2.2.1. Governing equation and boundary conditions

The fluid is assumed to be homogeneous, inviscid and incompressible, and the flow is assumed to be irrotational. The wave system can be described in terms of a velocity potential $\varphi (x,y,z,t)$ , where $(x,y,z)$ are spatial variables and $t$ denotes time. The velocity potential $\varphi (x,y,z,t)$ satisfies the Laplace equation

(2.1) \begin{equation} \nabla ^2\varphi =0, \qquad -d+h_1(x,y)+h_2(x,y)\lt z\lt \zeta (x,y,t), \end{equation}

where

(2.2) \begin{equation} \nabla ^2=\frac {\partial ^2}{\partial x^2}+\frac {\partial ^2}{\partial y^2}+\frac {\partial ^2}{\partial z^2}. \end{equation}

The kinematic and dynamic boundary conditions at the free surface $z=\zeta (x,y,t)$ are

(2.3) \begin{gather} \zeta _t-\varphi _z+(\varphi _x\zeta _x+\varphi _y\zeta _y)=0, \end{gather}

(2.4) \begin{gather} g\zeta +\varphi _t+\tfrac {1}{2}\boldsymbol {\nabla }\varphi \boldsymbol {\cdot }\boldsymbol {\nabla }\varphi =0, \end{gather}

where $g$ is the gravitational acceleration. Taking time and space derivatives of (2.4) and combining the resulting equations with (2.3) enables us to eliminate $\zeta$ and obtain

(2.5) \begin{equation} \varphi _{tt}+g\varphi _z+2\boldsymbol {\nabla }\varphi \boldsymbol {\cdot }\boldsymbol {\nabla }\varphi _t+\tfrac 12\boldsymbol {\nabla }\varphi \boldsymbol {\cdot }\boldsymbol {\nabla }(\boldsymbol {\nabla }\varphi \boldsymbol {\cdot }\boldsymbol {\nabla }\varphi )=0. \end{equation}

In this paper, the bottom is taken to be stable and impermeable. The bottom boundary condition at $z=-d+h_1(x,y)+h_2(x,y)$ can be expressed as

(2.6) \begin{equation} \varphi _{z}=(h_{1x}+h_{2x})\varphi _{x}+(h_{1y}+h_{2y})\varphi _{y}. \end{equation}

To facilitate a dimensionless treatment of the governing equation and boundary conditions, we introduce dimensionless variables $X=k_{1x}$ , $Y=k_{1y}$ , $\Omega _{1}=\sigma _{1}/\omega _{1}$ , $T=\sigma _{1}t/\Omega _{1}$ and $Z=k_{1z}$ , where $\sigma _{1}$ is the actual frequency with nonlinear effects taken into account (see the dimensionless treatment in Xu et al. Reference Xu, Zhao, Song and Zhang2023). We define three phase functions $\xi _1$ , $\xi _2$ and $\xi _3$ as

(2.7) \begin{align} \xi _1=\boldsymbol {k}_1\boldsymbol {\cdot }\boldsymbol {r}-\sigma _1t=X\cos \alpha &+Y\sin \alpha -\Omega _1T, \end{align}

(2.8) \begin{align} \xi _2=\boldsymbol {k}_{{b}1}\boldsymbol {\cdot }\boldsymbol {r}=K_{{b1}}X\cos \theta _1&+K_{{b1}}Y\sin \theta _1, \end{align}

(2.9) \begin{align} \xi _3=\boldsymbol {k}_{{b}2}\boldsymbol {\cdot }\boldsymbol {r}=K_{{b2}}X\cos \theta _2&+K_{{b2}}Y\sin \theta _2. \end{align}

The governing equation and boundary conditions are rewritten in the following dimensionless forms, with $\xi _1$ , $\xi _2$ and $\xi _3$ serving as variables:

(2.10) \begin{align} &\qquad\qquad\quad\hat {\nabla }^2\phi =0, \quad -\varepsilon _1+\varepsilon _3\,H(\xi _2,\xi _3)\lt Z\lt \varepsilon _2\,\eta (\xi _1,\xi _2,\xi _3),&& \end{align}

(2.11) \begin{align} &\qquad\eta (\xi _1,\xi _2,\xi _3)=\tanh (\varepsilon _1)\,\Omega _1\phi _{\xi _1}-\varepsilon _2\tanh (\varepsilon _1)\,f, \quad Z=\varepsilon _2\,\eta (\xi _1,\xi _2,\xi _3), \end{align}

