2.1 Governing equation

We assume that the fluid is inviscid and incompressible, the flow is irrotational and the surface tension is neglected. Here, $(x,y,z)$ represents the usual Cartesian coordinate system, in which $(x,y)$ defines a horizontal plane located at the mean water level and $z$ is measured vertically upwards. The governing equations for deep-water waves read

where $\unicode[STIX]{x1D711}$ denotes the velocity potential, $\unicode[STIX]{x1D702}$ is the free-surface elevation, $g$ is the gravity acceleration and $t$ is the time. We consider a wave system that consists of two primary progressive waves. Let $\boldsymbol{k}_{i}$ denote the wavenumber and $\unicode[STIX]{x1D70E}_{i}$ the actual angular frequency. For steady-state wave systems, all wave amplitudes $a_{i}$ , wavenumbers $\boldsymbol{k}_{i}$ and actual angular frequencies $\unicode[STIX]{x1D70E}_{i}$ are constant, i.e. independent of time. We write

where $\boldsymbol{r}=x\boldsymbol{i}+y\boldsymbol{j}$ . Therefore, in the new coordinate system $(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2},z)$ , the original initial/boundary-value problem (2.1)–(2.4) can be transformed into a boundary-value problem governed by

subject to the two boundary conditions on the unknown free surface $z=\unicode[STIX]{x1D702}(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2})$

and also the bottom condition

where ${\mathcal{N}}_{1}$ and ${\mathcal{N}}_{2}$ are the nonlinear differential operators and

The wave elevation $\unicode[STIX]{x1D702}$ and velocity potential $\unicode[STIX]{x1D711}$ can be expressed in the form

with the definition

where $C_{m,n}^{\unicode[STIX]{x1D702}}$ and $C_{m,n}^{\unicode[STIX]{x1D711}}$ are constants to be determined. The constant $C_{m,n}^{\unicode[STIX]{x1D702}}$ denotes the amplitude of the wave component $\cos (m\unicode[STIX]{x1D709}_{1}+n\unicode[STIX]{x1D709}_{2})$ . It should be noted that $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2},z)$ automatically satisfies the governing equation (2.6) and the bottom condition (2.9).

2.2 Solution procedure

The steady-state waves were obtained by means of the HAM for exact resonance (Liao Reference Liao2011) and near resonance (Liao et al. Reference Liao, Xu and Stiassnie2016) in deep water, and extended by others for more complicated cases of exact resonance (Xu et al. Reference Xu, Lin, Liao and Stiassnie2012; Liu & Liao Reference Liu and Liao2014). Detailed mathematical derivations can be found in these articles. For the sake of simplicity, we just give the most important formulae here. In the HAM-based approach, the solutions for the wave elevation $\unicode[STIX]{x1D702}$ and the velocity potential $\unicode[STIX]{x1D711}$ are approximated by the following two series:

which are governed by the high-order deformation equations

with the definition $\unicode[STIX]{x1D712}_{1}=0$ and $\unicode[STIX]{x1D712}_{m}=1$ for $m>1$ , where $\unicode[STIX]{x1D711}_{0}$ is an initial guess of the velocity potential $\unicode[STIX]{x1D711}$ , ${\mathcal{L}}$ is an auxiliary linear operator and $c_{0}$ is a convergence-control parameter. We choose the initial guess $\unicode[STIX]{x1D702}_{0}=0$ mainly because it is the simplest one and is also used by most traditional analytical methods. Up to the $m$ th order of approximation, all terms $\unicode[STIX]{x1D6E5}_{m-1}^{\unicode[STIX]{x1D702}}$ , $\unicode[STIX]{x1D6E5}_{m-1}^{\unicode[STIX]{x1D711}}$ , $\bar{S}_{m}$ and $S_{m-1}$ on the right-hand side of the high-order deformation equations (2.16) and (2.17) are already known and determined by $\unicode[STIX]{x1D702}_{j}$ and $\unicode[STIX]{x1D711}_{j}$ , with $j=0,1,2,\ldots ,m-1$ and $m\geqslant 1$ . The detailed expressions for $\unicode[STIX]{x1D6E5}_{m-1}^{\unicode[STIX]{x1D702}}$ and $\unicode[STIX]{x1D6E5}_{m-1}^{\unicode[STIX]{x1D711}}$ are shown in appendix A. It should be emphasized that, in the framework of the HAM, we have great freedom to choose the auxiliary linear operator ${\mathcal{L}}$ and the initial guess $\unicode[STIX]{x1D711}_{0}$ . The expressions for $\bar{S}_{m}$ and $S_{m-1}$ which depend on the choice of the auxiliary linear operator ${\mathcal{L}}$ are given in the next subsection.

