An analytical and numerical study of long wave run-up in U-shaped and V-shaped bays

By assuming the flow is uniform along the narrow long bays, 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. 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. (2003) and Didenkulova and Pelinovsky (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.