Summer 2016 Undergrad Research Program
This summer our team worked on the problem of tsunami run-up in piecewise sloping U-shaped bays and channels. This is a generalization of the work done by Synolakis for piecewise sloping plane beaches and was accomplished by using a combination of linear and nonlinear theory. The bays we analyzed are described in the figure below:
Our general solution strategy is as follows. (1) Linearize the nonlinear SWE’s, which effectively reduces the problem to one dimension. (2) Solve the linear problem for an arbitrary incoming wave in all regions by making use of spectral methods to match local solutions at the boundaries. (3) Use the generalized Carrier-Greenspan transform to find a general solution to the nonlinear problem in the sloped region containing the shoreline. (4) Use boundary conditions defined by the linear problem to get unique solution for an arbitrary incoming wave. Note the figure shows only 2 Regions for simplicity, but this method was derived for n slopes.
We were able to analytically relate our model to two well known solutions for simpler cases. We also saw excellent agreement when we compared our model to a numerical solution of the full nonlinear SWE. Evaluation of our model takes approximately one minute, while the solving the SWE’s takes up to 12 hours to compute. As a result, we have developed a superior method that is both accurate and fast in predicting the run-up of long waves in piecewise sloping U-shaped bays and channels.