## 3 Results for Waves Incident on a Linearly Sloping Beach
### Results with the Reflective Boundary Conditions  
We construct a multi-reflection system for the canonical case of Figure 1, where the reflective boundary conditions are used. Waves generated at the left seaward boundary propagate forward and are reflected at the right boundary. The reflected waves propagate back to the left boundary and are re-reflected, then the re-reflected waves propagate forward again and the process is repeated. A reflection cycle is defined as a cycle during which the wave travels the total domain back and forth. The time it takes is called the reflection period \(T_{reflect}\). For the canonical case of Figure 1, it can be expressed by  
\[T_{reflect}=2\left(\frac{L_{t}-L}{\sqrt{gh_{0}}}+\frac{2L}{\sqrt{gh_{0}}}\right)\;. \tag{35}\]  
In the simulations below, the total simulation time is always four reflection cycles, and the incident periodic wave at \(x=-L_{t}\) is expressed by  
\[\eta_{i}(t)=\eta_{0}\sin\left(\frac{2\pi t}{T}\right)\;, \tag{36}\]  
Figure 5: Maximum nondimensional run-up ratio as a function of nondimensional wavelength on a sloping beach. Line with stars: numerical solution of the nonlinear shallow water equations with the use of the left ITBC; solid line: analytical linear solution; dashed line: wave breaking criterion. The incident wave is a periodic monochromatic wave.  
where \(T\) is the wave period and \(\eta_{0}=0.1\) m.  
First, we considered three combinations of \(\alpha\), \(L\), and \(L_{t}\), thus fixing \(T_{reflect}\) for each combination. We tested the maximum run-up in a multi-reflection system for three cases by varying the nondimensional wave period within the range \(0.25<T/T_{reflect}<0.65\) (see Figure 6). Resonance effects are clearly visible. The maximum nondimensional run-ups are sharply amplified for certain wave periods, which lead to three peaks in Figure 6a, i.e., the maximum nondimensional run-up is up to 27 when \(\tan\alpha=0.02\), \(L_{t}=8000\) m, \(L=5000\) m, and \(T=220.8\) s \((T/T_{reflect}=0.2658)\). Second, we fixed \(\alpha\), \(L\), and \(T\) instead of \(L_{t}\), the horizontal distance from the left boundary to the undisturbed shoreline. This time, we tested the maximum run-up as a function of \(L_{t}\), which in turn made \(T_{reflect}\) and \(T/T_{reflect}\) vary through Equation (35). The results are shown in Figure 7. Waves with the same period propagating on the same bathymetry (\(\tan\alpha=0.02\), \(L=2000\) m) lead to different maximum run-ups with respect to various flat bottom lengths, which indicates that the wave behavior is affected by the location of the left boundary. This is different from the open boundary. However, resonance occurs when \(T/T_{reflect}=0.2658,0.3595\) in the range of \(0.25<T/T_{reflect}<0.4\). From these two figures, monochromatic forcing is found to lead to resonant run-up by non-leading waves when the nondimensional wave periods \(T/T_{reflect}\) are the roots of the determinant of system (17), which coincides with the resonance condition we gave above.  
Figure 6: (**a**) Maximum nondimensional run-up of monochromatic waves as a function of nondimensional wave periods for the canonical case with the use of reflective boundary conditions. (**b**) Plot of the determinant of the system (17). The circles show nondimensional resonant periods, which correspond to the zeros of the determinant.