Run-up of nonlinear long waves in U-shaped bays of finite length: analytical theory and numerical computations

Abstract

Run-up of long waves in sloping U-shaped bays is studied analytically in the framework of the 1-D nonlinear shallow water theory. By assuming that the wave flow is uniform along the cross section, the 2-D nonlinear shallow water equations are reduced to a linear semi-axis variable-coefficient 1-D wave equation via the generalized Carrier-Greenspan transformation (Rybkin et al. in J Fluid Mech 748:416–432, 2014). A spectral solution is developed by solving the linear semi-axis variable-coefficient 1-D equation via separation of variables and then applying the inverse Carrier-Greenspan transform. To compute the run-up of a given long wave, a numerical method is developed to find the eigenfunction decomposition required for the spectral solution in the linearized system. The run-up of a long wave in a bathymetry characteristic of a narrow canyon is then examined.