2.3 Choice of auxiliary linear operator and initial potential

The key to the HAM approach is the choice of the auxiliary linear operator and the initial guess. The choices for single exact and near resonances are reviewed in § 2.3.1 and § 2.3.2 respectively. Then, the auxiliary linear operator and the initial guess for multiple near resonances are considered in § 2.3.3.

3.1 Weakly nonlinear wave groups

For weakly nonlinear wave systems, Liao et al. (Reference Liao, Xu and Stiassnie2016) considered the single nearly resonant set

in the case of

with the generalized linear operator (2.28). Multiple solutions were obtained and the energy distribution was verified by Zakharov’s equation.

To investigate further whether nearby resonant components interact with one another at the end of the solution domain considered by Liao et al. (Reference Liao, Xu and Stiassnie2016), we consider two nearly resonant components in the auxiliary linear operator (2.38) and initial potential (2.46). We take the two groups of solutions found by Liao et al. (Reference Liao, Xu and Stiassnie2016) at $\unicode[STIX]{x1D70F}=0.89$ as an example; the calculations are carried out beyond the domain $0.89\leqslant \unicode[STIX]{x1D70F}\leqslant 0.895$ until one or more components have lost their energy and leave the resonance with one or two components. Specifically, we consider the components $\unicode[STIX]{x1D6F9}_{2,-1}$ and $\unicode[STIX]{x1D6F9}_{1,-2}$ in the initial potential (2.46) for $\unicode[STIX]{x1D70F}<0.89$ in group 1, and the components $\unicode[STIX]{x1D6F9}_{2,-1}$ and $\unicode[STIX]{x1D6F9}_{3,-2}$ in the initial potential (2.46) for $\unicode[STIX]{x1D70F}>0.895$ in group 2. The piecewise function $\unicode[STIX]{x1D707}_{2}(m,n)$ in the auxiliary linear operator (2.38) changes accordingly with the two nearly resonant components.

Figure 2.

The wave energy distribution of the six largest components $|C_{i,j}^{\unicode[STIX]{x1D702}}|^{2}/\unicode[STIX]{x1D6F1}$ of the resonant system (2.37) in the case of dimensionless frequency $\unicode[STIX]{x1D716}=1.0003$ and angle $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/36$ with various wavenumber ratios $\unicode[STIX]{x1D70F}$ : line with circles, $|C_{1,0}^{\unicode[STIX]{x1D702}}|^{2}/\unicode[STIX]{x1D6F1}$ ; line with triangles, $|C_{0,1}^{\unicode[STIX]{x1D702}}|^{2}/\unicode[STIX]{x1D6F1}$ ; line with squares, $|C_{2,-1}^{\unicode[STIX]{x1D702}}|^{2}/\unicode[STIX]{x1D6F1}$ ; line with pentagrams, $|C_{3,-2}^{\unicode[STIX]{x1D702}}|^{2}/\unicode[STIX]{x1D6F1}$ ; line with hexagrams, $|C_{4,-3}^{\unicode[STIX]{x1D702}}|^{2}/\unicode[STIX]{x1D6F1}$ ; line with diamonds, $|C_{-1,2}^{\unicode[STIX]{x1D702}}|^{2}/\unicode[STIX]{x1D6F1}$ .

