilde {D_1^\alpha }, \nonumber\\[4pt] D_1^\beta &=-\frac {i D^2 k_\beta \omega \, \mathrm{csch}\,k_\beta h \, \mathrm{sech}\,k_\beta h} {F_1 (2k_\alpha ,2\omega ) F_1 (2k_\alpha -2k_\beta ,2\omega )}\, \widetilde {D_1^\beta }, \ \ \! D_2^\alpha =D_2^\beta =\frac {i k_{\alpha } D \cosh 2 k_{\alpha} h} {4 g \omega \, F_1 (k_{\alpha } - k_{\beta }, \omega )\, F_1 (2 k_{\alpha }, 2 \omega )}\, \widetilde {D_2}, \nonumber\\[4pt] \sigma _1^\alpha &=\frac {i}{16 g^{2} \omega ^{7}}\, \widetilde {\sigma _1^\alpha },\quad \sigma _1^\beta =\frac {i \cosh 2 k_{\beta } h} {64 g^{2} \omega ^{3} \, F_1 (2 k_{\beta }, 4 \omega )}\, \widetilde {\sigma _1^\beta }, \nonumber\\[4pt] \sigma _2^\alpha &=\frac {i \! \cosh (k_{\alpha } - k_{\beta }) h \cosh (k_{\alpha } + k_{\beta }) h} {32 g^{2} \omega ^{3} \, F_1 (k_{\alpha } - k_{\beta }, \omega )\, F_1 (k_{\alpha } + k_{\beta }, 3 \omega )}\, \widetilde {\sigma _2^\alpha }, \nonumber\\[4pt] \sigma _2^\beta &=\frac {i \! \cosh (k_{\alpha } - k_{\beta }) h \cosh (k_{\alpha } + k_{\beta }) h} {16 g^{2} \omega ^{3} \, F_1 (k_{\alpha } - k_{\beta }, \omega )\, F_1 (k_{\alpha } + k_{\beta }, 3 \omega )}\, \widetilde {\sigma _2^\beta }, \end{align}

in which $ F_1(k,\omega )=\omega ^2 \cosh kh - gk \sinh kh$ . The expressions of the coefficients, e.g. $\widetilde {D_1^\alpha }$ , can be found in Appendix B.

To eliminate the second derivatives with respect to $x$ , we introduce the following differential transformations, derived from (2.24):

(2.33) \begin{equation} \frac {\partial ^2 A^\gamma }{\partial \xi _1^2} \to -\frac {1}{C_g^\gamma }\frac {\partial ^2 A^\gamma }{\partial \tau _1\,\partial \xi _1},\quad \frac {\partial ^2 A^\gamma }{\partial \tau _1\,\partial \xi _1} \to -\frac {1}{C_g^\gamma }\frac {\partial ^2 A^\gamma }{\partial \tau _1^2}, \end{equation}

to (2.30) and (2.31), and recover the temporal and spatial scales to $x$ and $t$ , using

(2.34) \begin{equation} \varepsilon \frac {\partial }{\partial \tau _1}+\varepsilon ^2\frac {\partial }{\partial \tau _2}\to \frac {\partial }{\partial t},\quad \varepsilon \frac {\partial }{\partial \xi _1}+\varepsilon ^2\frac {\partial }{\partial \xi _2}\to \frac {\partial }{\partial x}, \end{equation}

which enables $A^\gamma (\xi _1,\tau _1,\xi _2,\tau _2)$ to return to $A^\gamma (x,t)$ , and we denote

(2.35) \begin{equation} B^\gamma = B_1^\gamma - \left(C_g^\gamma \right)^{-1} B_2^\gamma + \left(C_g^\gamma \right)^{-2} B_3^\gamma , \end{equation}

and thus, a set of nonlinear envelope equations for $A^\alpha$ and $A^\beta$ can be obtained

(2.36) \begin{equation} \left \{ \begin{aligned} \!\!\frac {\partial A^\alpha }{\partial t} + C_g^\alpha \frac {\partial A^\alpha }{\partial x} + B^\alpha \frac {\partial ^2 A^{\alpha }}{\partial t^2} + \overbrace {\,D_1^\alpha A^\alpha +D_2^\alpha \overline {A^\alpha } A^\beta \,}& + \left\{\sigma _1^\alpha \left|A^\alpha \right|^2+\sigma _2^\alpha \left|A^\beta \right|^2\right\} A^\alpha = 0,\\[3pt] \!\!\frac {\partial A^\beta }{\partial t} + C_g^\beta \frac {\partial A^\beta }{\partial x} + B^\beta \frac {\partial ^2 A^{\beta }}{\partial t^2} + \underbrace {\,D_1^\beta A^\beta + D_2^\beta \left(A^\alpha \right)^2\,}_{\text{topographic effects}}& + \left\{\sigma _1^\beta \left|A^\beta \right|^2+\sigma _2^\beta \left|A^\alpha \right|^2\right\} A^\beta = 0. \end{aligned} \right . \end{equation}

Equations (2.36) are the newly derived MNLS equations, which incorporate the bottom effect ( $D_1^\gamma$ and $D_2^\gamma$ as topographic terms), the dispersion effect (represented by the $B^\gamma$ terms) and the nonlinear wave interaction (with $\sigma _1^\gamma$ and $\sigma _2^\gamma$ associated with third-order wave nonlinearity). It is worth noting that, unlike Class-I and Class-II Bragg resonance, which exhibit good symmetry between the two governing equations, the proposed equations for Class-III Bragg resonance shows a noticeable asymmetry in the topographic effects. Specifically, the first equation is driven by the term $D_2^\alpha \overline {A^\alpha } A^\beta$ , whereas the second is forced by $D_2^\beta \left(A^\alpha \right)^2$ . Another substantial difference lies in the form of the topographic forcing: for Class-I and Class-II resonance, it appears in the linear terms of the equations, while in the Class-III case it exhibits as a quadratic nonlinear term. Both features will introduce considerable complexity to obtaining an exact solution from (2.36).

While the remainder of the paper focuses on regular incident waves to investigate Bragg resonance, the MNLS equations are also applicable to a broader class of two-dimensional wave-related problems, such as wave focusing, breathers and rogue waves (Li et al. Reference Li, Zheng, Lin, Adcock and van den Bremer2021; Liu et al. Reference Liu, Tang, Mostert and Adcock2025), allowing topographic effects to be incorporated through appropriate choices of initial and boundary conditions.

3.1. Formulation of the BVP of Class-III Bragg resonance

As illustrated in figure 1, a train of nonlinear incident wave propagates from $x=-\infty$ and interacts with the sandbars over the region $0\lt x\lt L$ . Within this region, the incident waves are scattered, giving rise to either reflected (subharmonic) or transmitted (superharmonic) waves, which result in wave superposition for $x\lt 0$ or enhanced transmission for $x\gt L$ , respectively.

