## 4 Conditions for Run-Up Resonance in a Multi-Reflection System  
To investigate the influence of the boundary conditions on the resonance mechanism, we first consider the reflective boundary conditions. In Appendix A, we describe a technique to decompose a wave into left- and right-going waves in a multi-reflection system. Using that technique, we separated the recorded waves at a point \(x\) located \(2000\) m to the right of the left boundary, into left- and right-going waves with \(L_{t}=7000,8000,9000\) m. The other parameters were fixed at \(L=2000\) m, \(\tan\alpha=0.02\), \(T=268.5\) s, and \(\eta_{0}=0.1\) m (see Figure 9). The amplitudes of left- and right-going waves are constant in each reflection cycle. When \(L_{t}=8000\) m, the amplitude of right-going waves is the sum of amplitudes of the initial incident wave and of the left-going wave in the preceding reflection cycle, which are not observed in the other two cases (see Figure 9). The wave motion could develop as a standing wave pattern with the use of open boundary condition [26]. For a multi-reflection system, it can be considered as a superposition of a standing wave and traveling waves. When the phase difference between the right-going and left-going waves is \(\pi\), resonance can occur. When \(L_{t}=8000\) m, one can observe that the phase difference between the right-going and left-going waves is close to \(\pi\) (see Figure 9b,c), which leads to resonance and amplifies the free surface elevation significantly (see Figure 9a). The period \(T=268.5\) s (\(T/T_{reflect}=0.2658\)) used in Figure 9 is the resonant frequency that leads to the first peak in Figure 6a. Therefore, det(system) = 0 gives an analytical resonance condition, and Figure 9 reveals a wave motion resonance condition. We can conclude that the phase difference between the right-going and left-going waves must be \(\pi\) for the waves of the periods at the run-up peaks and det(system) = 0 in Figure 7.  
Due to the new incident component coming from the reflected component in the previous reflection cycle, the reflection coefficient \(r_{l}^{(m)}\) in the \(m\)th reflection cycle at the left boundary can be defined by  
\[r_{l}^{(m)}=\frac{A_{l}^{(1,m)}}{A_{R}^{(1,m-1)}}\,, \tag{37}\]  
where \(m>1\). The average value is considered as the reflection coefficient of the left boundary. Tables 1 and 2 list reflection coefficients corresponding to cases of resonant and non-resonant regimes for the three types of boundary conditions. The amplitudes of the harmonic components being small, tiny errors may lead to deviations, and the reflection coefficients have small discrepancies even with the use of the same boundary condition. The reflection coefficient of the reflective boundary condition is up to 95%, it can be regarded as \"full reflection\". With the use of a partial relaxation zone (\(L_{relax}=0.2\lambda_{0}\)) and the left ITBC, amplitudes of the harmonic components are equal to zero when \(m>1\), which indicates that there is no reflection at the left. The left boundary is open. Figure 8: Free-surface elevation time series of the recorded wave at \(x\) for the canonical case, when \(L=2000\) m, \(L_{t}=8000\) m, \(\tan\alpha=0.02\), \(\eta_{0}=0.1\) m, and the period \(T=268.5\) s (\(T/T_{reflect}=0.2658\)). The distance from \(x\) to the toe of the slope is 2000 m. The results with the relaxation zone are compared with those obtained with the other types of boundary conditions.