# The Behavior of Nonlinear Tsunami Waves Running on the Shelf
## 4 Solution of the Nonlinear Problem in the Case of a Homogeneous Wave  
Equation (8) in the case of small friction on the bottom compared to the horizontal turbulent friction \(R_{B}<<R_{L}\) has the following form:  
\[\frac{\partial Z}{\partial\xi}\ +\ \frac{1}{2}\bigg{(}\frac{\Phi r}{Sh} \bigg{)}\frac{\partial Z^{2}}{\partial\tau}\ =\ R_{L}\frac{\partial^{2}Z}{ \partial\tau^{2}}. \tag{13}\]  
In the case of horizontal quasi-homogeneity, changes in the horizontal coordinate can be neglected, assuming \(\partial/\partial\xi\approx 0\). Then,  
\[\frac{d}{d\tau}\ \bigg{[}\frac{1}{2}\bigg{(}\frac{\Phi r}{Sh}\bigg{)}Z^{2}-R_{L} \frac{dZ}{d\tau}\bigg{]}=0. \tag{14}\]  
The first integral of this equation is  
\[\frac{1}{2}\bigg{(}\frac{\Phi r}{Sh}\bigg{)}\bigg{(}Z^{2}-U^{2}\bigg{)}-R_{L} \frac{dZ}{d\tau}=0. \tag{15}\]  
Here, \(U\) is the extreme value of \(Z\). In Equation (15), the variables are separated:  
\[\frac{dZ}{(U^{2}-Z^{2})}=(-1)\bigg{(}\frac{\Phi r}{2 ShR_{L}}\bigg{)}d\tau \tag{16}\]  
and it integrates:  
\[\frac{1}{2U}ln\bigg{|}\frac{U+Z}{U-Z}\bigg{|}=(-1)\bigg{(}\frac{\Phi r}{2 Sh \ R_{L}}\bigg{)}\tau. \tag{17}\]If \(Z^{2}<U^{2}\), then the solution of Equation (17) can be written as [34]  
\[\frac{1}{U}\mathrm{artanh}\bigg{(}\frac{Z}{U}\bigg{)}=(-1)\bigg{(}\frac{\Phi r}{2 Sh\ R_{L}}\bigg{)}\tau. \tag{18}\]  
Here, \(\mathrm{artanh}\) is the inverse hyperbolic tangent. From Equation (18), it follows that  
\[Z=-Utanh\bigg{(}\frac{\Phi rU}{2\ ShR_{L}}\tau\bigg{)}. \tag{19}\]  
Returning to dimensional variables, we write the solution to the problem in the following form:  
\[\xi\ =\ \ \ \xi_{\mathrm{max}}\ \tanh\bigg{(}\frac{\xi_{\mathrm{max}}\ \xi_{0}}{(H\ -\ r)\ A_{L}}\ (x\ -\ c_{0}\ t)\bigg{)}. \tag{20}\]  
The level disturbance \(\zeta\), is counted down from the calm water level \(z=0\) (Figure 2). It is convenient to introduce the level \(D=-\zeta\), which is counted from \(z=0\). At the initial moment, a tsunami wave from the open ocean arrives at the edge of the shelf \(x=0\) with amplitude \(D_{\mathrm{max}}=-\xi_{\mathrm{max}}=5\) m. We also set the shelf depth \(H=101\) m and the height of the roughness projections on the bottom \(r=1\) m. Then, \((H-r)=100\) m and \(c_{0}=[g\ (H-r)]^{1/2}=31.3\) m/s. Suppose that we are at a distance \(x=100\) m from the shelf edge and observe the arrival of a tsunami wave in time. Figure 3 shows the arriving waveforms calculated by the formula (20) for different values of the turbulent viscosity coefficient \(A_{L}\): blue curve at \(A_{L}=10\) m\({}^{2}\)/s, red curve at \(A_{L}=100\) m\({}^{2}\)/s, and green curve at \(A_{L}=1000\) m\({}^{2}\)/s. A wave of this shape is sometimes called a kink. As seen, the tsunami wave arrives in the phase of lowering the level, and then the wave height rises to the maximum value. At low values of turbulent viscosity \(A_{L}\), the wave height \(D\) overgrows, and the wavefront turns out to be steep. As the viscosity A\({}_{\mathrm{L}}\) increases, the wavefront stretches in time, and the shelf flooding occurs more slowly. Figure 3: Calculations of a switching wave (kink) arriving at a given shelf point \(x=100\) m. The curves correspond to different values of the turbulent viscosity coefficient \(A_{L}\). The blue curve is calculated at \(A_{L}=10\) m\({}^{2}\)/s, the red curve is calculated at \(A_{L}=100\) m\({}^{2}\)/s, and the green curve is calculated at \(A_{L}=1000\) m\({}^{2}\)/s. Solutions (17), (19), or (20) also describe the shock wave structure in gas dynamics [35; 36]. It depends on the value of the coefficient of horizontal turbulent viscosity \(A_{L}\). At high values of viscosity \(A_{L}\), the shock wave is smoothed out and stretched horizontally. On the contrary, as the viscosity decreases, \(A_{L}\to 0\) and the shock wave twists, turning into a step (level difference). Sometimes, at the difference in the level of the water surface, additional level fluctuations occur, known as undulations.