(2.12) \begin{align} &\Omega _1^2\phi _{\xi _1\xi _1}+\frac 1{\tanh (\varepsilon _1)}\,\phi _Z-2\varepsilon _2\Omega _1f_{\xi _1}+\varepsilon _2^2\,\hat {\boldsymbol {\nabla }}\phi \boldsymbol {\cdot }\hat {\boldsymbol {\nabla }}f=0, \quad Z=\varepsilon _2\,\eta (\xi _1,\xi _2,\xi _3), \end{align}

(2.13) \begin{align} &\qquad\qquad\quad\qquad\phi _Z-\varepsilon _3\,\hat {\boldsymbol {\nabla }}\phi \boldsymbol {\cdot }\hat {\boldsymbol {\nabla }}H=0, \quad Z=-\varepsilon _1+\varepsilon _3\,H(\xi _2,\xi _3), \end{align}

where

(2.14) \begin{align} H(\xi _2,\xi _3)={}&\cos (\xi _2)+\varepsilon _4\cos (\xi _3), \end{align}

(2.15) \begin{align} \hat {\boldsymbol {\nabla }}={}&\left (\cos \alpha\, \frac \partial {\partial \xi _1}+K_{{b1}}\cos \theta _1\,\frac \partial {\partial \xi _2}+K_{{b2}}\cos \theta _2\,\frac \partial {\partial \xi _3}\right )\boldsymbol {i}\nonumber \\ &{}+\left (\sin \alpha\, \frac \partial {\partial \xi _1}+K_{{b1}}\sin \theta _1\,\frac \partial {\partial \xi _2}+K_{{b2}}\sin \theta _2\,\frac \partial {\partial \xi _3}\right )\boldsymbol {j} +\left (\frac \partial {\partial z}\right )\boldsymbol {k}, \end{align}

(2.16) \begin{align} f={}&\tfrac 12(\phi _{\xi _1}^2+K_{{b1}}^2\phi _{\xi _2}^2+K_{{b2}}^2\phi _{\xi _3}^2+\phi _{Z}^2)+K_{{b1}}\cos (\alpha -\theta _1)\,\phi _{\xi _1}\phi _{\xi _2}\nonumber \\ &{}+K_{{b2}}\cos (\alpha -\theta _2)\,\phi _{\xi _1}\phi _{\xi _3}+K_{{b1}}K_{{b2}}\cos (\theta _1-\theta _2)\,\phi _{\xi _2}\phi _{\xi _3}. \end{align}

Here, $\varepsilon _{1}=k_{1}d$ , $\varepsilon _{2}=a_{1}k_{1}$ , $\varepsilon _{3}=b_{1}k_{1}$ and $\varepsilon _{4}=b_{2}/b_{1}$ are dimensionless parameters, and $a_1$ represents the measured (dimensional) wave amplitude of the primary transmitted wave with wavenumber $\boldsymbol {k}_1$ .

We apply the HAM to solve (2.10)–(2.13). The core principle of the HAM is to deform the unknown function from an initial guess to precise solutions through homotopy, achieved by introducing a physically meaningless embedding parameter. We first construct the zeroth-order deformation equations on the basis of the surface and bottom boundary conditions by introducing three mapping functions $\hat {\phi }(\xi _{1},\xi _{2},\xi _{3},Z;q)$ , $\hat {\eta }(\xi _1,\xi _2,\xi _3;q)$ and $\hat {\Omega }_1(q)$ . As $q$ changes from 0 to 1, the three mapping functions transition from their corresponding initial guesses to the precise solutions. A detailed explanation of the solution procedure can be found in Xu et al. (Reference Xu, Zhao, Song and Zhang2023).

