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By assuming the flow is uniform along the narrow long bays, the 2-D nonlinear shallowwater equations are reduced to a linear semi-axis variable-coefficient 1-D wave equation via the generalized Carrier-Greenspan transformation. The run-up of long waves in constantly sloping U-shaped and V-shaped bays is studied both analytically and numerically within the framework of the 1-D nonlinear shallow-water theory. An analytic solution, in the form of a double integral, to the resulting linear wave equation is obtained by utilizing the Hankel transform, and consequently the solution to the tsunami run-up problem is developed by applying the inverse generalized Carrier-Greenspan transform. The presented solution is a generalization of the solutions found by Carrier et al. [7] (2003) and Didenkulova and Pelinovsky [9, 10] (2011) for the case of a plane beach and a parabolic bay respectively. The shoreline dynamics in U-shaped and V-shaped bays are computed via a double integral through standard integration techniques.

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.