## 2 Materials and Methods
### Theoretical Model for Run-Up Resonance Based on the Linear Shallow Water Equations  
Resonance phenomena play an important role in run-up amplification. They can explain extreme run-up heights and later arrival of waves with larger amplitude. For the geometry of Figure 1, [PERSON] and [PERSON] [25] used linear shallow water theory to show that a resonance occurs when \(2\omega\sqrt{L/(g\tan\alpha)}\) is a zero of the Bessel function \(J_{0}\). This resonance can also be found both in the numerical results of [PERSON] et al. [13] and in the experimental results of [PERSON] et al. [22], because of the presence of a reflective boundary condition seaward. Even though the numerical simulations shown in the next section are based on the 1D NSWE (1) and (2), we use the 1D linear shallow water equations (LSWE) to explain the resonance phenomenon analytically. The 1D LSWE read  
\[\frac{\partial\eta}{\partial t}+\frac{\partial}{\partial x}(h_{0} u) =0\,, \tag{3}\] \[\frac{\partial u}{\partial t}+g\frac{\partial\eta}{\partial x} =0\,. \tag{4}\]  
The phase velocity of the wave is denoted by \(c(x)=\omega/k(x)=\sqrt{gh_{0}(x)}\). Equations (3) and (4) can be combined into a single equation:  
\[\frac{\partial^{2}\eta}{\partial t^{2}}-g\frac{\partial}{\partial x}\left(h_ {0}\frac{\partial\eta}{\partial x}\right)=0\,. \tag{5}\]  
For a beach with a constant slope, we look for a harmonic solution of the form [26]  
\[\eta(x,t)=\mathcal{A}(x)e^{-i\omega t}\,. \tag{6}\]  
Inserting Equation (6) into Equation (5) leads to  
\[\frac{d^{2}\mathcal{A}}{d\sigma^{2}}+\frac{1}{\sigma}\frac{d\mathcal{A}}{d \sigma}+\mathcal{A}=0\,, \tag{7}\]  
where \(\sigma=\int_{0}^{x}k(x)\,dx\) with \(k(x)\) the wave number. Equation (7) is the Bessel equation of the first kind, whose solution can be represented by  
\[\mathcal{A}=A_{1}J_{0}(\sigma)+A_{2}Y_{0}(\sigma)\,, \tag{8}\]  
Figure 1: The geometry of a sloping beach connected to a region of constant depth. The still water depth is given by \(h_{0}(x)\), with \(h_{0}(x)=h_{0}=L\tan\alpha\) for \(-L_{t}\leq x\leq-L\) and \(h_{0}(x)=-x\tan\alpha\) for \(-L\leq x\leq 0\). where \(A_{1}\) and \(A_{2}\) are linear coefficients, and \(J_{0}\), resp. \(Y_{0}\), are the Bessel functions of the first and second kind. As \(Y_{0}\) tends to infinity at the shore, the coefficient of \(Y_{0}\) must be zero in the nearshore region if we assume that the amplitude of the free-surface elevation is finite. Therefore, \(\mathcal{A}\) is expressed only in terms of the Bessel function of the first kind:  
\[\mathcal{A}=R_{max}J_{0}(\sigma)\,, \tag{9}\]  
where \(R_{max}\) denotes the run-up. [PERSON] and [PERSON] [25] found the resonance mentioned above by imposing a fixed amplitude at \(x=L\). Indeed, \(\sigma(x=L)=2\omega\sqrt{L/(g\tan\alpha)}\). For a constant-depth region, the wave field can be described as a sum of incident and reflected waves:  
\[\eta(x,t)=(A_{i}e^{-ikx}+A_{r}e^{ikx})e^{-i\omega t}\,, \tag{10}\]  
where \(A_{i}\) and \(A_{r}\) are the amplitudes of the incident and reflected waves, respectively. Next, we discuss run-up due to the more realistic bathymry profile that consists of a constant depth region connected to a sloping beach. Hereafter, it will be referred to as the canonical case (see Figure 1). Following [PERSON] and [PERSON] [27] and [PERSON] et al. [13], we use the continuity of the free-surface elevation and of the horizontal fluxes to match surface elevations (Equation (11)) and particle velocities (according to Equation (4), it is equivalent to Equation (12)) at the toe of the sloping beach from two adjacent segments, and match the surface elevations at the left boundary Equation (13), respectively.