2.2.2. Convergence verification

According to the convergence theorem of the HAM (Liao Reference Liao2012), three residual squares are defined as follows:

(2.17) \begin{align} \varepsilon _M^\phi &=\frac {1}{{\unicode[Arial]{x03C0}} ^3}\int _0^{{\unicode[Arial]{x03C0}} }\int _0^{{\unicode[Arial]{x03C0}} }\int _0^{{\unicode[Arial]{x03C0}} }\left (\sum _{n=0}^M\Delta _n^\phi \right )^3\,\mathrm {d}\xi _1\,\mathrm {d}\xi _2\,\mathrm {d}\xi _3, \end{align}

(2.18) \begin{align} \varepsilon _M^\eta &=\frac {1}{{\unicode[Arial]{x03C0}} ^3}\int _0^{{\unicode[Arial]{x03C0}} }\int _0^{{\unicode[Arial]{x03C0}} }\int _0^{{\unicode[Arial]{x03C0}} }\left (\sum _{n=0}^M\Delta _n^\eta \right )^3\,\mathrm {d}\xi _1\,\mathrm {d}\xi _2\,\mathrm {d}\xi _3, \end{align}

(2.19) \begin{align} \varepsilon _M^b&=\frac {1}{{\unicode[Arial]{x03C0}} ^3}\int _0^{{\unicode[Arial]{x03C0}} }\int _0^{{\unicode[Arial]{x03C0}} }\int _0^{{\unicode[Arial]{x03C0}} }\left (\sum _{n=0}^M\Delta _n^b\right )^3\,\mathrm {d}\xi _1\,\mathrm {d}\xi _2\,\mathrm {d}\xi _3, \end{align}

where $M$ denotes the homotopy approximation order. The terms $\Delta _n^{\phi }$ , $\Delta _n^{\eta }$ and $\Delta _{n}^{b}$ arise in solving the high-order deformation equations, and this theorem provides us with a convenient way to check the convergence and accuracy of the homotopy-series solution. All of $\Delta _n^{\phi }$ , $\Delta _n^{\eta }$ and $\Delta _{n}^{b}$ are functions of the convergence control parameter. Specific expressions for $\Delta _n^{\phi }$ , $\Delta _n^{\eta }$ and $\Delta _{n}^{b}$ can be found in the appendix of Xu et al. (Reference Xu, Zhao, Song and Zhang2023).

Figure 3.

Relationships between $c_0$ and residual squares for $k_{1}d=1.5$ , $a_{1}k_{1}=0.02$ , $b_{1}k_{1}=0.02$ , $b_{2} / b_{1}=0.5$ , $k_{{b1}} / k_{{b2}}=0.8$ , $\alpha =70^{\circ }$ , $\theta =0^{\circ }$ and $\delta \omega =-0.00001$ . (a–c) Relationships between $c_0$ and $\log _{10}\varepsilon _{M}^{b}$ , $\log _{10}\varepsilon _M^\phi$ and $\log _{10}\varepsilon _M^\eta$ , respectively: dash-double-dotted line, first-order approximation; dash-dotted line, third-order approximation; dotted line, fifth-order approximation; dashed line, seventh-order approximation; solid line, ninth-order approximation. (d) Variation of squared residuals with order $M$ : solid line, $\log _{10}\varepsilon _{M}^{b}$ ; dash-dotted line, $\log _{10}\varepsilon _M^\phi$ ; dotted line, $\log _{10}\varepsilon _M^\eta$ .

To illustrate the selection of the convergence parameter and the verification of the homogeneous approximate solution, we take the following case as an example:

(2.20) \begin{equation} \begin{gathered} k_{1}d = 1.5, \quad a_{1}k_{1} = 0.02, \quad b_{1}k_{1} = 0.02, \\ b_{2} / b_{1} = 0.5, \quad k_{{b1}} / k_{{b2}} = 0.8, \quad \alpha = 70^{\circ }, \\ \theta = 0^{\circ }, \quad \delta \omega = -0.00001. \end{gathered} \end{equation}

Figures 3(a–c) display the relationships between the convergence control parameter $c_0$ and the three residual squares. It is evident that the three average residuals exhibit their steepest decreases when $c_0=1.2$ . Therefore, $c_0=1.2$ is chosen for subsequent computations of the residual squares. As depicted in figure 3(d), the residual squares exhibit a monotonic decrease as the approximation order $M$ is increased. This reduction reaches a minimum of 11 orders of magnitude by the 20th order, indicating successful convergence of the approximate solution obtained at $c_0=1.2$ . All results presented in this paper meet the criteria for convergence: (i) the three residual squares decrease by at least five orders of magnitude; (ii) as the approximation order $M$ increases, the three residual squares maintain a downward trend; (iii) all three residual squares are less than $1\times 10^{-6}$ ; (iv) all results in this paper retain at least five significant figures.