To investigate the behaviour in the vicinity of Bragg resonance, let the angular frequencies of the incident and scattered waves be slightly detuned from the standard resonance condition, namely $\omega _\beta =2\omega _\alpha =2(\omega +\omega _D)$ . The magnitude of the angular-frequency detuning $(\omega_D)$ represents the corresponding wavenumber deviations $k_{\alpha D}$ and $k_{\beta D}$ for the incident and scattered waves, respectively. Without loss of generality, we consider the derivation for the subharmonic resonance; the formulation for the superharmonic case is analogous, with the only differences lying in the sign of $k_{\beta D}$ and in the boundary conditions. The solutions can thus be expressed as

(3.1) \begin{equation} A^\alpha =A_0\,T(x)\,e^{-i\omega _D t},\qquad A^\beta =A_0\,R(x)\,e^{-2i\omega _D t}. \end{equation}

On the side where $x\lt 0$ , the topographic terms in (2.36) vanish, leading to the dispersion relations for $k_{\alpha D}$ and $k_{\beta D}$ ,

(3.2) \begin{equation} k_{\alpha D}=\big(C_g^\alpha \big)^{-1}\Big \{\omega _D - i B^\alpha \omega _D^2 + i A_0^2\left [\sigma _1^\alpha +\sigma _2^\alpha |R(0)|^2\right ]\Big \}, \end{equation}

and

(3.3) \begin{equation} k_{\beta D}= \big(C_g^\beta \big)^{-1}\Big \{2\omega _D - 4i B^\beta \omega _D^2 + i A_0^2\left [\sigma _1^\beta |R(0)|^2+\sigma _2^\beta \right ]\Big \}. \end{equation}

For subharmonic resonance, no reflected waves exist in the region $x\gt L$ , and in $x\lt 0$ the amplitude of the incident waves remains spatially constant; for superharmonic resonance, no reflected waves exist in the region $x\lt 0$ . Accordingly, the boundary conditions are given as

(3.4) \begin{equation} \begin{aligned} R(L)&=0,\quad T(0)=1, \ \text{(subharmonic)};\\ R(0)&=0,\quad T(0)=1, \ \text{(superharmonic)}. \end{aligned} \end{equation}

By substituting (3.1) into (2.36), a set of coupled ordinary differential equations for $T$ and $R$ is obtained,

(3.5) \begin{equation} \left \{ \begin{aligned} & i\,C_g^\alpha \,\dot T + \big (\varOmega ^\alpha + iA_0^2\sigma _1^\alpha |T|^2 + iA_0^2\sigma _2^\alpha |R|^2\big )T + iA_0 D_2^\alpha \, \overline {T} \,R=0 ,\\[3pt] & i\,C_g^\beta \,\dot R + \big (\varOmega ^\beta + iA_0^2\sigma _1^\beta |R|^2 + iA_0^2\sigma _2^\beta |T|^2\big )R + iA_0 D_2^\beta \, T^2=0, \end{aligned} \right . \end{equation}

in which $\dot T$ and $\dot R$ denote the derivatives of $T(x)$ and $R(x)$ with respect to $x$ . Here $\varOmega ^\alpha$ and $\varOmega ^\beta$ include the detuned angular frequency and the topographic effects, defined as

(3.6) \begin{equation} \varOmega ^\alpha =\omega _D - i B^\alpha \omega _D^2 + i D_1^\alpha ,\quad \varOmega ^\beta =2\omega _D - 4i B^\beta \omega _D^2 + i D_1^\beta . \end{equation}

3.2. A perturbation analysis of the MNLS equations

In this section, we will employ the perturbation method to (3.5) to analyse the orders of the amplitudes of the incident and scattered waves, for the purpose of facilitating the derivation of closed-form solutions in the next section. Note that in (3.5), $A_0$ is of the order $\mathcal{O}(\varepsilon )$ , and the solutions $T(x)$ and $R(x)$ can therefore be expanded as series in terms of $\varepsilon$ , namely

(3.7) \begin{equation} T(x)=T_0(x)+T_1(x)+{\cdots} ,\quad R(x)=R_0(x)+R_1(x)+{\cdots} , \end{equation}

in which $T_j(x)$ and $R_j(x)$ are of the same order as $\mathcal{O}(\varepsilon ^{j})$ . Firstly, we consider the subharmonic resonance. The zero-order problem consists of homogeneous equations and the inhomogeneous boundary conditions in (3.4), which can be readily solved as follows:

(3.8) \begin{equation} T_0(x)=\exp \!\left (\frac {i\,\varOmega ^\alpha }{C_g^\alpha }\,x\right )\!,\quad R_0(x)=0. \end{equation}

The first-order equations can be expressed in terms of $T_0(x)$ and $R_0(x)$ ,

(3.9) \begin{equation} i C_g^\alpha \,\dot T_1+\varOmega ^\alpha T_1=0,\qquad iC_g^\beta \,\dot R_1+\varOmega ^\beta R_1=-\,iA_0 D_2^\beta \,\exp {\left (\frac {2i\,\varOmega ^\alpha }{C_g^\alpha }x\right )}, \end{equation}

with the homogeneous boundary conditions $R_1(L)=0$ and $T_1(0)=0$ , the solutions are

(3.10) \begin{align} T_1(x)&=0, \nonumber \\ R_1(x)=\frac {iA_0 D_2^\beta }{C_g^\beta } \left (\frac {\varOmega ^\beta }{C_g^\beta }-\frac {2\varOmega ^\alpha }{C_g^\alpha }\right )^{-1} & \exp\! \left ({\frac {2i\varOmega ^\alpha L}{C_g^\alpha }}\right ) \left \{ \exp\! \left [{\frac {i\varOmega ^\beta (x-L)}{C_g^\beta }}\right ] - \exp\! \left [{\frac {2i\varOmega ^\alpha (x-L)}{C_g^\alpha }}\right ] \right \}\!. \end{align}

The reflection coefficient is

(3.11) \begin{equation} |R(0)|=\sqrt {2}\left |\frac {A_0 D_2^\beta }{C_g^\beta }\right |\, \sqrt { {\left(\frac {\varOmega ^\beta }{C_g^\beta }-\frac {2\varOmega ^\alpha }{C_g^\alpha }\right)^{-2}}\left [{1-\cos \left(\frac {\varOmega ^\beta }{C_g^\beta }-\frac {2\varOmega ^\alpha }{C_g^\alpha }\right)L}\right ]}. \end{equation}

In the vicinity of Class-III subharmonic Bragg resonance, where ${\varOmega ^\beta }/{C_g^\beta }-{2\varOmega ^\alpha }/{C_g^\alpha }\to 0$ , we have

(3.12) \begin{equation} |R(0)|\sim \left |\frac {A_0 D_2^\beta }{C_g^\beta }\right |L \to \infty , \quad (\text{when}\, L \to \infty ). \end{equation}

When the sandbar length tends to infinity, the solution will blow up, yielding an unphysical, infinite reflection coefficient. This is an inherent limitation of the perturbation method, making it unsuitable for cases with sufficiently large sandbar lengths. It is worth noting that the solution in (3.12) is similar to that of Liu & Yue (Reference Liu and Yue1998) in the vicinity of resonance, and the two are of the same infinitesimal order with respect to the detuned angular frequency. Although the perturbation solutions proposed here are not in closed form, the analysis further reveals the orders of magnitude of $T(x)$ and $R(x)$ , namely $T(x)=\mathcal{O}(1)$ and $R(x)=\mathcal{O}(\varepsilon )$ , with the latter being the higher-order small quantity and the former dominant. We analyse only the subharmonic resonance; for the superharmonic case, the incident and transmitted wave amplitudes are of the same order, and thus it is not discussed here.

