Maximum run-up produced by tsunami wave trains entering bays of variable cross section

Investigation of the behavior of various types of tsunami wave trains entering bays is of practical importance for coastal hazard assessments. Furthermore, a mathematical algorithm for quick analysis of the run-up in bays for a large number of incident wave scenarios is also a practical need for tsunami hazard assessments. The linear shallow water equations admit two types of solutions inside an inclined bay with parabolic cross sections: Energy-transmitting modes and modes with spatial decay towards the inland tip of the bay. In the low-frequency limit there is only one mode susceptible of transmitting energy to the inland tip. A full solution for the run-up requires taking into account these two types of modes and the scattered field outside, leading to mathematical complications. However, in the long wave limit, this complication can be avoided if one imposes the free surface at the bay mouth being equal to twice the disturbance associated with the incident wave in the open sea. The run-up produced by the solution obtained from this Dirichlet boundary condition can be easily calculated using a series of images. In this model no energy is allowed to escape from the bay; therefore the error arising from the simplification of the boundary condition at the bay mouth grows with time. Nevertheless the maximum run-up occurs before this error becomes significant. If the standard deviation of a Gaussian-shaped incident wave is 8 times the square root of the width of the bay, then this simple solution overestimates the first maximum of the run-up only by 12% compared to the exact solution calculated by means of an integral equation. This overestimation is partly due to the fact that the Dirichlet boundary conditions violate the continuity of the fluxes at the bay mouth. The solution associated with the Dirichlet boundary condition is perturbed in order to match fluxes inside and outside of the bay. The height of the first maximum of the run-up coming from the perturbation theory is in excellent agreement with the "exact"solution. This perturbation theory can also be applied to narrow bays with an arbitrary cross section as long as their depth does not change significantly in the longitudinal direction. The method developed here can also be used to calculate maximum run-up in noninclined bays of arbitrary cross section.