3.3.1. Amplitudes of bottom waveforms

We first examine how $b_1k_1$ and $b_2/b_1$ affect the energy distribution in the resonant wave system. As shown in figures 5(a,c), in subtype 2, the energy contribution of the resonant wave increases with increasing $b_1k_1$ and $b_2/b_1$ , while that of the primary transmitted wave decreases. Changes in these parameters have less impact in subtype 1.

Figure 5.

Influence of bottom topography on wave formation and propagation for $k_{1}d=1.5$ , $\alpha =70^{\circ }$ , $k_{{b1}} / k_{{b2}}=0.8$ , $\theta =0^{\circ }$ and $\delta \omega =-0. 0001$ . (a,b) Dependences of energy contributions of primary transmitted and resonant components ( $a_{1,0,0}^{2} / \Pi$ , $a_{1,-1,-1}^{2} / \Pi$ ) and combined energy contribution of primary transmitted and resonant components $\Sigma _{1}$ , respectively, on $b_{1}k_{1}$ for $a_{1}k_{1}=0.04$ and $b_{2} / b_{1}=0.5$ . (c,d) Dependences of energy contributions of primary transmitted and resonant components ( $a_{1,0,0}^{2} / \Pi$ , $a_{1,-1,-1}^{2} / \Pi$ ) and combined energy contribution of primary transmitted and resonant components $\Sigma _{1}$ , respectively, on $b_{2}/b_{1}$ for $a_{1}k_{1}=0.06$ and $b_{1}k_{1}=0.03$ .

The combined contribution of the primary transmitted and resonant wave components to the total energy is shown in figures 5(b,d). It is apparent that there is a significant decrease in $\Sigma _{1}$ with increasing $b_1k_1$ and $b_2/b_1$ , indicating that the influence of these wave components on the overall wave energy diminishes. As $b_1k_1$ and $b_2/b_1$ increase, the number of wave components in the system grows gradually, decreasing the percentage of primary transmitted and resonant wave energies, and this decrease is reflected in $\Sigma _{1}$ . A similar trend is observed as the angle between the topographic waveforms is varied, with more pronounced effects (see § 3.3.2). This is in agreement with the results of previous work on exact resonance by Xu et al. (Reference Xu, Zhao, Song and Zhang2023). Similar findings have also been reported by Xu & Liu (Reference Xu and Liu2020). All these studies highlight how multiple-wave nonlinear effects can introduce additional wave components, leading to a redistribution of energy within the wave system.

We now investigate how the values of $b_1k_1$ and $b_2/b_1$ affect the wave steepness $h_{{s}}$ and the dimensionless angular frequencies $\sigma _{1}/\omega _{1}$ and $\sigma _{2}/\omega _{2}$ of the primary transmitted and resonant waves in the wave system. The relationships between $b_1k_1$ and $b_2/b_1$ and the wave properties in the steady-state wave system are shown in figure 6. It can be seen that the wave properties are positively correlated with the variable parameters. Additionally, we see that the wave steepness and angular frequency in subtype 1 are higher than those in subtype 2, indicating differences between these two types of steady-state wave system. This is in agreement with the findings of Xu et al. (Reference Xu, Zhao, Song and Zhang2023) from their study of exact resonance. Overall, the influence of topographic waveform amplitude on wave steepness and angular frequency remains consistent and positively correlated, regardless of the type of resonance being considered.

Figure 6.

Influence of bottom topography on wave systems for $k_{1}d=1.5$ , $\alpha =70^{\circ }$ , $k_{{b1}} / k_{{b2}}=0.8$ , $\theta =0^{\circ }$ and $\delta \omega =-0.0001$ . (a,b) Dependences of wave steepness $h_{{s}}$ and angular frequencies ( $\sigma _{1}/\omega _{1}$ , $\sigma _{2}/\omega _{2}$ ), respectively, on $b_1k_1$ for $a_{1}k_{1}=0.04$ and $b_{2} / b_{1}=0.5$ . (c,d) Dependences of wave steepness $h_{{s}}$ and angular frequencies ( $\sigma _{1}/\omega _{1}$ , $\sigma _{2}/\omega _{2}$ ), respectively, on $b_{2}/b_{1}$ for $a_{1}k_{1}=0.06$ and $b_{1}k_{1}=0.03$ .

