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A note on tsunamis: their generation and propagation in an ocean of uniform depth

  • Joseph L. Hammack (a1)

The waves generated in a two-dimensional fluid domain of infinite lateral extent and uniform depth by a deformation of the bounding solid boundary are investigated both theoretically and experimentally. An integral solution is developed for an arbitrary bed displacement (in space and time) on the basis of a linear approximation of the complete (nonlinear) description of wave motion. Experimental and theoretical results are presented for two specific deformations of the bed; the spatial variation of each bed displacement consists of a block section of the bed moving vertically either up or down while the time-displacement history of the block section is varied. The presentation of results is divided into two sections based on two regions of the fluid domain: a generation region in which the bed deformation occurs and a downstream region where the bed position remains stationary for all time. The applicability of the linear approximation in the generation region is investigated both theoretically and experimentally; results are presented which enable certain gross features of the primary wave leaving this region to be determined when the magnitudes of parameters which characterize the bed displacement are known. The results indicate that the primary restriction on the applicability of the linear theory during the bed deformation is that the total amplitude of the bed displacement must remain small compared with the uniform water depth; even this restriction can be relaxed for one type of bed motion.

Wave behaviour in the downstream region of the fluid domain is discussed with emphasis on the gradual growth of nonlinear effects relative to frequency dispersion duringpropagationand the subsequent breakdown of the linear theory. A method is presented for finding the wave behaviour in the far field of the downstream region, where the effects of nonlinearities and frequency dispersion have become about equal. This method is based on the use of a model equation in the far field (which includes both linear and nonlinear effects in an approximate manner) first used by Peregrine (1966) and morerecently advocated by Ben jamin, Bona & Mahony (1972) as a preferable model to the more commonly used equation of Korteweg & de Vries (1895). An input-output approach is illustrated for the numerical solution of this equation where the input is computed from the linear theory in its region of applicability. Computations are presented and compared with experiment for the case of a positive bed displacement where the net volume of the generated wave is finite and positive; the results demonstrate the evolution of a train of solitary waves (solitons) ordered by amplitude followed by a dispersive train of oscillatory waves. The case of a negative bed displacement in which the net wave volume is finite and negative (and the initial wave is negative almost everywhere) is also investigated; the results suggest that only a dispersive train of waves evolves (no solitons) for this case.

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Benjamin, T. B., Bona, J. L. & Mahony, J. J. 1972 Model equations for long waves in nonlinear dispersive systems Phil. Trans. Roy. Soc. 272, 4778.
French, J. A. 1969 Wave uplift pressures on horizontal platforms. W. M. Keck Lab. Hydraul. & Water Res. Calif. Inst. Tech. Rep. KH-R-19.
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1967 Method for solving the Korteweg–de Vries equation Phys. Rev. Lett. 19, 10951097.
Hammack, J. L. 1972 Tsunamis: a model of their generation and propagation. W. M. Keck Lab. Hydraul. & Water Res., Calif. Inst. Tech. Rep. KH-R-28.
Honda, H. & Nakamura, K. 1951 The waves caused by one-dimensional deformation of the bottom of shallow sea of uniform depth Sci. Rep. Tohoku University, Sendai, Japan, 3, 133137.
Hwang, L. S. & Divoky, D. 1970 Tsunami generation J. Geophys. Res. 75, 68026817.
Ichiye, T. 1950 On the theory of tsunami Oceanograph. Mag. 2, 83100.
Ichiye, T. 1958 A theory of the generation of tsunami by an impulse at the sea bottom J. Oceanograph. Soc. Japan, 14, 4144.
Jeffreys, H. & Jeffreys, B. S. 1946 Methods of Mathematical Physics, 1st edn. Cambridge University Press.
Kajiura, K. 1963 The leading wave of a tsunami Bull. Earthquake Res. Inst. Tokyo University, 41, 535571.
Keller, J. B. 1963 Tsunamis: water waves produced by earthquakes. Int. Un. Geodesy & Geophys. Monograph, no. 24, pp. 154166.
Keulegan, G. H. 1948 Gradual damping of solitary waves J. Res. Nat. Bur. Stand. 40, 487498.
Korteweg, D. J. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag. 39 (5), 422443.
Lax, P. D. 1968 Integrals of nonlinear equations of evolution and solitary waves Comm. Pure & Appl. Math. 21, 457490.
Meyer, R. E. 1967 Note on the undular jump J. Fluid Mech. 28, 209221.
Momoi, T. 1964 Tsunami in the vicinity of a wave origin Bull. Earthquake Res. Inst. Tokyo University, 42, 133146.
Nakamura, K. 1953 On the waves caused by the deformation of the bottom of the sea: I. Sci. Rep. Tohoku University, Sendai, Japan, 5 (5), 167176.
Peregrine, D. H. 1966 Calculations of the development of an undular bore J. Fluid Mech. 25, 321330.
Plafker, G. 1969 Tectonics of the March 27, 1964 Alaska earthquake. Geol. Survey Prof. Paper, no. 543–1.
Stokes, G. G. 1847 On the theory of oscillatory waves Trans. Camb. Phil. Soc. 8, 441455.
Takahasi, R. 1963 On some model experiments on tsunami generation Int. Un. Geodesy & Geophys. Monograph, 24, pp. 235248.
Takahasi, R. & Hatori, T. 1962 A model experiment on the tsunami generation from a bottom deformation area of elliptic shape Bull. Earthquake Res. Inst. Tokyo University, 40, 873883.
Tuck, E. O. & Hwang, L. S. 1972 Long wave generation on a sloping beach J. Fluid Mech. 51, 449461.
Ursell, F. 1953 The long-wave paradox in the theory of gravity waves Proc. Camb. Phil. Soc. 49, 685694.
Van Dorn, W. G. 1964 Source mechanism of the tsunami of March 28, 1964 in Alaska. Proc. 9th Conf. Coastal Engng, Lisbon, pp. 166190.
Van Dorn, W. G. 1966 Boundary dissipation of oscillatory waves J. Fluid Mech. 24, 769779.
Webb, L. M. 1962 Theory of waves generated by surface and sea-bed disturbances. In The Nature of Tsunamis, Their Generation and Dispersion in Water of Finite Depth, Tech. Rep. no. SN 57–2. Nat. Engng. Sci. Co.
Zabusky, N. J. 1968 Solitons and bound states of the time-independent Schrödinger equation Phys. Rev. 168, 124128.
Zabusky, N. J. & Kruskal, M. D. 1965 Interactions of ‘solitons’ in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240243.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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