Figure 2 shows the wave energy distribution of the six largest components (two primary components together with four nearly resonant ones) $|C_{i,j}^{\unicode[STIX]{x1D702}}|^{2}/\unicode[STIX]{x1D6F1}$ for the resonant system (2.37) in the case of dimensionless frequency $\unicode[STIX]{x1D716}=1.0003$ and angle $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/36$ with various wavenumber ratios $\unicode[STIX]{x1D70F}$ . The total wave energy $\unicode[STIX]{x1D6F1}=\sum _{m=0}^{+\infty }\sum _{n=-\infty }^{+\infty }(C_{m,n}^{\unicode[STIX]{x1D702}})^{2}$ is mainly contained by three components belonging to one resonant set. Energy transfer due to resonant interaction among different components is obvious, but no coupled resonances can be identified. In group 1, the energy contained by the component $\cos (\unicode[STIX]{x1D709}_{1}-2\unicode[STIX]{x1D709}_{2})$ ( $|C_{1,-2}^{\unicode[STIX]{x1D702}}|^{2}/\unicode[STIX]{x1D6F1}$ ) decreases from $17.2\,\%$ at $\unicode[STIX]{x1D70F}=0.87075$ to $0\,\%$ at $\unicode[STIX]{x1D70F}=0.895$ ; meanwhile, the energy contained by the component $\cos (2\unicode[STIX]{x1D709}_{1}-\unicode[STIX]{x1D709}_{2})$ ( $|C_{2,-1}^{\unicode[STIX]{x1D702}}|^{2}/\unicode[STIX]{x1D6F1}$ ) increases from $0\,\%$ to $88.6\,\%$ . In group 2, the energy contained by the component $\cos (2\unicode[STIX]{x1D709}_{1}-\unicode[STIX]{x1D709}_{2})$ ( $|C_{2,-1}^{\unicode[STIX]{x1D702}}|^{2}/\unicode[STIX]{x1D6F1}$ ) decreases from $26.1\,\%$ at $\unicode[STIX]{x1D70F}=0.89$ to $0.6\,\%$ at $\unicode[STIX]{x1D70F}=0.89669$ ; meanwhile, the energy contained by the component $\cos (3\unicode[STIX]{x1D709}_{1}-2\unicode[STIX]{x1D709}_{2})$ ( $|C_{3,-2}^{\unicode[STIX]{x1D702}}|^{2}/\unicode[STIX]{x1D6F1}$ ) increases from $0.2\,\%$ to $56.2\,\%$ . This transfer of energy between nearby nearly resonant components for weakly nonlinear wave groups indicates that coupled resonances among multiple nearly resonant components may occur for finite amplitude wave groups.

More calculations were carried out in the wavevector space $(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D6FC})$ for weakly nonlinear cases ( $\unicode[STIX]{x1D716}=1.0003$ ). The related energy distribution analyses confirm the continuum of steady-state resonances in the wavevector space $(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D6FC})$ and the phenomenon of energy transfer among nearby nearly resonant components.

3.2 Wave groups with increased components

More nearly resonant components are considered in wave groups with increased nonlinearity to further check the existence of steady-state multiple near resonances. Without loss of generality, we consider the near resonance case