3.3.2. Angle between topographic waveforms

One characteristic that distinguishes wave–bottom resonances is the number of waveforms in the undulating bottom topography. When the bottom consists of two waveforms, variations in the angle $\theta$ between these two waveforms will have an effect on the propagation of surface waves. However, to the best of our knowledge, previous studies have paid little attention to the influence of this angle on wave behaviour, and to remedy this defect, we examine this influence in detail. In particular, our present investigation has been prompted by the identification by Zhang et al. (Reference Zhang, Tao, Tu, Su and Xie2021) of a focusing effect at the bottom of the V shape formed between two topographic waveforms.

We consider angles in the range $\theta =0^{\circ }$ – $100^{\circ }$ . As we have seen from figures 5(b,d), when the two topographic waveforms are collinear ( $\theta =0^{\circ }$ ), the combined energy contribution of the primary transmitted and resonant wave components $\Sigma _{1}$ decreases with increasing $b_1k_1$ and $b_2/b_1$ . This indicates the presence of additional wave components in the system at larger values of $b_1k_1$ and $b_2/b_1$ . For example, in subtype 1, the combined energy contribution $\Sigma _{1}$ is $99.64\,\%$ at $b_1k_1=0.02$ , and decreases to $95.84\,\%$ at $b_1k_1=0.1$ (figure 5 b). It must be emphasised that the steady-state wave systems analysed in this study exhibit constant amplitudes and frequencies that remain consistent over time. This is similar to what was found in the study of the exact resonance by Xu et al. (Reference Xu, Zhao, Song and Zhang2023). Comparison of different wave systems with various values of physical parameters indicates the presence of additional wave components, beyond the resonant and primary transmitted components, and such a comparison enables us to identify which wave components contribute to the behaviour of these different steady-state wave systems.

As demonstrated in figure 7, variations in the angle $\theta \gt 0^{\circ }$ have significant effects on $\Sigma _{1}$ for both subtype 1 and subtype 2 wave systems. It can also be seen that these effects are more obvious as $b_2/b_1$ increases. This can be attributed to the presence of a greater number of wave components in the wave system when $\theta \gt 0^{\circ }$ , along with a greater energy contribution from components other than primary transmitted and resonant waves. It can be seen from figure 7 that the dependence of $\Sigma _{1}$ on $\theta$ differs between subtypes 1 and 2. First, in subtype 1, $\Sigma _{1}$ exhibits two local minima around $\theta =6^{\circ }$ and $40^{\circ }$ , respectively, whereas in subtype 2, there are two local minima around $\theta =14^{\circ }$ and $40^{\circ }$ . Second, the absolute minimum of $\Sigma _{1}$ in subtype 1 is approximately $91.38\,\%$ and corresponds to the local minimum at the lower value $\theta =6^{\circ }$ , whereas the absolute minimum of $\Sigma _{1}$ in subtype 2 is approximately $88.87\,\%$ and corresponds to the local minimum at the higher value $\theta =40^{\circ }$ . It is also noteworthy that in both subtypes, the values of $\theta$ corresponding to the local minima of $\Sigma _{1}$ remain consistently stable and do not change with variations in $b_2/b_1$ .

Figure 7.

Influence of the angle $\theta$ between the waveforms of the bottom topography on the combined energy contribution of primary transmitted and resonant components $\Sigma _{1}$ for $k_{1}d=1.5$ , $a_{1}k_{1}=0.06$ , $b_{1}k_{1}=0.02$ , $k_{{b1}} / k_{{b2}}=0.8$ , $\alpha =78^{\circ }$ and $\delta \omega =0.00004$ : (a) subtype 1; (b) subtype 2.

Figure 8.

Influence of the angle $\theta$ between the waveforms of the bottom topography in subtype 1 on the energy contribution of each wave component in the wave system for $k_{1}d=1.5$ , $a_{1}k_{1}=0.06$ , $b_{1}k_{1}=0.02$ , $b_{2} / b_{1}=1.0$ , $k_{{b1}} / k_{{b2}}=0.8$ , $\alpha =78^{\circ }$ and $\delta \omega =0.00004$ . (a) Energy contributions of all wave components. (b) Energy contributions of wave components other than primary transmitted and resonant waves.

