# Run-Up of Long Waves in Piecewise Sloping U-Shaped Bays

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## Abstract

We present an analytical study of the propagation and run-up of long waves in piecewise sloping, U-shaped bays using the cross-sectionally averaged shallow water equations. The nonlinear equations are transformed into a linear equation by utilizing the generalized Carrier–Greenspan transform (Rybkin et al. J Fluid Mech 748:416–432, 2014). The solution of the linear wave propagation is taken as the boundary condition at the toe of the last sloping segment, as in Synolakis (J Fluid Mech 185:523–545, 1987). We then consider a piecewise sloping bathymetry, and as in Kanoglu and Synolakis (J Fluid Mech 374:1–28, 1998), find the linear solution in the near shore region, which can be used as the boundary condition for the nonlinear problem. Our primary results are an analytical run-up law for narrow channels and breaking criteria for both monochromatic waves and solitary waves. The derived analytical solutions reduce to well-known solutions for parabolic bays and plane beaches. Our analytical predictions are verified in narrow bays via a comparison to direct numerical simulation of the 2-D shallow water equations.

## Keywords

Long-wave run-up shallow water wave equations carrier–greenspan transformation piecewise sloping bays## Notes

### Acknowledgements

D. Anderson, H. Hartle, and A. Raz were supported by the National Science Foundation Research Experience for Undergraduate program (Grant #1411560) and the Geophysical Institute, University of Alaska Fairbanks. D. Nicolsky acknowledges support from the Geophysical Institute, University of Alaska Fairbanks. E. Pelinovsky acknowledges support from the Grant of the RF President for state support of leading scientific schools (NSH-6637.2016.5) and RFBR grant (15-45-02061). A. Rybkin acknowledges support from the National Science Foundation Grant DMS-1411560. We would also like to thank Research Computing Systems at the Geophysical Institute for providing supercomputing resources required for the FUNWAVE trials.

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