3.3. Closed-form solutions via the frozen-coefficient method

First, the solution for $T_0(x)$ needs to be rederived, as the effects of third-order wave nonlinearity were not accounted for in the leading-order solution. By introducing $|T|^2\sim 1$ and $|R|^2\sim \mathcal{O}(\varepsilon ^2)$ , the leading-order equation can be rewritten as follows:

(3.13) \begin{equation} \left \{ \begin{aligned} & iC_g^\alpha \,\dot T_0+\big (\varOmega ^\alpha +iA_0^2 \sigma _1^\alpha \big ) T_0=0,\\[3pt] & T_0(0)=1, \end{aligned} \right . \end{equation}

which leads to a modified expression for $T_0(x)$ ,

(3.14) \begin{equation} T_0(x)=\exp \!\left [\frac {i\big (\varOmega ^\alpha +iA_0^2 \sigma _1^\alpha \big )}{C_g^\alpha }\,x\right ]\!. \end{equation}

To apply the frozen-coefficient method (Tang et al. Reference Tang, Jiang and Zhou2016), the leading-order solutions $T_0(x)$ and $R_0(x)$ are substituted into the nonlinear terms in (3.5). Specifically, $|T|^2\sim 1$ and $|R|^2\sim \mathcal{O}(\varepsilon ^2)$ are used to replace the cubic nonlinear terms, while we substitute $T\sim \exp [{ix (\varOmega ^\alpha +iA_0^2 \sigma _1^\alpha )}/{C_g^\alpha } ]$ and $\overline {T}\sim \exp [-\,{ix (\varOmega ^\alpha +iA_0^2 \sigma _1^\alpha )}/{C_g^\alpha } ]$ to the quadratic nonlinear terms. By neglecting higher-order terms, the following equations are obtained:

(3.15) \begin{equation} \left \{ \begin{aligned} & i C_g^\alpha \,\dot T+\big (\varOmega ^\alpha +iA_0^2 \sigma _1^\alpha \big )T = -\,iA_0 D_2^\alpha \,\exp \!\left [-\frac {i\left(\varOmega ^\alpha +iA_0^2 \sigma _1^\alpha \right)}{C_g^\alpha }\,x\right ] R,\\[3pt] & i C_g^\beta \,\dot R+\big (\varOmega ^\beta +iA_0^2 \sigma _2^\beta \big )R = -\,iA_0 D_2^\beta \,\exp \!\left [\frac {i\left(\varOmega ^\alpha +iA_0^2 \sigma _1^\alpha \right)}{C_g^\alpha }\,x\right ] T, \end{aligned} \right . \end{equation}

which exhibit a symmetric structure similar to that reported by Fang, Tang and Lin (Reference Fang, Tang and Lin2024a ,Reference Fang, Tang and Lin b ). Accordingly, (3.15) can be transformed into

(3.16) \begin{equation} \frac {{\rm d}\tilde T}{{\rm d}x}=iK_\alpha e^{2iKx}\,\tilde R,\quad \frac {{\rm d}\tilde R}{{\rm d}x}=iK_\beta e^{-2iKx}\,\tilde T, \end{equation}

where

(3.17) \begin{equation} \tilde T=T\,\exp \!\left [-\frac {i\left(\varOmega ^\alpha +iA_0^2 \sigma _1^\alpha \right)}{C_g^\alpha }\,x\right ]\!,\quad \tilde R=R\,\exp \!\left [-\frac {i\left(\varOmega ^\beta +iA_0^2 \sigma _2^\beta \right)}{C_g^\beta }\,x\right ]\!. \end{equation}

Here $K_\alpha$ , $K_\beta$ and $K$ are defined as

(3.18) \begin{equation} K_\alpha =\frac {iA_0 D_2^\alpha }{C_g^\alpha },\quad K_\beta =\frac {iA_0 D_2^\beta }{C_g^\beta },\quad K=\frac {\varOmega ^\beta +iA_0^2 \sigma _2^\beta }{2C_g^\beta }-\frac {\varOmega ^\alpha +iA_0^2 \sigma _1^\alpha }{C_g^\alpha }, \end{equation}

in which $K_\alpha$ and $K_\beta$ are influenced by the wave amplitude $A_0$ and the sandbar amplitude $D$ , whereas $K$ is further related to the detuned angular frequency $\omega _D$ , which is included in $\varOmega ^\alpha$ and $\varOmega ^\beta$ .

The transformed equations, (3.16), together with the boundary conditions, can be solved to obtain exact solutions. Based on (3.17), the corresponding expressions for $R$ and $T$ are then derived as follows:

(3.19) \begin{equation} R(x)=\frac {-iK_\beta \, \sinh K_\mu (L-x)} {K_\mu \cosh K_\mu L + iK \sinh K_\mu L }\, \exp \!\left \{ i\!\left [\frac {\varOmega ^\beta +iA_0^2 \sigma _2^\beta }{C_g^\beta }-K\right ]x \right \}\!, \end{equation}

(3.20) \begin{equation} T(x)=\frac {K_\mu \cosh K_\mu (L-x) + iK \sinh K_\mu (L-x)} {K_\mu \cosh K_\mu L + iK \sinh K_\mu L }\, \exp \!\left \{ i\!\left [\frac {\varOmega ^\alpha +iA_0^2 \sigma _1^\alpha }{C_g^\alpha }+K\right ]x \right \}\!, \end{equation}

in which $K_\mu$ is defined as

(3.21) \begin{equation} K_\mu =\sqrt {-K^2-K_\alpha K_\beta }. \end{equation}

The corresponding reflection and transmission coefficients are obtained

(3.22) \begin{equation} |R(0)|=\sqrt {\frac {K_\beta ^2}{\,|K_\mu \coth K_\mu L|^2+K^2\,}}, \quad |T(L)|=\sqrt {\frac {|K_\mu \,\text{csch}\, K_\mu L|^2}{\,|K_\mu \coth K_\mu L|^2+K^2}}, \end{equation}

which are the closed-form solutions of reflection and transmission coefficients for subharmonic resonance. Compared with the reflection coefficient in (3.11), $|R(0)|$ in (3.22) remains bounded for any $\omega _D$ (included in $K_\mu$ ) and $L$ , and a rough upper bound can be expressed as $|R(0)|\lt |K_\beta /K|$ .

For superharmonic resonance, the derivation of the solutions is analogous, differing only in the boundary conditions, where reflection is absent and replaced by reinforced transmission. To avoid confusion, a new notation $U(x)$ is introduced in place of $R(x)$ in (3.15), and thus the boundary conditions in (3.4) become

(3.23) \begin{equation} T(0)=1,\quad U(0)=0, \end{equation}

which, combined with (3.15), leads to the closed-form solutions for superharmonic resonance

(3.24) \begin{equation} U(x)=\frac {iK_\beta \sin K_\nu x}{K_\nu }\, \exp \left \{{\,i\left [\frac {\varOmega ^\beta +iA_0^2 \sigma _2^\beta }{C_g^\beta }-K\right ]x}\right \}\!, \end{equation}

(3.25) \begin{equation} T(x)=\left (\cos K_\nu x-\frac {iK \sin K_\nu x}{K_\nu }\right )\, \exp \left \{{\,i\left [\frac {\varOmega ^\alpha +iA_0^2 \sigma _1^\alpha }{C_g^\alpha }+K\right ]x}\right \}\!, \end{equation}

in which

(3.26) \begin{equation} K_\nu =\sqrt {K^2+K_\alpha K_\beta }\gt 0, \end{equation}

and the two transmission coefficients are

(3.27) \begin{equation} |U(L)|=\left |\frac {K_\beta }{K_\nu }\sin K_\nu L\right |\!, \quad |T(L)|=\sqrt {\,1-\frac {K_\alpha K_\beta }{K_\nu ^2}\left (\sin K_\nu L\right )^2}. \end{equation}

It is worth noting that although a singularity point appears to exist at $K_\nu =0$ , this point is essentially unachievable, as $K_\nu$ remains strictly positive. This arises from the fact that, in the case of superharmonic resonance, the two interacting waves propagate in the same direction, yielding group velocities of the same sign, a key distinction from subharmonic cases. The product $K_\alpha K_\beta$ remains positive, and $K_\nu$ must be real and positive. Therefore, both transmission coefficients, $|U(L)|$ and $|T(L)|$ , remain bounded and valid as the sandbar length $L$ increases, with a rough estimate given as $|U(L)|\leqslant |K_\beta /K_\nu |\leqslant |\sqrt {K_\beta /K_\alpha }|$ .

