IG wave generation with 2nd-order correction

The wavemaker can take into account bound IG waves based on the weakly-nonlinear second-order finite-depth wave theory of Hasselmann (1962). A time series of wave surface elevation (\(\eta\)) can be described by the linear superposition of free wave components, as in

(137)\[\eta = \sum_{i=1}^N a_i \cos(2 \pi f_i t + \phi_i),\]

where \(f_i\) is the frequency and \(\phi_i\) is the phase of \(i^{th}\) wave component. \(N\) is the total number of free wave components. The sub-harmonic component (\(\eta_{ij}\)) generated by two free waves \(i\) and \(j\), is given by

(138)\[\eta_{ij} = D_{ij} a_i a_j \cos(f_{ij} t + \phi_{ij}),\]

where \(f_{ij} = f_j - f_i\) and \(\phi_{ij}= \phi_j - \phi_i\) (note \(f_j > f_i\)), which represent the frequency and the phase of the sub-harmonic component, respectively. \(D_{ij}\) is the interaction coefficient given by

(139)\[D_{ij} = -\frac{gk_i k_j}{2 \omega_i \omega_j} + \frac{\omega^2_i - \omega_i \omega_j + \omega^2_j}{2g} - C \frac{g(\omega_i - \omega_j)}{\omega_i \omega_j [gk_{ij}\tanh(k_{ij}h) - (\omega_i - \omega_j)^2]},\]

where \(\omega = 2 \pi f\) is the radial frequency, and \(k_{ij} = k_i - k_j\) is the wave number of the sub-harmonic component. \(C\) is a coefficient given by

(140)\[C = (\omega_i - \omega_j) \left( \frac{\omega_i^2 \omega_j^2}{g^2} + k_i k_j \right) - \frac{1}{2} \left (\frac{\omega_i k_j^2}{\cosh^2(k_j h)} - \frac{\omega_j k_i^2}{\cosh^2(k_i h)} \right).\]

The total surface elevation including the sub-harmonics can be described by

(141)\[\eta = \sum_{i=1}^n a_i \cos(2 \pi f_i t + \phi_i) + \sum_{i=n_1}^{n_2-2} \sum_{i+1}^{n_2} D_{ij} a_i a_j \cos(2 \pi f_{ij} + \phi_{ij}),\]

where \((n_1, n_2)\) represent the range of free wave components for the sub-harmonic generation. In Li et al. (2020), for example, the range was selected from \(f(n_1)=0.5 f_p\) to \(f(n_2)=1.5 f_p\), where \(f_p\) was the spectral peak frequency. The range selected in Malej (2021) is from \(f(n_1)=0.7 f_p\) to \(f(n_2)=1.3 f_p\), smaller than Li et al. (2020).