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

Hammack, Joseph L. (1973) A note on tsunamis: their generation and propagation in an ocean of uniform depth. Journal of Fluid Mechanics, 60 (4). pp. 769-799. ISSN 0022-1120. http://resolver.caltech.edu/CaltechAUTHORS:HAMjfm73

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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 during propagation and 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 more recently advocated by Benjamin, 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.


Item Type:Article
Additional Information:Copyright © 1973 Cambridge University Press. Reprinted with permission. (Received 21 March 1973) The author is greatly indebted to Professor Fredric Raichlen for his kind guidance through every phase of this study. Gratitude is also extended to other members of the faculty and staff of the W.M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology. This work was supported by the National Science Foundation under Grants GK-2370 and GK-24716.
Record Number:CaltechAUTHORS:HAMjfm73
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:HAMjfm73
Alternative URL:http://dx.doi.org/10.1017/S0022112073000479
Official Citation:Joseph L. Hammack (1973). A note on tsunamis: their generation and propagation in an ocean of uniform depth. Journal of Fluid Mechanics Digital Archive, 60, pp 769-799 doi:10.1017/S0022112073000479
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:11022
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:23 Jun 2008
Last Modified:26 Dec 2012 10:07

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