For subharmonic resonance, the existence of an upper bound for the reflection coefficient and the presence of roots of $K_\mu =0$ with respect to $\omega _D$ , implying a transition in the form of the reflection coefficient from the hyperbolic cotangent to the cotangent, suggest that a cutoff frequency and a finite resonance bandwidth emerge when full reflection occurs. In contrast, such characteristics are absent in superharmonic resonance, as $K_\nu$ remains strictly positive and possesses no real roots. This difference potentially leads to a substantial distinction between subharmonic and superharmonic resonance, which will be further discussed in §§ 4.3 and 4.4.

Note that the solutions in (3.22) and (3.27) are approximate to the MNLS equations, where part of the higher-order wave nonlinearity is neglected by substituting the nonlinear terms with their leading-order expressions. Consequently, slight discrepancies from the exact solutions of the MNLS equations are expected. Nevertheless, the analytical solutions not only provide essential quantitative measures, such as the reflection and transmission coefficients, resonance detuning and resonance bandwidth, forming a mathematical foundation for engineering design, but also reveal new phenomena, as will be shown later.

3.4. Energy conservation law derived from the MNLS equations

In this section, we derive the energy conservation law and establish a more precise upper bound for the solutions in (3.22) and (3.27). Subsequently, we demonstrate that the present solutions strictly satisfy the proposed energy conservation law, thereby elucidating, through the principle of energy conservation, why the proposed solutions remain in closed form and confirming the rationality of the formulated symmetric structure in (3.15).

It is worth indicating that the MNLS equations intrinsically satisfy energy conservation, without any approximation needed. To formulate the energy conservation law, we begin the analysis with the MNLS equations, (3.5). By multiplying the two equations in (3.5) by $\overline {T}$ and $\overline {R}$ , respectively, and then taking the complex conjugate of (3.5) and multiplying again by $T$ and $R$ , four equations are obtained. By performing algebraic addition and subtraction among them, the following equation is derived:

(3.28) \begin{equation} \left \{ \begin{aligned} \frac {{d}}{\mathrm{d}x} \bigl ( C_g^{\alpha } |T|^{2} \bigr ) &= - A_{0} D_{2}^{\alpha } \bigl ( \overline {T}^{\,2} R - T^{2} \overline {R} \bigr ), \\[5pt] \frac {{d}}{\mathrm{d}x} \bigl ( C_g^{\beta } |R|^{2} \bigr ) &= A_{0} D_{2}^{\beta } \bigl ( \overline {T}^{\,2} R-T^{2} \overline {R} \bigr ). \end{aligned} \right . \end{equation}

Equation (3.28) expresses the energy-conservation property of the proposed MNLS equations and is derived without introducing any approximation. The left-hand sides represent the spatial gradients of the wave-energy fluxes associated with the incident and scattered components, whereas the forcing terms on the right-hand sides describe the energy exchange between the two wave envelopes. Since $D_2^{\alpha } = D_2^{\beta }$ , these two terms are equal in magnitude and opposite in sign, implying that the energy flux extracted from the incident wave is entirely transferred to the scattered wave. Consequently, the energy-conservation law is rigorously ensured by the fundamental equality $D_2^{\alpha } = D_2^{\beta }$ .

Equation (3.28) can be further combined to give

(3.29) \begin{equation} \frac {{d}}{\mathrm{d}x} \bigl (C_g^{\alpha } |T|^{2} + C_g^{\beta } |R|^{2} \bigr ) = A_{0} \bigl ( D_2^{\beta } - D_2^{\alpha } \bigr ) \bigl ( \overline {T}^{\,2} R - T^{2} \overline {R} \bigr ) = 0, \end{equation}

which can be solved over the region $[0,L]$ to obtain the relation between the reflection and transmission coefficients. We can obtain the following relation:

(3.30) \begin{equation} -C_g^\beta |R(0)|^2 + C_g^\alpha |T(L)|^2 = C_g^\alpha . \end{equation}

From a physical perspective, energy conservation corresponds to the conservation of energy flux, which is proportional to the product of the group velocity and the squared wave amplitude. Accordingly, (3.30) shows that the outgoing energy flux carried by the reflected and transmitted waves balances the incoming flux of the incident wave. The negative sign accounts for the opposite propagation direction of the reflected component. Based on (3.30), a precise upper bound of the reflection coefficient can be derived as

(3.31) \begin{equation} \max |R(0)|=\sqrt {\frac {-C_g^\alpha }{C_g^\beta }}, \end{equation}

which depends solely on the group velocities of the two wave components and is independent of the amplitudes of the wave and the topography.

Similarly, for superharmonic cases, the energy conservation law can be derived as

(3.32) \begin{equation} C_g^\beta |U(L)|^2 + C_g^\alpha |T(L)|^2 = C_g^\alpha . \end{equation}

Equation (3.32) represents the energy-conservation law for superharmonic resonance and likewise reflects the conservation of energy flux. In this case, the outgoing energy flux carried by the two transmitted wave components balances the incoming flux of the incident wave. A key difference lies in the sign of $C_g^{\alpha }$ : for superharmonic resonance, the scattered waves propagate in the forward direction. Based on (3.32), the upper bound of the transmission coefficient for the superharmonic resonance can be obtained

(3.33) \begin{equation} \max |U(L)|=\sqrt {\frac {C_g^\alpha }{C_g^\beta }}. \end{equation}

One can directly verify that the present solutions in (3.22) and (3.27) rigorously satisfy the energy-conservation laws given in (3.30) and (3.32), respectively. On the other hand, the frozen-coefficient method introduces approximations only in the right-hand side terms of (3.28), where $\bigl ( \overline {T}^{\,2} R - T^{2} \overline {R} \bigr ) \sim \bigl ( \overline {T}\,\overline {T}_{0} R - T T_{0} \overline {R} \bigr )$ . With this approximation, one obtains ${\mathrm{d}} \bigl ( C_g^{\beta } |R|^{2} + C_g^{\alpha } |T|^{2} \bigr )/{\mathrm{d}x} = A_{0} \bigl ( D_2^{\beta } - D_2^{\alpha } \bigr ) \bigl ( \overline {T}\,\overline {T}_{0} R - T T_{0} \overline {R} \bigr ) = 0.$ This result demonstrates that energy conservation is preserved after applying the frozen-coefficient method. Consequently, the symmetric formulation introduced in (3.15) ensures a balance between the input and output energy fluxes, and thus strictly satisfies the energy-conservation relations in (3.29). This structural consistency guarantees that the resulting solutions remain bounded and admit closed-form expressions.