Figure 9.

Influence of the angle $\theta$ between the waveforms of the bottom topography in subtype 2 on the energy contribution of each wave component in the wave system for $k_{1}d=1.5$ , $a_{1}k_{1}=0.06$ , $b_{1}k_{1}=0.02$ , $b_{2} / b_{1}=1.0$ , $k_{{b1}} / k_{{b2}}=0.8$ , $\alpha =78^{\circ }$ and $\delta \omega =0.00004$ . (a) Energy contributions of all wave components. (b) Energy contributions of wave components other than primary transmitted and resonant waves.

For a clearer understanding of the energy distribution in the wave system, all wave components in subtypes 1 and 2 with energy contributions exceeding $0.1\,\%$ have been identified and are presented in detail in figures 8 and 9, as well as in tables 2 and 3, from which it is evident that there is a pronounced increase in the energy contribution of the remaining wave components within the wave system for relatively low values of $\Sigma _{1}$ . This suggests that the energy of the wave system is distributed across a greater number of wave components, with some being allocated to higher-order components. Taking subtype 1 as an example, figures 8(b) and 9(b) demonstrate that the energy contributions of each of the wave components other than the resonant and primary transmitted wave components vary with the angle $\theta$ . Furthermore, the wave components that make the greatest contribution are not always the same. In particular, the component $\cos (\xi _{1}-\xi _{3})$ makes the greatest contribution in the vicinity of $\theta =6^{\circ }$ , whereas the component $\cos (\xi _{1}+\xi _{3})$ makes the greatest contribution in the vicinity of $\theta =40^{\circ }$ . This phenomenon occurs consistently in both subtype 1 and subtype 2.

Table 2.

Energy contribution (%) of each wave component in a steady-state wave system at near resonance for different angles $\theta$ between the waveforms of the bottom topography (subtype 1).

Table 3.

Energy contribution (%) of each wave component in a steady-state wave system at near resonance for different angles $\theta$ between the waveforms of the bottom topography (subtype 2).

To explore the reason behind the energy distribution mentioned above, we substitute the wave components shown in figures 8 and 9 into the following general resonance conditions:

(3.5) \begin{gather} i\boldsymbol {k}_1 +p\boldsymbol {k}_{{b}1}+q\boldsymbol {k}_{{b}2}=\boldsymbol {k}_{i,p,q}, \end{gather}

(3.6) \begin{gather} \omega _1=\delta \omega _{i,p,q}+\omega _{i,p,q}. \end{gather}

We denote these conditions as $\mathrm {Res}(i,p,q)$ . Thus, for example, $\mathrm {Res}(1,-1,-1)$ corresponds to conditions (1.1) and (1.2). The corresponding $\delta \omega _{i,p,q}$ are listed in table 4. It can be seen that at the two peaks ( $\theta =6^{\circ }$ and $40^{\circ }$ for subtype 1; $\theta =14^{\circ }$ and $40^{\circ }$ for subtype 2), the values of $\delta \omega _{i,p,q}$ for the wave components $\cos (\xi _1-\xi _3)$ and $\cos (\xi _1+\xi _3)$ , i.e. $\delta \omega _{1,0,-1}$ and $\delta \omega _{1,0,1}$ , are relatively small, indicating that near resonances also occur among these components, the primary transmitted wave $\cos (\xi _1)$ and the seabed. In other words, multiple resonances exist simultaneously. For example, in subtype 1, when $\theta =6^{\circ }$ , the near-resonance conditions $\mathrm {Res}(1,-1,-1)$ and $\mathrm {Res}(1,0,-1)$ are satisfied. Thus except for the component $\cos (\xi _1-\xi _2-\xi _3)$ , the energy of the component $\cos (\xi _1-\xi _3)$ cannot be ignored. The same phenomenon also occurs in the other three scenarios shown in table 4. As mentioned above, multiple resonances also occur among surface gravity waves propagating over a flat seabed (Liu et al. Reference Liu, Xu and Liao2018).

Table 4.

Values of $\delta \omega _{i,p,q}$ in the resonance conditions (3.5) and (3.6) corresponding to the wave components shown in figures 8 and 9.

