An asymptotically numerical description of tsunami waves propagation in a basin with non-uniform depth in a neighborhood of wavefronts that can have caustics is proposed. The piston model and the long wave approximation are used. It is assumed that the size of the area of the initial disturbance is small in comparison both with the characteristic length of interval of the varying of the bottom depth and the distance from the observation point. The description is based on a generalization of asymptotic approach known as the Maslov canonical operator. We find formulas that are relatively simple and can be transformed in a computer program for fast calculating wave profiles. Some features of the tsunami waves propagation in basins of non-uniform depth are illustrated by graphics.
tsunami, asymptotics, dispersion-less piston model, wavefront, wave-profile.
1. Arnold, Mathemetics and Its Applications, vol. 62, 2001.
2. Babich, Short-Wavelength Diffraction Theory, 1991.
3. Berry, New Journal of Physics, v. 7, no. 129, 2005.
4. Borovikov, Fluid Dynamics, v. 31, no. 4, 1996.
5. Brekhovskikh, Acoustics of layered media II, Point source and bounded beams, 2006.
6. Dobrokhotov, Russ. J. Math. Phys., v. 10, 2003.
7. Dobrokhotov, Math. Notes, v. 53, no. 6, 1993., doi:https://doi.org/10.1007/BF01212605
8. Dobrokhotov, Math. Notes, v. 49, no. 4, 1991.
9. Dobrokhotov, Dokl. Mathematics, v. 74, no. 1, 2006., doi:https://doi.org/10.1134/S1064562406040326
10. Dobrokhotov, Publications of the Research Institute for Mathematical Science, vol. 4, 2006.
11. Kiselev, 11, Zapiski nauchnyh seminarov LOMI, v. 104, 1980.
12. Kowalik, Numerical modeling of ocean dynamics, 1993.
13. Kravtsov, Geometrical Optics of Inhomogeneous Media, 1990.
14. Lewis, The Tsunami Society Monograph Series, v. 1, 1983.
15. Maslov, Perturbation Theory and Asymptotic Methods, 1965.
16. Maslov, Operational Methods, 1973.
17. Maslov, Mathematical Physics and Applied Mathematics 7, Contemporary Mathematics 5, 1981.
18. Maslov, Math. Notes, v. 45, no. 5, 1989.
19. Pelinovski, Hydrodynamics of Tsunami Waves, 1996.
20. Shokin, Numerical Experiment in Tsunami Problem, 1989.
21. Tinti, Tsunami in the world, 1993.
22. Vishik, Uspekhi Mat. Nauk, Engl. translation: American Math. Society Transl., vol. 20, (2), 1962, pp. 239--364, v. 12, no. 5, 1957.
23. Whitmore, Science of Tsunami Hazards, v. 4, no. 3, 1996.