Essentially, energy conservation in the present framework is ensured by the conserved quantities inherent in NLS-type and other integrable equations (Adcock & Taylor Reference Adcock and Taylor2009; Tang et al. Reference Tang, Taylor, Chen and Adcock2025). It is worth indicating that the present MNLS equations should also admit certain number of conserved quantities, which is an aspect that can be further studied in future work.

Table 1.

Parameters for wave scattering by sinusoidal sandbars.

4.1. Reflection and transmission coefficients

The concept of Class-III Bragg resonance was introduced by Liu & Yue (Reference Liu and Yue1998), and they derived analytical solutions using the regular perturbation method and then conducted numerical simulations based on the high-order spectral method. Their study examined the nonlinear effects on wave reflection in subharmonic resonance, showing that nonlinearity significantly enhances reflection and intensifies the downshift of the resonance frequency. To validate the proposed solutions for subharmonic cases, we first compare the reflection and transmission coefficients in (3.22) with both the perturbation and numerical results of Liu & Yue (Reference Liu and Yue1998), and the corresponding wave and topographic parameters are summarized in table 1. Since the present solutions in (3.22) and (3.27) are approximate solutions to the MNLS equations, neglecting part of the wave nonlinearity to preserve the closed-form structure. Therefore, numerical solutions based on the MNLS equations (regarded as exact solutions) are also obtained for comparison, computed using the procedure described in Appendix C.

Figure 2.

Comparison of reflection ( $\lvert R(0)\rvert$ ) and transmission coefficients ( $\lvert T(L)\rvert$ ) among the present solutions ((3.22), blue solid line), the numerical solution from the MNLS equations ((3.5), red dashed line), the numerical results of Liu & Yue (Reference Liu and Yue1998) (black dots) and their perturbation solution (purple dash–dotted line). The leftward arrow indicates the downshift from the standard resonance condition ( $k_\beta =2k_\alpha -k_d$ ). Here (a) Case a; (b) Case b.

In figure 2, the reflection coefficients of subharmonic cases under different wave nonlinearity conditions are compared among the present solution, (3.22), the numerical solution from the MNLS equations, (3.5), and the regular perturbation and numerical results of Liu & Yue (Reference Liu and Yue1998). The good agreement between the present analytical and numerical solutions verifies the validity of the frozen-coefficient method, both of which compare well with the numerical results of Liu & Yue (Reference Liu and Yue1998), thereby further supporting the accuracy of the present theory.

Moreover, as reported by Liu & Yue (Reference Liu and Yue1998), a downshift from the standard resonant condition is observed, and the present solutions can accurately capture the magnitude of this frequency downshift. It can be demonstrated the when the effects of wave dispersion ( $B^\alpha$ and $B^\beta$ ), wave modulation ( $\sigma _1^\alpha$ and $\sigma _2^\beta$ ) and rescattering ( $D_1^\alpha$ and $D_1^\beta$ ) vanish in the $K$ term in (3.18), the downshift behaviour will disappear. Therefore, the frequency downshift is attributed to the combined influence of these three mechanisms, which is consistent with the conclusion of Liu & Yue (Reference Liu and Yue1998) that the downshift originates from third-order wave and bottom nonlinearity.

Furthermore, experimental evidence confirming the occurrence of Class-III Bragg resonance was provided by Peng et al. (Reference Peng, Tao, Liu, Zheng, Zhang and Wang2019), in which both subharmonic and superharmonic resonance were examined through laboratory experiments. To further validate the present solution, particularly for the superharmonic cases, we compare it with the experimental data and numerical results reported by Peng et al. (Reference Peng, Tao, Liu, Zheng, Zhang and Wang2019), with the corresponding parameters summarized in table 1 (Cases c and d).

Figure 3.

Comparison of reflection (subharmonic) and transmission (superharmonic) coefficients among the present analytical solutions ((3.22) for (a) and (3.27) for (b), blue solid line), the numerical solution from the MNLS equations ((3.5), red dashed line), the numerical results of Ning et al. (Reference Ning, Zhang, Chen, Liu and Teng2022) (yellow diamonds), the numerical results (black triangles) of Peng et al. (Reference Peng, Tao, Liu, Zheng, Zhang and Wang2019) and their experimental data (green dots). The arrows indicate the downshift or upshift from the standard resonance conditions (subharmonic, $k_\beta =2k_\alpha -k_d$ ; superharmonic, $k_\beta =2k_\alpha +k_d$ ). Here (a) Case c; (b) Case d.

For the superharmonic case shown in figure 3(b), an upshift of the resonance can be observed. The resonance wavenumber predicted by the present solution agrees well with both the experimental and numerical results of Peng et al. (Reference Peng, Tao, Liu, Zheng, Zhang and Wang2019) and the numerical solutions of Ning et al. (Reference Ning, Zhang, Chen, Liu and Teng2022), demonstrating the present solution’s ability on capturing the detuning behaviour. For the subharmonic case shown in figure 3(a), good agreement is obtained among the present analytical solution, the MNLS-based numerical results, and the numerical solutions of Ning et al. (Reference Ning, Zhang, Chen, Liu and Teng2022), as well as the numerical results of Peng et al. (Reference Peng, Tao, Liu, Zheng, Zhang and Wang2019).

While the present solutions slightly overestimate the magnitude of the frequency downshift relative to the experiments. One possible reason for the overestimation of the resonance detuning may lie in the sandbar length, which, similar to Class-I and -II resonance, can significantly influence the detuning, as demonstrated by Fang, Tang and Lin (Reference Fang, Tang and Lin2024a ,Reference Fang, Tang and Lin b ). It should be noted that the MNLS equations are lower than fourth-order accuracy. Unlike Class-I resonance, where third-order theory can account for the length effect, Class-II and -III resonance, as higher-order resonance phenomena, can incorporate the length effect into detuning only in fourth- or higher-order theories, as shown by Fang et al. (Reference Fang, Tang and Lin2024b ). Nevertheless, similar to Class-II resonance (Fang et al. Reference Fang, Tang and Lin2024b ), the present method provides increasingly accurate predictions as the sandbar length increases.

As illustrated in figures 2 and 3, a detuning of the resonance from the standard resonance condition is observed for both subharmonic and superharmonic cases, exhibiting downward and upward shifts, respectively. Moreover, as wave nonlinearity increases, the magnitude of resonance frequency detuning becomes more pronounced. It is therefore beneficial to develop an analytical formula that not only enhances understanding of the distinct detuning behaviours between subharmonic and superharmonic resonance, but also provides a theoretical basis for practical engineering applications, such as the design of coastal protection structures and soil erosion mitigation measures. In addition, the present solutions can capture the wave frequencies at which true resonance occurs, providing a mathematical foundation for further analysis of the detuning behaviour.

Through systematic validation, the MNLS equations and the present solutions, (3.22) and (3.27), are shown to perform well within a certain range of wave and topography nonlinearity, at least for $k_{\alpha } A_{0} \lt 0.1$ and $D/h \lt 0.2$ . Essentially, Class-III Bragg resonance is dominated by the second-order bound wave. Although a higher-order expansion may further refine this wave mode, it would also introduce additional modes and hence more complex resonance features; in this sense, the third-order MNLS formulation is sufficient for the present problem.