For further clarity, let us delve deeper into the occurrence of resonance and the wave components involved. For resonance to occur, there are two requirements: (i) the exact or near resonance conditions must be satisfied, and (ii) the corresponding resonant wave components must have a non-negligible amount of energy. The data on $\delta \omega$ in table 4 show that there are one or two wave components that meet both of these requirements, and their energy contribution is higher than that of other higher-order wave components. Taking subtype 1 as an example, at $\theta =6^{\circ }$ , $\delta \omega _{1,-1,-1}$ and $\delta \omega _{1,0,-1}$ are equal to $4\times 10^{-5}$ and $6.5\times 10^{-5}$ , respectively. Table 2 also presents the energy contributions of these two wave components in the entire wave system: $6.48\,\%$ and $44.93\,\%$ , respectively. This indicates that $\cos (\xi _1-\xi _2-\xi _3)$ and $\cos (\xi _1-\xi _3)$ are both resonant wave components. For subtype 2, at $\theta =40^{\circ }$ , $\delta \omega _{1,-1,-1}$ and $\delta \omega _{1,0,1}$ are equal to $4\times 10^{-5}$ and $5.17\times 10^{-3}$ , respectively, while the energy contributions of these two wave components are 0 and $5.84\,\%$ , respectively. This allows us to confirm that resonance does indeed occur, and the resonant component is $\cos (\xi _1+\xi _3)$ . In the cases shown in figures 8 and 9, in addition to the resonance components, there are actually other high-order wave components involved in the interaction. For example, in the case $\theta =6^{\circ }$ in subtype 1, apart from $\cos (\xi _1-\xi _2-\xi _3)$ and $\cos (\xi _1-\xi _3)$ , the energies of $\cos (\xi _1-\xi _2)$ , $\cos (\xi _1+\xi _3)$ , $\cos (2\xi _1-\xi _2-\xi _3)$ and $\cos (\xi _1-2\xi _2-\xi _3)$ also account for a certain proportion. In fact, wave components of even higher orders are present, but their energy is much smaller. It should be noted that the number of wave components involved in the interaction increases with increasing wave steepness. Usually, as the steepness of the waves increases, higher-order wave components involved in the nonlinear interaction will make a more substantial energy contribution, thus a greater number of wave components will participate in the interaction. It is worth noting that this conclusion has been corroborated in the case of waves propagating in deep water with multiple near resonances occurring (Liu et al. Reference Liu, Xu and Liao2018).

Further investigations reveal that the angle $\theta$ also has a significant influence on wave steepness and angular frequency, as illustrated in figures 10–12. It can be seen that both the wave steepness and angular frequency exhibit pairs of peaks, and that these occur at the same angles as the local minima in the combined energy contribution of the primary transmitted and resonant components $\Sigma _{1}$ . Furthermore, we also find that in subtype 2, for $\theta \gt 50^{\circ }$ , the wave steepness $h_{{s}}$ is almost unaffected by changes in $\theta$ . This is in contrast to subtype 1, where $h_{{s}}$ exhibits a continuously decreasing trend for $\theta \gt 40^{\circ }$ . Moreover, in both subtypes, the steepness of the wave system for $\theta \gt 50^{\circ }$ is smaller than for $\theta =0^{\circ }$ . In this angular range, changes in $b_{2}/b_{1}$ and $k_{{b1}}/k_{{b2}}$ have a more limited effect on the wave system than in the small-angle case. However, increasing $k_{{b1}}/k_{{b2}}$ in subtype 1 does not necessarily increase the wave steepness of the wave system, but actually decreases it at some angles.

In addition, we examine the influence of various parameters on the peak values, namely the ratios $b_2/b_1$ and $k_{{b1}}/k_{{b2}}$ , and the propagation angle $\alpha$ of the primary transmitted wave component. The peaks are not affected at all by $b_2/b_1$ (see figure 10), but are slightly affected by $k_{{b1}}/k_{{b2}}$ (see figure 11) and significantly affected by $\alpha$ (see figure 13). In particular, as shown in figure 13, we examine the wave steepness of the whole wave system for different values of $\alpha$ . When $\alpha$ varies from $70^{\circ }$ to $78^{\circ }$ , both the numbers of the peaks and their values change. It should be noted that the ratio $b_2/b_1$ does not appear in the resonance conditions (1.1) and (1.2). Its value does not affect the magnitude and direction of the wave components that participate in the resonance. However, changes in $k_{{b1}}/k_{{b2}}$ and $\alpha$ will directly affect those wave components, and further change the wave systems. Thus both of them will affect the peaks, but the impact of $\alpha$ is more significant.