with dimensionless frequency $\unicode[STIX]{x1D716}$ increased from 1.0003 to 1.005. Different combinations of the four largest nearly resonant components at $\unicode[STIX]{x1D716}=1.0003$ are considered as non-trivial components in the initial guess (2.46) to search for possible steady-state solutions. As shown in table 1, the number of nearly resonant components $l$ increases step by step from one to four. The term $\unicode[STIX]{x1D707}_{2}(m,n)$ in the auxiliary linear operator (2.38) changes accordingly. When three or fewer resonant components are considered as initial non-trivial ones, the coupled nonlinear algebraic equations for the coefficients $A_{0,i}$ are solved globally in the calculation of the first-order approximation. For four initial non-trivial nearly resonant components, the resulting algebraic equations are too complicated and time consuming to solve globally. Instead, a root searching procedure based on the partial solver ‘FindRoot’ in Mathematica is applied. Table 1 shows that the number of real solutions for the coefficients $A_{0,i}$ increases with the dimensionless frequency $\unicode[STIX]{x1D716}$ , especially when three or four nearly resonant components ( $l=3,4$ ) are considered. For example, for four nearly resonant components, the number of algebraic solutions increases from 1 at dimensionless frequency $\unicode[STIX]{x1D716}=1.0003$ to 32 at $\unicode[STIX]{x1D716}=1.005$ . The increased number of algebraic solutions for larger values of the dimensionless frequency $\unicode[STIX]{x1D716}$ indicates that more steady-state nearly resonant waves may exist when the nonlinearity increases, if more non-trivial resonant components are considered in the solution procedure.

Table 1.

The number of real solutions for the initial guess coefficients $A_{0,i}$ in (2.46), for angle $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/36$ and wavenumber ratio $\unicode[STIX]{x1D70F}=0.89$ .

Table 2.

The number of convergent solutions for resonant wave systems (2.37), for angle $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/36$ and wavenumber ratio $\unicode[STIX]{x1D70F}=0.89$ .

Table 2 shows the number of convergent solutions based on the initial guesses listed in table 1. When three or fewer initial non-trivial resonant components are considered, the number of convergent solutions first increases and then decreases with respect to the dimensionless frequency $\unicode[STIX]{x1D716}$ . When four initial non-trivial resonant components are considered, the number of convergent solutions increases with $\unicode[STIX]{x1D716}$ up to $1.005$ . It should be noted that at $\unicode[STIX]{x1D716}=1.005$ , no convergent solution is obtained for $l=1$ (corresponding to only one resonant component) and the number of solutions increases for $l=2,3,4$ (corresponding to two, three and four resonant ones). Steady-state waves with multiple near resonances indeed exist, and more nearly resonant components need to be considered as initial non-trivial ones in the solution procedure when the nonlinearity increases. The last column in table 2 shows that the total number of convergent solutions increases from $2$ at $\unicode[STIX]{x1D716}=1.0003$ to $17$ at $\unicode[STIX]{x1D716}=1.005$ , i.e. the number of steady-state wave groups increases with respect to the nonlinearity of wave groups when more components join the near resonance. This indicates that the probability of existence of steady-state resonant waves increases with the nonlinearity of wave groups.

Figures 3 and 4 show the amplitude spectra of steady-state waves for dimensionless frequencies of $\unicode[STIX]{x1D716}=1.003$ and $1.005$ respectively. The wave spectra are ordered based on the maximum dimensionless amplitude $|C_{i,j}^{\unicode[STIX]{x1D702}}k_{i,j}|$ , from smallest to largest. Moreover, within each spectrum, the dominant frequency $f_{i,j}$ that is surrounded by other smaller peaks is indicated ( $f_{i,j}$ denotes the frequency of the component $\cos (i\unicode[STIX]{x1D709}_{1}+j\unicode[STIX]{x1D709}_{2})$ ). The spectral shape, especially the dominant frequency $f_{i,j}$ together with the related amplitude $|C_{i,j}^{\unicode[STIX]{x1D702}}k_{i,j}|$ , changes among different groups. The evidence of multiple resonances in steady-state waves is clearly shown. Compared with the spectra at $\unicode[STIX]{x1D716}=1.003$ , two additional dominant frequencies $f_{0,1}$ and $f_{4,-3}$ appear in the spectra at $\unicode[STIX]{x1D716}=1.005$ . This suggests that more components indeed join the resonance. Besides, the maximum amplitudes $|C_{i,j}^{\unicode[STIX]{x1D702}}k_{i,j}|$ increase and five more time-independent spectra are obtained at $\unicode[STIX]{x1D716}=1.005$ . Spectral analysis confirms that both the number of components comprising the resonance and the number of steady-state waves increase with the nonlinearity of the wave groups.