4.2. The detuning of Bragg resonance

The frequency at which real resonance occurs corresponds to the peak of the reflection or transmission coefficient curve. At this point, the reflection or transmission coefficient reaches a stationary value, implying that its derivative with respect to $\omega _D$ is zero. The magnitude of the wave angular frequency, $\omega _\theta$ , can then be obtained from the following equations for the subharmonic and superharmonic cases, respectively:

(4.1) \begin{equation} \left .\frac {\partial }{\partial \omega _D}\,|R(0)|\right |_{\omega _D=\omega _\theta }=0, \quad \left .\frac {\partial }{\partial \omega _D}\,|U(L)|\right |_{\omega _D=\omega _\theta }=0, \end{equation}

in which $|R(0)|$ and $|U(L)|$ are expressed in (3.22) and (3.27), respectively. The two equations can be solved exactly following the method of Fang et al. (Reference Fang, Tang and Lin2024b ). After algebraic manipulation, both yield the same condition, $K=0$ , which can be further solved by substituting (3.6) into (3.18) to obtain

(4.2) \begin{equation} \omega _\theta =\sqrt {\varXi ^2+\varPi }-\varXi \;\propto \; \mathcal{O}\left(D^2,A_0^2\right), \end{equation}

in which

(4.3) \begin{equation} \varXi = i\left ( \frac {2B^\alpha }{C_g^\alpha } - \frac {4B^\beta }{C_g^\beta } \right )^{-1} \left ( \frac {1}{C_g^\alpha } - \frac {1}{C_g^\beta } \right )\!, \quad \varPi = \varPi _1 D^2 + \varPi _2 A_0^2, \end{equation}

in which $\varPi _1$ and $\varPi _2$ are coefficients defined as

(4.4) \begin{equation} \varPi _1=\left ( \frac {2B^\alpha }{C_g^\alpha } - \frac {4B^\beta }{C_g^\beta } \right )^{-1} \! \left ( \frac {2D_1^\alpha }{D^2 C_g^\alpha } - \frac {D_1^\beta }{D^2 C_g^\beta } \right )\!, \! \quad \varPi _2=\left ( \frac {2B^\alpha }{C_g^\alpha } - \frac {4B^\beta }{C_g^\beta } \right )^{-1} \left ( \frac {2\sigma _1^\alpha }{C_g^\alpha } - \frac {\sigma _2^\beta }{C_g^\beta } \right )\!. \end{equation}

According to the dispersion relation (3.2), the magnitude of wavenumber detuning can be expressed as

(4.5) \begin{equation} k_{\alpha \theta }= \begin{cases} \displaystyle \left({C_g^\alpha }\right)^{-1}\left \{ \omega _\theta - i B^\alpha \omega _\theta ^2 + i A_0^2 \left [ \sigma _1^\alpha + \sigma _2^\alpha |R(0)|^2 \right ] \right \}\!, & \text{(subharmonic)},\\[9pt] \displaystyle \left({C_g^\alpha }\right)^{-1}\left \{ \omega _\theta - i B^\alpha \omega _\theta ^2 + i A_0^2 \sigma _1^\alpha \right \}\!, & \text{(superharmonic)}. \end{cases} \end{equation}

which includes additional effects from wave self-modulation and cross-modulation processes.

To validate the proposed theoretical formulae and examine the effects of the amplitudes of the topography and the incident wave on the detuning of Bragg resonance, we follow the configurations of Cases c and d, varying the sandbar amplitude $D$ from $0$ to $0.2$ m, and $k_\alpha A_0$ from $0.02$ to $0.08$ , respectively.

Figure 4.

Comparison of the resonant wavenumber with increasing bottom amplitude between the analytical solution (4.5) and the numerical solution from the MNLS equations (3.5) for different wave nonlinearities: $k_\alpha A_0=0.02$ (solid line and squares); $0.05$ (dashed line and triangles); $0.08$ (dash–dotted line and circles). Here (a) subharmonic resonance; (b) superharmonic resonance.

Figure 4 compares the wavenumbers of subharmonic and superharmonic resonance, showing good agreement between the analytical solution (4.5) and the numerical solution from (3.5). Moreover, the influences of wave nonlinearity and bottom amplitude are examined, both of which amplify the magnitude of resonance detuning. In these cases, the closed-form solutions are shown to be applicable within the parameter range $D/h \lt 0.4$ (corresponds to $D\lt 0.2$ m) and $k_{\alpha } A_{0} \lt 0.08$ , where the resonance detuning is well captured.

It can be demonstrated that $\varXi \gt 0$ , since $B^\alpha$ and $B^\beta$ are even functions whose imaginary parts remain positive with respect to the wavenumber and are not affected by the direction of wave propagation. The sign of $\omega _\theta$ is determined by the sign of $\varPi$ , which is a quadratic function of $D$ and $A_0$ . The direction of resonance detuning is mainly attributed to $\varPi _1$ and $\varPi _2$ , whose the signs are not fixed and will vary with $k_\alpha h$ .

Figure 5.

Comparison of how coefficients $\varPi _1$ (a) and $\varPi _2$ (b) vary with $k_\alpha h$ for subharmonic (blue solid line) and superharmonic resonance (red dashed line).

In figure 5(a), $\varPi _1$ is positive for all values of $k_\alpha h$ in the superharmonic case, whereas for subharmonic resonance, $\varPi _1$ is positive when $k_\alpha h\lt 0.35$ and becomes negative when $k_\alpha h\gt 0.35$ . In the former scenario, the topographic effects on resonance detuning lead to a upshift of the resonance frequency, which is different from Class-I and Class-II Bragg resonance. In contrast, in the latter case, both upward and downward shifts may occur, depending on the value of $k_\alpha h$ .

Figure 5(b) compares the variation of $\varPi _2$ with $k_\alpha h$ . In this case, the superharmonic resonance exhibits opposite signs on the sub side (negative) and super side (positive) of a critical value $k_\alpha h\approx 0.67$ , whereas the subharmonic case displays the opposite behaviour on either side of $k_\alpha h\approx 0.62$ . Thus, under shallower water conditions, wave nonlinearity results in an upshift or a downshift for the subharmonic or superharmonic cases, respectively, whereas the opposite trend is observed under deeper water conditions.

Therefore, in Cases a and b, where $k_\alpha h=0.6$ , while an upshift of the resonance frequency is expected with increasing wave nonlinearity, an intensified downshift of the resonant wavenumber is observed in figures 2(a) and 2(b). This phenomenon can be explained by (4.5), in which $i\sigma _1^\alpha \lt 0$ ; hence, increasing wave nonlinearity contributes to a decrease in the resonance wavenumber.

Thus, compared with Class-I and Class-II resonance, Class-III Bragg resonance exhibits a more complex detuning behaviour in both resonance frequency and wavenumber, where topography and wave nonlinearity can each induce either upward or downward shifts. For a given wave condition, the magnitudes of the detuning of wave angular frequency and wavenumber can be predicted from (4.2) and (4.5), respectively.

Figure 6.

