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IUGG XXIV General Assembly July 2-13, 2007 Perugia, Italy
(S) - IASPEI - International Association of Seismology and Physics of the Earth's
Interior
JSS002 Poster presentation 1832
Analytical solution for long wave directivity
Prof. Utku Kanoglu
Department of Engineering Sciences Middle East Technical University IASPEI
Vasily Titov, Baran Ayd N, Costas Synolakis
The spatial and temporal evolution of a finger-like pattern from a finite-dimension source is referred to
as its directivity. Ben-Menahem (1961, Bull. Seismol. Soc. Am. 51, 401-435) recognized it and defined a
directivity function generated by the source length and the rupture velocity. Later, Ben-Menahem and
Rosenman (1972, J. Geophys. Res. 77, 3097-3128) used linear theory to calculate the two-directional
radiation pattern from a moving source. Ben-Menahem and Rosenman (1972) showed that tsunami
energy radiates primarily at a right angle to a rapturing fault and showed the dependence of directivity
on the seafloor rupture and wave celerity, using the path of 1964 Great Chilean tsunami as an example.
One example of their work is presented in Synolakis and Bernard (2006, Phil. Trans. R. Soc. A 364,
2231-2265) for the 2004 Boxing Day Tsunami. Okal (2005, Pers. Comm.) quickly eliminated the shorter
source proposed in the immediate aftermath of this tsunami, in favour of the larger source, based on
qualitative assessment of the radiation field on the basis of the directivity of the source. Okal (2003,
Pure Appl. Geophys. 160, 2189-2221) discussed the details of the analysis of Ben-Menahem (1961) and
the distinct difference between the directivity patterns of landslide and dislocation generated tsunamis.
Okal (2003) considered directivity pattern differences with the field observation of 1964 Alaska event
concluding that a large slow earthquake and landslide must occur concurrently. Carrier and Yeh (2005,
CMES-Computer Modeling in Engineering and Science, 10(2), 113-121) presented an analytical study for
the tsunami propagation with finite crest length over a flat bathymetry and discussed directivity.
However, the solution of Carrier and Yeh (2005) appears to have two drawbacks. One, they were not
able to compute the integrals numerically for the farfield. Therefore, Carrier and Yeh (2005) proposed a
solution using complete elliptic integral of the first kind, with a singularity and, in continuation of
conservation of difficulty, they proposed a self similar approximate solution for large times. Two, Carrier
and Yeh (2005) analytically modelled only the propagation of Gaussian shaped finite-crest wave profiles,
involving approximations in their solution. However, most realistic initial waveform are N-wave like,
Tadepalli and Synolakis (1994, Proc. R. Soc. Lond. A 445, 99112). We introduce a new exact analytical
solution for the linear shallow water-wave equation over a flat bottom for a finite-crest length source.
Our solution can be applied any given initial wave profile with finite-crest length. We discuss several
features of directivity along with field observations of the December 26, 2004 Boxing Day Tsunami.
Keywords: tsunami directivity, finite tsunami source