Figure 10.

Influence of the angle $\theta$ between the waveforms of the bottom topography on the wave steepness $h_{{s}}$ of the wave system for $k_{1}d=1.5$ , $a_{1}k_{1}=0.06$ , $b_{1}k_{1}=0.02$ , $k_{{b1}} / k_{{b2}}=0.8$ , $\alpha =78^{\circ }$ and $\delta \omega =0.00004$ , and different values of $b_2/b_1$ : (a) subtype 1; (b) subtype 2.

Figure 11.

Influence of the angle $\theta$ between the waveforms of the bottom topography on the wave steepness $h_{{s}}$ of the wave system for $k_{1}d=1.5$ , $a_{1}k_{1}=0.06$ , $b_{1}k_{1}=0.02$ , $b_2 / b_1=0.5$ , $\alpha =78^{\circ }$ and $\delta \omega =0.00004$ , and different values of $k_{{b1}}/k_{{b2}}$ : (a) subtype 1; (b) subtype 2.

Figure 12.

Influence of the angle $\theta$ between the waveforms of the bottom topography on the angular frequencies ( $\sigma _{1}/\omega _{1}$ , $\sigma _{2}/\omega _{2}$ ) of subtype 1 and subtype 2 wave systems for $k_{1}d=1.5$ , $a_{1}k_{1}=0.06$ , $b_{1}k_{1}=0.02$ , $b_{2} / b_{1}=1.0$ , $k_{{b1}} / k_{{b2}}=0.8$ , $\alpha =78^{\circ }$ and $\delta \omega =-0.0005$ .

Figure 13.

Influence of the angle $\theta$ between the waveforms of the bottom topography on the wave steepness $h_{{s}}$ of the wave system for $k_{1}d=1.5$ , $a_{1}k_{1}=0.06$ , $b_{1}k_{1}=0.02$ , $b_2 / b_1=0.5$ , $k_{{b1}}/k_{{b2}}=0.8$ and $\delta \omega =0.00004$ and different values of $\alpha$ : (a) subtype 1; (b) subtype 2.

It is important to note that according to the definition (3.4) of the wave steepness $h_{{s}}$ adopted in this paper, and the assumption that wave energy is directly proportional to wave amplitude, the magnitude of $h_{{s}}$ provides some indication of the total energy of the wave system. It should also be noted that primary conversion of wave energy is attained by an oscillating system, and this is able to store some kinetic and/or potential energy (Brooke Reference Brooke2003). Most wave energy converters (WECs) are oscillating-body devices (Falcao Reference Falcao2010). Since energy production by oscillating bodies is governed by wave-induced motion, the dimensions of such WECs are constrained significantly by the incident wavelengths.

The design of a WEC typically involves consideration of the local primary wave frequencies. When the natural frequency of the WEC is close to the primary local wave frequency, resonance occurs between the WEC and the waves, resulting in significant relative motion between the two. The WEC then converts this mechanical energy into electricity (Xu, Stuhlmeier & Stiassnie Reference Xu, Stuhlmeier and Stiassnie2018). On the other hand, according to the wave–bottom resonance studied in this paper, and depending on the local primary wave frequency, the seabed topography can be adjusted and designed to enhance the wave amplitude after resonance with the surface waves, thereby allowing the WEC to undergo greater relative motion and capture more wave energy. Thus wave–bottom resonance can be exploited to design the appropriate seabed topography for a given marine area, and determine the geometrical parameters of a WEC wave energy harvesting device so that it captures as much wave energy as possible.

In this paper, a dual-frequency seabed topography has been considered, and for certain specific angles, the studied steady-state resonant wave system exhibits a larger amplitude. Thus the use of wave energy collection devices based on bottom topographies characterised by angles $\theta$ corresponding to the peak values of $h_{{s}}$ revealed in the above analysis should help to maximise the amount of wave energy that can be harvested. Such an approach to improving the effectiveness of wave energy harvesting closely resembles the approaches of Zhang et al. (Reference Zhang, Tao, Tu, Su and Xie2021) and Hao et al. (Reference Hao, Li, Liu and Wang2022).