Comparison of the absolute values of the coefficients $|\varPi _1|$ (red line) and $|\varPi _2|$ (blue line) as functions of $k_{\alpha } h$ . The shaded red and blue regions indicate $|\varPi _1| \gt |\varPi _2|$ and $|\varPi _1| \lt |\varPi _2|$ , respectively: (a) subharmonic resonance; (b) superharmonic resonance.

Figure 6 compares the absolute values of $|\varPi _1|$ and $|\varPi _2|$ to assess the relative importance of wave modulation and topography-induced rescattering under different $k_{\alpha } h$ conditions. For subharmonic resonance, the topographic rescattering effect dominates the detuning magnitude when $k_{\alpha } h \approx 0.56$ – $0.71$ ; outside this range, nonlinear wave modulation becomes the primary contributor. In contrast, for superharmonic resonance, the detuning behaviour is governed by wave modulation over the entire range of $k_{\alpha } h$ . This difference likely arises because superharmonic resonance generates higher-frequency (or relatively deeper water) transmitted waves, for which topographic effects are weaker compared with surface-wave nonlinearity.

Dispersion effects are incorporated through the term $\varXi$ in (4.2) and the coefficients appearing in $|\varPi _1|$ and $|\varPi _2|$ . These effects act in combination with wave nonlinearity and bottom-induced scattering and are therefore not treated as an independent mechanism in the present analysis. The ability of the present solutions to capture detuning behaviour stems from intrinsic properties of NLS-type equations, which, as narrow-banded models, faithfully describe wave dynamics in the vicinity of the carrier frequency. Future work may consider the influence of full dispersion by extending the broad-banded model of Li (Reference Li2021) to consider topographic effects.

4.3. Cutoff frequency and bandwidth

As discussed in § 3.3, the solution for subharmonic resonance in (3.22) and that for superharmonic resonance in (3.27) are governed by the parameters $K_\mu$ and $K_\nu$ , respectively. Since $K_\nu$ is proven to be strictly positive, which implies that, as the sandbar length tends to infinity, the sine-function form in (3.27) is preserved, without presenting singular behaviours and transitions. Therefore, it is expected that the cutoff frequency and bandwidth will be zero in superharmonic resonance.

However, for subharmonic resonance, $K_\mu$ becomes zero at certain values of $\omega _D$ , and on either side of the root, $K_\mu$ changes from a purely real to a purely imaginary value. As a result, the hyperbolic cotangent function in (3.22) transitions into a cotangent function, indicating a geometric transformation in the curve of the reflection coefficient. For a real $K_\mu$ , taking the limit of $\coth K_\mu L$ as $L\to \infty$ , we obtain $|\coth K_\mu L|\to 1$ . Therefore, $|R(0)|$ tends towards its maximum value, as shown in (3.31). This implies that the two roots of $K_\mu =0$ correspond to the two ends of the cutoff angular frequencies, which can be obtained from

(4.6) \begin{equation} K^2+K_\alpha K_\beta =0, \end{equation}

which is an algebraic equation with respect to $\omega _D$ and can be solved analytically to yield $\omega _{{cf1}}$ and $\omega _{{cf2}}$ ,

(4.7) \begin{equation} \omega _{{cf1}}=\sqrt {\varXi ^2+\varPi -\varDelta }-\varXi ,\quad \omega _{{cf2}}=\sqrt {\varXi ^2+\varPi +\varDelta }-\varXi , \end{equation}

in which $\varXi$ and $\varPi$ are defined in (4.3) and (4.4), and $\varDelta$ denotes

(4.8) \begin{equation} \varDelta = A_0\left |\frac {2iB^\beta }{C_g^\beta }-\frac {iB^\alpha }{C_g^\alpha }\right |^{-1} \sqrt {\frac {D_2^\alpha }{C_g^\alpha }\,\frac {D_2^\beta }{C_g^\beta }} \gt 0. \end{equation}

Note that the angular frequency of resonance, $\omega _\theta$ , falls within the interval $\left[\omega +\omega _{{cf1}},\omega +\omega _{{cf2}}\right]$ . When the $\varDelta$ term is removed, this interval reduces to the angular frequency of resonance $\omega +\omega _\theta$ .

Therefore, the cutoff frequency for subharmonic resonance can be expressed as

(4.9) \begin{align} \bigl [f_{{cf1}},\,f_{{cf2}}\bigr ] = \Big [ ({2\pi })^{-1}\Big ({\omega -\varXi +\sqrt {\varXi ^2+\varPi -\varDelta }}\Big ), \, ({2\pi })^{-1}\Big ({\omega -\varXi +\sqrt {\varXi ^2+\varPi +\varDelta }}\Big ) \Big]. \end{align}

The bandwidth can be obtained accordingly,

(4.10) \begin{equation} B=\frac {\varDelta }{\pi }\!\left (\sqrt {\varXi ^2+\varPi +\varDelta }+\sqrt {\varXi ^2+\varPi -\varDelta }\right )^{-1} \;\propto \; A_0 D. \end{equation}

It can be demonstrated that $B\propto A_0 D$ , indicating that the bandwidth is directly proportional to the amplitudes of both the incident wave $A_0$ and the sandbars $D$ .

The analytical solution (3.22) indicates that full reflection can occur when the sandbar length becomes sufficiently large, thereby maximizing the sandbar’s capacity as an energy barrier. The existence of a cutoff frequency and bandwidth suggests that waves with frequencies falling within the interval (4.9) can resonate with the sandbars and be fully reflected when the sandbar length is sufficiently long, leaving minimal wave energy transmitted into coastal areas and thus protecting the shorelines. Furthermore, the proposed formula for the bandwidth in (4.10) demonstrates that increasing the amplitudes of both the sandbar and the waves enhances the capacity for coastal protection, as it allows waves with frequencies farther from the resonance frequency to also fall within the cutoff range and be fully reflected. While wave amplitude is difficult to control artificially, sandbar amplitude can be achieved through engineering measures.

To further validate these findings and elucidate the differences between subharmonic and superharmonic resonance, we compare the associated reflection and transmission coefficients as functions of the sandbar length. The corresponding configurations are listed in table 2, where the number of sandbars is set to $N_d = 10$ , $20$ and $40$ .

Table 2.

Parameters for studying the influence of sandbar number on resonance bandwidth.

Figure 7.

Comparison of the reflection (a), Case e, (3.22) and transmission (b), Case f, (3.27) coefficients with varying sandbar length: $N_d=10$ (red dash–dotted line), $20$ (purple dashed line), $40$ (black solid line) and infinite (blue shaded region). The blue lines represent the envelopes: (4.11) for (a) and (4.12) for (b).