Figure 3.

The amplitude spectrum $|C_{i,j}^{\unicode[STIX]{x1D702}}k_{i,j}|$ of wave groups in the case of dimensionless frequency $\unicode[STIX]{x1D716}=1.003$ , angle $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/36$ and wavenumber ratio $\unicode[STIX]{x1D70F}=0.89$ . Here, $f_{i,j}$ denotes the dominant frequency.

Figure 4.

The same as figure 3, except that $\unicode[STIX]{x1D716}=1.005$ .

3.3 Finite amplitude wave groups

The dimensionless frequency $\unicode[STIX]{x1D716}$ is further increased to search for finite amplitude steady-state nearly resonant wave groups.

When the wavevectors of the primary components are given, the eigenvalue $\unicode[STIX]{x1D706}_{m_{\ast ,\unicode[STIX]{x1D704}},n_{\ast ,\unicode[STIX]{x1D704}}}^{a}$ of every nearly resonant component is known and is determined by the resonance criterion (2.37), while the amplitude related constant $C$ is unknown and can only be determined during the calculations. It is hard to predetermine which nearly resonant component should be considered as the initial non-trivial one in the solution procedure that determines the initial guess (2.46) and the auxiliary linear operator (2.38). To guide the choice of the initial guess and the auxiliary linear operator, the magnitude of the angular frequency mismatch $\text{d}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D704}}$ is analysed. Table 3 shows the 10 nearly resonant components with the smallest angular frequency mismatch $\log _{10}(|\text{d}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D704}}|/\unicode[STIX]{x1D714}_{1})$ in the case of angle $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/36$ and wavenumber ratio $\unicode[STIX]{x1D70F}=0.89$ . Results based on combinations of the first four components in § 3.2 show that the number of initial non-trivial components increases with the dimensionless frequency when $\unicode[STIX]{x1D716}\in [1.0003,1.005]$ . As the dimensionless frequency $\unicode[STIX]{x1D716}$ further increases, more nearly resonant components are suggested to be considered as the initial non-trivial ones. The remaining six components in table 3 could be provided as the possible non-trivial ones in the initial guess (2.46).

Table 3.

The 10 near-resonant components with the smallest angular frequency mismatch $\log _{10}(|\text{d}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D704}}|/\unicode[STIX]{x1D714}_{1})$ in the case of angle $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/36$ and wavenumber ratio $\unicode[STIX]{x1D70F}=0.89$ .

Taking the first two groups of solutions (at $\unicode[STIX]{x1D716}=1.0003$ ) in table 2 as an example (groups 5 and 6), the number of resonant components $l$ considered for different dimensionless frequencies $\unicode[STIX]{x1D716}$ is shown in table 4. Within each group, the number of resonant components $l$ increases, so that three significant digits can be obtained for the unknown constants $C_{m,n}^{\unicode[STIX]{x1D702}}$ and $C_{m,n}^{\unicode[STIX]{x1D711}}$ in (2.11) and (2.12). It should be noted that in group 5, the number of resonant components $l$ increases up to 12 when $\unicode[STIX]{x1D716}\geqslant 1.008$ . For each dimensionless frequency $\unicode[STIX]{x1D716}$ , the corresponding coupled nonlinear algebraic equations for $A_{0,i}$ in the initial guess (2.46) are solved by the partial root solver described in § 3.2, with the solution of $A_{0,i}$ at a slightly smaller value of $\unicode[STIX]{x1D716}$ serving as the initial solution. Detailed components together with related coefficients in the initial potential (2.46) for groups 5 and 6 are shown in tables 5 and 6 in appendix B. The convergence of the related high-order approximations obtained by the HAM for multiple nearly resonant wave groups is shown in appendix C.

Table 4.

The number of resonant components in the initial guess (2.46) with increased dimensionless frequency $\unicode[STIX]{x1D716}$ for groups 5, 6 and 7 in the case of angle $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/36$ and wavenumber ratio $\unicode[STIX]{x1D70F}=0.89$ .

