Identification of Forerunners and Transmission of Energy to Tsunami Waves Generated by Instanteneous Ground Motion on a Non-Uniformly Sloping Beach

Arghya Bandyopadhyay

Department of Mathematics, Khalisani College, Chandannagar, India.

DOI: 10.4236/ijg.2013.42042                

Abstract

The problem of generation and propagation of tsunami waves is mainly focused on plane beach, there are very few analytical works where wave generation is considered on non-uniformly sloping beach and as a result those works might have failed to capture important facts which are influenced by bottom-slope of the beach. Some researchers provided solution to the forced long linear waves but on a beach with uniform slope while the importance of including variable bottom topography is mentioned by few researchers but they also stayed away from considering continuous variability of the ocean bed as they were studying runup problem. This paper analyzes tsunami waves which are generated by instantaneous bottom dislocation on a ocean floor with variable slope of the form y=-qx^r, q > 0, r > 0. Attempts are made to find analytical solution of the problem and along the way tsunami forerunners are identified while investigating the short time wave behavior, not found even with constant slope beaches. In our study a rather significant phenomenon with regard to energy transmission to the waves at steady-state are observed with some notable features.

Keywords

Tsunami Waves; Shallow Water Equations; Hankel Transform; Hankel Functions; Asymptotic Expansion

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	C. E. Synolakis and E. N. Bernard, “Tsunami Science before and beyond Boxing Day 2004,” Philosophical Transactions of the Royal Society A, Vol. 364, No. 1845, 2006, pp. 2231-2265. doi:10.1098/rsta.2006.1824
[2]	E. O. Tuck and L. S. Hwang, “Long Wave Generation on a Sloping Beach,” Journal of Fluid Mechanics, Vol. 51, No. 3, 1972, pp. 449-461. doi:10.1017/S0022112072002289
[3]	P. L.-F. Liu, P. Lynett and C. E. Synolakis, “Analytical Solution for Forced Long Waves on a Sloping Beach,” Journal of Fluid Mechanics, Vol. 478, 2003, pp. 101-109. doi:10.1017/S0022112002003385
[4]	U. Kanoglu and C. E. Synolakis, “Long Wave Runup on Piecewise Linear Topographies,” Journal of Fluid Mechanics, Vol. 374, 1998, pp. 1-28.
[5]	C. E. Synolakis, “Tsunami Runup on Steep Slopes: How Good Linear Theory Is,” Natural Hazards, Vol. 4, No. 2-3, 1991, pp. 221-234. doi:10.1007/BF00162789
[6]	J. J. Stoker, “Water Waves,” Interscience Pulishers, New York, 1957.
[7]	S. Tinti and R. Tonini, “Evolution of Tsunamis Induced by Near-Shore Earthquakes on a Constant Slope Ocean,” Journal of Fluid Mechanics, Vol. 535, 2005, pp. 33-64. doi:10.1017/S0022112005004532
[8]	E. Pelinovsky, T. Talipova, et al., “Nonlinear Mechanism of Tsunami Wave Generation by Atmospheric Disturbances,” Natural Hazards and Earth System Sciences, Vol. 1, 2001, pp. 243-250. doi:10.5194/nhess-1-243-2001
[9]	J. V. Weahausen and E. V. Laitone, “Surface Waves,” In: Handbuch der Physik, Vol. 9, Springer, Berlin, 1960, pp. 446-778.
[10]	B. Edward, “Tsunami an Underrated Hazard,” Cambridge University Press, Cambridge, 2005.
[11]	Erdélyi, et al., “Higher Transcendental Functions,” Vol. 2, McGraw-Hill, New York, 1953, p. 58.
[12]	Erdélyi, et al., “Tables of Integral Transforms,” Vol. 2, McGraw-Hill, New York, 1954, p. 29.
[13]	F. Oberhettinger, “Tables of Bessel Transforms,” Springer, Berlin, 1970.
[14]	Y. Okada, “Internal Deformation Due to Shear and Tensile Faults in a Half-Space,” Bulletin of the Seismological Society of America, Vol. 82, No. 2, 1992, pp. 1018-1040.
[15]	A. Bandyopadhyay, “Mathematical Modeling of Tsunami Waves Generated by Bottom Motion on a Non-Uniformly Sloping Beach,” Proceedings of 22nd ISOPE Conference, Rhodes, June 2012, pp. 68-71.