Figure 7(a) shows the variation of the reflection coefficient with increasing sandbar length under subharmonic resonance conditions. As the sandbar length tends to infinity, the results converge to the blue shaded region, forming a resonance band, with a bandwidth $B$ , bounded by the two cutoff frequencies, $f_{{cf1}}$ and $f_{{cf2}}$ . Within the band, waves cannot propagate through the sandbar region and are therefore reflected towards the offshore region. The blue line represents the envelope, obtained by taking the limit superior of $L$ for the reflection coefficient in (3.22),

(4.11) \begin{equation} \mathrm{En}_{\textit{sub}} =\overline {\lim _{L\to \infty }}\,|R(0)| =\begin{cases} \displaystyle \sqrt {\frac {K_\beta ^2}{K_\mu ^2+K^2}}=\sqrt {-\frac {C_g^\alpha }{C_g^\beta }}, & \displaystyle \frac {\omega +\omega _D}{2\pi }\in \bigl [f_{{cf1}},f_{{cf2}}\bigr ],\\[15pt] \displaystyle \left |\frac {K_\beta }{K}\right |\,\ \propto \,\ \omega _D^{-1}, & \displaystyle \frac {\omega +\omega _D}{2\pi }\notin \bigl [f_{{cf1}},f_{{cf2}}\bigr ]. \end{cases} \end{equation}

Here $\text{En}_{\textit{sub}}$ represents the maximum reflection coefficient at a specific frequency, indicating that the reflection coefficient does not exceed the envelope for any sandbar length. Within the resonance band, full reflection occurs, whereas on both sides of the cutoff frequencies, the reflection coefficient decreases monotonically to zero at a rate proportional to $\omega _D^{-1}$ . The local maxima of the reflection coefficient lie on the envelope, as illustrated in figure 7(a).

Similarly, the envelope of the transmission coefficient for superharmonic resonance can be derived by taking the limit superior of $L$ for the transmission coefficient in (3.27),

(4.12) \begin{equation} \mathrm{En}_{\textit{sup}} =\overline {\lim _{L\to \infty }}\,|U(L)| =\left |\frac {K_\beta }{K_\nu }\right | \,\ \propto \,\ \left(C_{\textit{sup}}+\omega _D^{2}\right)^{-\frac {1}{2}}, \end{equation}

which, in contrast to the subharmonic case, is not a piecewise function, as illustrated in figure 7(b). The transmission coefficient decreases to zero on both sides of the superharmonic resonance, with a decay rate proportional to $\left( C_{\textit{sup}}+\omega _D^{2}\right)^{-1/2}$ , where $C_{\textit{sup}}$ is a positive constant. As the sandbar length increases, the curve converges to the shaded region, and cutoff frequency is zero in the superharmonic resonance, exhibiting a zero-bandwidth structure.

4.4. Zero-bandwidth behaviour of superharmonic resonance

Section 4.3 illustrated the substantial differences between subharmonic and superharmonic resonance. The former exhibits a finite width resonance band, a feature commonly observed in Class-I and Class-II Bragg resonance. In contrast, the latter represents an unusual phenomenon characterized by a zero-bandwidth structure, marking a significant difference from conventional Bragg scattering behaviour. To verify the existence of this zero-bandwidth feature, numerical calculations based on the MNLS equations are performed to further support the present findings for Case f under varying sandbar numbers $N_d$ , ranging from $40-500$ .

Figure 8.

Comparison of $\text{En}_{\textit{sup}}$ from the analytical solution (blue line, (4.12)) and numerical results for different sandbar number: $N_d=40$ (red line), $50$ (purple line), $250$ (green line) and $500$ (black dashed line).

Figure 8 compares the envelopes of the transmission coefficients obtained from the analytical and numerical solutions, with the latter derived from the local peaks of the transmission curves. The envelopes derived from the numerical solutions converge as $N_d$ increases from 40 to 500. The good agreement between the envelopes for $N_d=250$ and 500 indicates that the envelope has already reached its asymptotic form.

The numerical solutions show slight discrepancies compared with the analytical solution (4.12). At the peak, the curve exhibits sharp corners with discontinuous derivatives, and the envelope curve displays noticeable asymmetry. Specifically, on the right-hand side of the resonance, the curve drops rapidly, indicating pronounced singular behaviour. In addition, the resonance frequency predicted by the numerical solution exhibits a slight upward shift from the analytical prediction. These differences are demonstrated to arise from the approximation of the cubic terms (i.e. third-order wave–wave interactions) in the MNLS equations introduced by the frozen-coefficient method. A quantitative assessment of the discrepancy is conducted, and a modified solution is derived to account for the additional frequency upshift by reapplying the closed-form solutions to incorporate the spatially averaged cubic terms representing wave–wave interactions in the limit $L \to \infty$ . The modified solution is shown to capture the difference observed in figure 8. A detailed theoretical analysis can be found in Appendix D.

Additionally, the envelopes obtained numerically present a zero-bandwidth structure similar to that predicted analytically, thereby confirming the existence of this unique feature and validating the capability of the present closed-form solutions to predict the zero-bandwidth structure.

To better understand this phenomenon, the characteristic equation of the system, (3.16), can be written as

(4.13) \begin{equation} (\varsigma \mp K)^2 = K^2 + K_{\alpha } K_{\beta }, \end{equation}

where $\varsigma$ denotes the characteristic root, and $K$ , $K_{\alpha }$ and $K_{\beta }$ are real quantities defined in (3.18). The wave-frequency dependence is contained in $K$ , while

(4.14) \begin{equation} K_{\alpha } K_{\beta } = A_0^{2} \frac {\bigl ( i D_2^{\alpha } \bigr )^{2}}{C_g^{\alpha } C_g^{\beta }}, \end{equation}

with $i D_2^{\alpha }$ being real. For superharmonic resonance, the incident and scattered waves propagate in the same direction, such that $C_g^{\alpha } C_g^{\beta } \gt 0$ , and hence $K_{\alpha } K_{\beta } \gt 0$ . In this case, the characteristic roots remain real for all wave frequencies, implying that no transition occurs. Consequently, the sine-form solution for $|U(L)|$ in (3.27) persists for any wave frequency of the incident wave.

In contrast, for subharmonic resonance, the two wave components propagate in opposite directions, yielding $C_g^{\alpha } C_g^{\beta } \lt 0$ and thus $K_{\alpha } K_{\beta } \lt 0$ . The characteristic roots then undergo a transition from real to imaginary at specific frequencies, causing $|R(0)|$ in (3.22) to change from a hyperbolic cotangent form to a cotangent form. The cutoff frequency therefore corresponds to the critical point at which the nature of the characteristic roots changes. For superharmonic resonance, the two cutoff frequencies collapse into a single point, resulting in a zero-bandwidth structure. While the detailed physical mechanisms underlying this zero-bandwidth behaviour needs further investigation, the present analysis demonstrates that it originates from the identical propagation directions, namely the same sign of group velocities, of the incident and scattered waves.

From an engineering perspective, subharmonic and superharmonic resonances generally coexist. Subharmonic resonance can be exploited for coastal protection by reflecting wave energy to offshore region. Consequently, the range of effective reflection can be enlarged by increasing the sandbar length, allowing waves over a broader frequency range to be reflected and effectively creating an energy barrier. Superharmonic resonance, on the other hand, enhances the transmission of higher-frequency waves, which may contribute to seabed erosion and pose risks to coastal infrastructure. However, the zero-bandwidth structure implies that full transmission occurs only at the resonance frequency (or in its immediate vicinity); outside this narrow range, the transmitted energy remains small, even for an infinitely long sandbar. In practice, this limited transmitted energy can be efficiently dissipated by rough seabeds, porous structures or breakwaters, particularly since superharmonic resonance generates relatively short waves. Moreover, the concentration of transmitted energy near the resonance frequency, being narrow-banded, could be advantageous for targeted wave-energy conversion.

Overall, the present results suggest that artificial periodic structures can be designed to function as effective energy barriers via subharmonic resonance, while the unavoidable wave transmission associated with superharmonic resonance is narrow-banded owing to its zero-bandwidth nature, and can therefore be either mitigated through additional engineering measures or potentially harnessed for energy applications.