Figure 5 shows the amplitude spectra of groups 5 and 6 when the dimensionless frequency $\unicode[STIX]{x1D716}$ increases from $1.0003$ to $1.010$ . More components indeed join the near resonances and the frequency bands broaden with respect to the increased dimensionless frequency $\unicode[STIX]{x1D716}$ . The wave group energy is mainly contained by components near the first-order primary ones, with some by second-order bound ones. No spectral symmetry is found for all values of $\unicode[STIX]{x1D716}$ considered. The dimensionless amplitudes of components with $|C_{i,j}^{\unicode[STIX]{x1D702}}k_{i,j}|>0.01$ in groups 5 and 6 are shown in tables 9 and 10 in appendix D.

Figure 5.

The amplitude spectra $|C_{i,j}^{\unicode[STIX]{x1D702}}k_{i,j}|$ in the case of angle $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/36$ and wavenumber ratio $\unicode[STIX]{x1D70F}=0.89$ with increased dimensionless frequency $\unicode[STIX]{x1D716}$ .

Within each group, the amplitude of each component may either increase or decrease with the dimensionless frequency $\unicode[STIX]{x1D716}$ , whereas the amplitude of the whole wave group increases continuously. We define the wave steepness as

where $k_{d}$ is the wavenumber corresponding to the dominant frequency. It is found that, at $\unicode[STIX]{x1D716}=1.010$ , the wave steepness $H_{steepness}$ reaches $0.22$ and $0.21$ in groups 5 and 6 respectively. For wave groups with discrete spectra, the large-amplitude case with $H_{steepness}=0.21$ was investigated by Shemer et al. (Reference Shemer, Jiao, Kit and Agnon2001) for the evolution of the nonlinear wave field along a tank. Therefore, it is reasonable to conclude that finite amplitude steady-state nearly resonant wave groups are obtained in groups 5 and 6 at $\unicode[STIX]{x1D716}=1.010$ .

Figure 6.

The same as figure 5.

More calculations have been carried out for other groups in table 2. In most groups, the spectrum broadens and the asymmetry is more pronounced when the dimensionless frequency $\unicode[STIX]{x1D716}$ increases. Only in a few groups does the energy shrink to one or two components. Taking the first group in figure 3 (group 7) as an example, the amplitude spectra in figure 6 show that the number of components joining the near resonance increases considerably with $\unicode[STIX]{x1D716}$ . At $\unicode[STIX]{x1D716}=1.014$ , the wave steepness reaches $H_{steepness}=0.26$ . The wave energy is mainly contained by the components between the first-order primary components and the second-order bound ones. No spectral symmetry is found. The components together with the related coefficients in the initial potential (2.46) are shown in table 7 in appendix B, and the dimensionless amplitudes of the components with $|C_{i,j}^{\unicode[STIX]{x1D702}}k_{i,j}|>0.01$ are shown in table 11 in appendix D.

Table 5.

Detailed components together with the related coefficients in the initial guess (2.46) for different dimensionless frequencies $\unicode[STIX]{x1D716}$ in group 5, for angle $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/36$ , wavenumber ratio $\unicode[STIX]{x1D70F}=0.89$ and $k_{2}=\unicode[STIX]{x03C0}/5$ .

Table 6.

Detailed components together with the related coefficients in the initial guess (2.46) for different dimensionless frequencies $\unicode[STIX]{x1D716}$ in group 6, for angle $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/36$ , wavenumber ratio $\unicode[STIX]{x1D70F}=0.89$ and $k_{2}=\unicode[STIX]{x03C0}/5$ .

The spectra in figures 5 and 6 illustrate that the auxiliary linear operator (2.38) and the initial guess (2.46) proposed in (§ 2.3) work well for multiple near resonances. Small divisors are indeed resolved when more components join the resonance. Theoretically, no limit is set on the number of components joining the near resonance, and it is reasonable to predict the existence of larger-amplitude steady-state wave groups until the waves break.