Simple estimation of linear 1+1 D long wave run-up

Abstract

An analytical solution is derived concerning the linear run-up for any given initial wave generated over a sloping bathymetry. Due to the simplicity of the linear formulation, complex transformations are unnecessary, hence the shoreline motion is directly obtained in terms of the initial wave. This result supports not only maximum run-up invariance between linear and nonlinear theories but also the time evolution of shoreline motion and velocity, exhibiting good agreement with the nonlinear theory. The present formulation also allows quantifying the shoreline motion numerically from a customized initial waveform, including non-smooth functions. This is useful for numerical tests, laboratory experiments or realistic cases in which the initial disturbance might be retrieved from seismic data rather than using a theoretical model. It is also shown that the run-up calculation for the real case studied is consistent with the field observations.