Discipline: Mathematics and Statistics
William Noland - North Central College
Co-Author(s): Dylan Smith, University of Connecticut, Storrs, CT Seth Selken, Iowa State University, Ames, IA Marcus Battraw and Sergei Fomin, Chico State University, Chico, CA Monica Swartz, Smith College, Northampton, MA Peter Gerodette, North Valley High School, Anderson, CA
A computational model of tsunami behavior on the beach is essential in effectively protecting coastlines from their devastating effects. Previous literature has investigated the run-up and draw-down of tsunami waves on a one-dimensional, constant-sloped beach, but the existing solutions are highly complex and computationally unwieldy. Our research aims to establish a simpler and more usable model while still obtaining accurate results. We do so using a quasi-linear theory which we derive from the nonlinear shallow-water wave equations. These equations are considered over a constant-sloped linear beach, with properly imposed initial and boundary conditions as well as physically meaningful relationships between variables. The main difficulty in solving this problem is the moving boundary of the water associated with the shoreline motion resulting from wave run-up. To eliminate this difficulty, we apply an appropriate algebraic substitution to the spatial variable, and thus replace the moving boundary of the computational domain with a stationary boundary, effectively transferring the movement of the shoreline into the equations themselves. Real-world tsunamis are difficult to detect due to their extremely small amplitude (often a mere 30 centimeters), in contrast to wavelength significantly greater than the depth of the ocean. As a result, a key feature of any tsunami problem is the presence of a small parameter (epsilon) representing the characteristic amplitude of the wave divided by the characteristic depth of the ocean. Due to the presence of this small parameter, the problem can be essentially linearized using the method of perturbations, expanding each variable as a power series in terms of epsilon and eliminating terms with high powers of epsilon. The resulting system of equations is transferred using an integral transformation into a Sturm-Liouville problem, leading to a solution in the form of a series involving cosine and Bessel functions. Using Mathematica, we were able to determine that the behavior of our series can be approximated with great accuracy using only 100 terms (the difference between 100 and 2000, for example, is roughly 0.0001). This truncated solution has the advantage of explicit formulation and can be plotted quickly and relatively easily. To test the accuracy of our results, we obtained and plotted a numerical solution to our equations using standard Mathematica packages. This numerical solution was found to match our truncated solution very well, with the only deviation occurring during the wave draw-down and being extremely minor. That deviation is shown to be easily explained as a natural consequence of the known tendencies of numerical methods to smoothen extreme behaviors. In the future we hope to extend our model to a modified beach configuration that more accurately reflects real-world shoreline topography.
Funder Acknowledgement(s): This work was performed under the direction of Sergei Fomin at the 2016 NSF REU on Statistics and Mathematical Modelling.
Faculty Advisor: Sergei Fomin, firstname.lastname@example.org
Role: I was directly involved in all parts of the project, but my personal focus was on obtaining and solving the theoretical problem described in the middle portion of the abstract. I was also heavily involved in the approximation and numerical solution described in the final paragraph.