Tsunami focusing

Tsunamis are long waves generated by impulsive disturbances of the seafloor or coastal topography caused by earthquakes, submarine/subaerial mass failures. They evolve substantially through three dimensional - 2 spatial+1 temporal - spreading as the initial surface deformation propagates. This is referred to as its directivity and focusing. A directivity function was first defined by Ben-Menahem (1961, Bull. Seismol. Soc. Am. 51, 401-435) using the source length and the rupture velocity. Okal (2003, Pure Appl. Geophys. 160, 2189-2221) discussed the details of the analysis of Ben-Menahem (1961) and demonstrated the distinct difference between the directivity patterns of landslide and earthquake generated tsunamis. Marchuk and Titov (1989, Proc. IUGG/IOC International Tsunami Symposium, July 31 - August 3, 1989, Novosibirsk, USSR. p.11-17) described the process of tsunami focusing for a rectangular initial deformation combining positive and negative surface displacements. They showed the existence of a focusing point where abnormal tsunami wave height can be registered. Here, first, we describe and quantify numerically tsunami focusing processes for a combined positive and negative - N-wave type - strip source representing the 17 July 1998 Papua New Guinea and 17 July 2006 Java events. Specifically, considering field observations and tsunami focusing, we propose a source mechanism for the 17 July 2006 Java event. Then, we introduce a new analytical solution for a strip source propagating over a flat bottom using the linear shallow-water wave equation. The analytical solution of Carrier and Yeh (2005, Computer Modeling In Engineering & Sciences, 10(2), 113-121) appears to have two drawbacks. One, the solution involves singular complete elliptic integral of the first kind which results in a self-similar approximate solution for the far-field at large times. Two, only the propagation of Gaussian shaped finite-crest wave profiles can be modeled. Our solution is not only exact but also more general and allows the use of realistic initial waveform such as N-waves defined by Tadepalli and Synolakis (1994, Proc. R. Soc. Lond. A 445, 99-112). We explore parametric dependence of the focusing height and distance on the initial wave parameters through the analytical solution.