Seismic-generated unsteady motions in shallow basins and channels. Part I: Smooth analytical solutions

In this paper time-dependent water motions generated by seismic-type horizontal excitation in shallow basins and channels are modelled by the two-dimensional depth-averaged shallow water equations in which a specific source term is added in order to include an earthquake-induced forcing effect. Sinusoidal excitation is considered as a first approximation, and the response of shallow basins and channels to this simple external forcing is characterized. The nondimensional form of the governing equations shows that the Strouhal number and a ratio representing the amplitude of the forcing acceleration are the influential dimensionless parameters. Novel exact solutions of sinusoidally-forced smooth waves in a prismatic tank, a rectangular open channel, and a parabolic basin are presented. In the first two cases, a sway motion occurs, and reflections take place at the side walls. In the last case, the water sloshes back and forth flowing up the sloping sides of the basin; the free surface remains planar and a moving circular shoreline is present. These analytical solutions provide useful standards for assessing the accuracy of the numerical models used to solve the two-dimensional shallow water equations with source terms.