Abstract

We present a theoretical solution for the water wave elevation along the shoreline and a closed formula is derived for the runup distribution in the 2+1 dimensional (2 spatial + 1 temporal) linear case. An integral transform method is used to solve the linear shallow water wave equations for a sloping beach model extended along the coastline. At large regional scale, this simple bathymetry model gives a good approximation for the seafloor geometry along the Chilean subduction zone. In the bidimensional problem, for special cases, we found an analytic expression for the runup distribution along the shoreline that includes the obliquity of the incident incoming wave approaching the coast. Our solution establishes that obliquity can reduce maximum runup in important ways. It suggests that the distribution along the shoreline is determined mainly by the maximum unidimensional runup modulated, by a function controlling the horizontal extent of the incident wave parallel to the shoreline. This function was introduced to include the physical expected boundedness of the leading wave approaching the shore. Our theoretical solution was applied to model the runup generated by the 2010, M-w 8.8 Maule mega-thrust earthquake. The model fits rather well the runup to first-order. The theoretical solution proposed in this study also provides new insights to the near-field tsunami discriminant formula proposed by Okal & Synolakis.