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Transient and impulse responses of a one-dimensional linearly attenuating medium – I. Analytical results

Minster, J. B. (1978) Transient and impulse responses of a one-dimensional linearly attenuating medium – I. Analytical results. Geophysical Journal of the Royal Astronomical Society, 52 (3). pp. 479-501. ISSN 0016-8009. doi:10.1111/j.1365-246X.1978.tb04245.x.

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The transient and impulse responses (Green's function) for one-dimensional wave propagation in a standard linear solid are calculated using a Laplace Transform method. The spectrum of relaxation times is chosen so as to model a constant Q medium within an absorption band covering a broad frequency range which may be chosen so as to include the seismic frequencies. The inverse transform may be evaluated asymptotically in the limit of very long propagation times using the saddle point method. For shorter propagation times the method of steepest descent may be modified so as to yield an accurate first motion approximation. The character of the small amplitude precursor to the large amplitude ‘visible’ signal is investigated analytically. It is shown that the signal velocity is intermediate between the high-frequency (‘unrelaxed’) and the low-frequency (‘relaxed’) limits of the phase velocity.

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Additional Information:Copyright © 1978 The Royal Astronomical Society. Received 1977 August 19; in original form 1977 May 18. Numerous and fruitful discussions with H. Kanamori, D. Anderson, G. Mellman, and A. Suteau about various aspects of this work, as well as a detailed review by V. Cormier, are gratefully acknowledged. This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Air Force Office of Scientific Research under Contract No. AFOSR F49620-77-C-0022. Contribution No. 2898, Division of Geological and Planetary Sciences, Seismological Laboratory, California Institute of Technology, Pasadena, California. Since the completion of this work, I became familiar with the independent work of R. Chin & L. Thigpen (1977). These authors obtained results which are generally compatible with the present work and provide some interesting extensions, particularly with regard to the construction of uniformly asymptotic expansions. Very instructive discussions with these authors are gratefully acknowledged.
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Advanced Research Projects Agency (ARPA)UNSPECIFIED
Air Force Office of Scientific Research (AFOSR)F49620-77-C-0022
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Caltech Division of Geological and Planetary Sciences2898
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Official Citation:J. B. Minster Transient and impulse responses of a one-dimensional linearly attenuating medium – I. Analytical results Geophys. J. Int. (1978) 52 (3): 479-501 doi:10.1111/j.1365-246X.1978.tb04245.x
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:51616
Deposited By: George Porter
Deposited On:12 Nov 2014 15:35
Last Modified:10 Nov 2021 19:13

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