Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T06:30:27.321Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  29 March 2006

Joseph L. Hammack
Joseph L. Hammack
Affiliation:
W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 60 , Issue 4 , 9 October 1973 , pp. 769 - 799
Copyright
© 1973 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benjamin, T. B., Bona, J. L. & Mahony, J. J. 1972 Model equations for long waves in nonlinear dispersive systems Phil. Trans. Roy. Soc. 272, 47–78.Google Scholar
French, J. A. 1969 Wave uplift pressures on horizontal platforms. W. M. Keck Lab. Hydraul. & Water Res. Calif. Inst. Tech. Rep. KH-R-19.Google Scholar
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, 1095–1097.Google Scholar
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.Google Scholar
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, 133–137.Google Scholar
Hwang, L. S. & Divoky, D. 1970 Tsunami generation J. Geophys. Res. 75, 6802–6817.Google Scholar
Ichiye, T. 1950 On the theory of tsunami Oceanograph. Mag. 2, 83–100.Google Scholar
Ichiye, T. 1958 A theory of the generation of tsunami by an impulse at the sea bottom J. Oceanograph. Soc. Japan, 14, 41–44.Google Scholar
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, 535–571.Google Scholar
Keller, J. B. 1963 Tsunamis: water waves produced by earthquakes. Int. Un. Geodesy & Geophys. Monograph, no. 24, pp. 154–166.Google Scholar
Keulegan, G. H. 1948 Gradual damping of solitary waves J. Res. Nat. Bur. Stand. 40, 487–498.Google Scholar
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), 422–443.Google Scholar
Lax, P. D. 1968 Integrals of nonlinear equations of evolution and solitary waves Comm. Pure & Appl. Math. 21, 457–490.Google Scholar
Meyer, R. E. 1967 Note on the undular jump J. Fluid Mech. 28, 209–221.Google Scholar
Momoi, T. 1964 Tsunami in the vicinity of a wave origin Bull. Earthquake Res. Inst. Tokyo University, 42, 133–146.Google Scholar
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), 167–176.Google Scholar
Peregrine, D. H. 1966 Calculations of the development of an undular bore J. Fluid Mech. 25, 321–330.Google Scholar
Plafker, G. 1969 Tectonics of the March 27, 1964 Alaska earthquake. Geol. Survey Prof. Paper, no. 543–1.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Takahasi, R. 1963 On some model experiments on tsunami generation Int. Un. Geodesy & Geophys. Monograph, 24, pp. 235–248.Google Scholar
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, 873–883.Google Scholar
Tuck, E. O. & Hwang, L. S. 1972 Long wave generation on a sloping beach J. Fluid Mech. 51, 449–461.Google Scholar
Ursell, F. 1953 The long-wave paradox in the theory of gravity waves Proc. Camb. Phil. Soc. 49, 685–694.Google Scholar
Van Dorn, W. G. 1964 Source mechanism of the tsunami of March 28, 1964 in Alaska. Proc. 9th Conf. Coastal Engng, Lisbon, pp. 166–190.Google Scholar
Van Dorn, W. G. 1966 Boundary dissipation of oscillatory waves J. Fluid Mech. 24, 769–779.Google Scholar
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.Google Scholar
Zabusky, N. J. 1968 Solitons and bound states of the time-independent Schrödinger equation Phys. Rev. 168, 124–128.Google Scholar
Zabusky, N. J. & Kruskal, M. D. 1965 Interactions of ‘solitons’ in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243.Google Scholar