Exact solutions of the nonlinear shallow water wave equations for forced flow involving linear bottom friction in a region with quadratic bathymetry have been found. These solutions also involve moving shorelines. The motion decays over time. In the solution of the three simultaneous nonlinear partial differential shallow water wave equations it is assumed that the velocity is a function of time only and along one axis. This assumption reduces the three simultaneous nonlinear partial differential equations to two simultaneous linear ordinary differential equations. The analytical model has been tested against a numerical solution with good agreement between the numerical and analytical solutions. The analytical model is useful for testing the accuracy of a moving boundary shallow water numerical model. References Balzano, A., Evaluation of methods for numerical simulation of wetting and drying in shallow water flow models, Coastal Engineering, 34, 1998, 83--107. Carrier, G. F. and Greenspan, H. P., Water waves of finite amplitudes on a sloping beach, Journal of Fluid Mechanics, 4, 1958, 97--109. Holdahl, R., Holden, H., and Lie,K-A., Unconditionally Stable Splitting Methods For the Shallow Water Equations, BIT, 39, 1998, 451--472. Johns, B., Numerical integration of the shallow water equations over a sloping shelf, International Journal for Numerical Methods in Fluids, 2, 1982, 253--261. Kawahara, M., Hirano, H., and Tsubota, K., Selective lumping finite element method for shallow water flow, {International Journal for Numerical Methods in Fluids}, 2, 1982, 89--112. Lewis, C. H. III and Adams, W. M., Development of a tsunami-flooding model having versatile formation of moving boundary conditions, The Tsunami Society Monograph Series, 1983, No. 1, 128 pp. Parker, B. B., Frictional Effects on the Tidal Dynamics of a Shallow Estuary, PhD thesis, The John Hopkins University, 1984. Peterson P., Hauser J., Thacker W. C., Eppel D., An Error-Minimizing Algorithm for the Non-Linear Shallow-Water Wave Equations with Moving Boundaries. In Numerical Methods for Non-Linear Problems, editors C. Taylor, E. Hinton, D. R. J. Owen and E. Onate, 2, Pineridge Press, 1984, 826--836, http://www.cle.de/hpcc/publications/ Sachdev, P. L., Paliannapan, D. and Sarathy, R., Regular and chaotic flows in paraboloidal basins and eddies, Chaos, Solitons and Fractals, 7, 1996, 383--408 . Sampson, J., Easton, A., and Singh, M., Moving Boundary Shallow Water Flow in Circular Paraboloidal Basins. Proceedings of the Sixth Engineering Mathematics and Applications Conference, 5th International Congress on Industrial and Applied Mathematics, at the University of Technology, Sydney, Australia, editors R. L. May and W. F. Blyth, 2003, 223--227. Sampson, J., Easton, A., and Singh, M., Moving boundary shallow water flow in parabolic bottom topography, Australian and New Zealand Industrial and Applied Mathematics Journal, 47 (EMAC2005), C373--C387, 2006, http://anziamj.austms.org.au/V47EMAC2005/Sampson Sampson, Joe, Easton, Alan and Singh, Manmohan, A New Moving Boundary Shallow Water Wave Numerical Model, Australian and New Zealand Industrial and Applied Mathematics Journal, 48 (CTAC2006), C605--C617, 2007, http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/78 Thacker, W. C., Some exact solutions to the nonlinear shallow-water wave equations, J. Fluid. Mech., 107, 1981, 499--508. Vreugdenhil, C. B., Numerical Methods for Shallow-Water Flow, Kluwer Academic Publishers, 1998. Yoon S. B., and Cho J. H., Numerical simulation of Coastal Inundation over Discontinuous Topography, Water Engineering Research, 2(2), 2001, 75--87
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