Quantitative study of nonstationary interference wave fields in layered homogeneous elastic media with plane-parallel interfaces. I. Statements of the problems and efficient methods for their solution

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1. When preparing for publication the results of our research on propagation of nonstationas'y wave fields in media comprising thin layers, we intended to start with general ideas concerning: (1) the formulation of' problems on wave propagation in layered media; (2) methods for constructing mathematical representations of wave fields excited in them; (3) a presentation of approaches and methods of quantitatively studying wave fields with a view to deriving physical consequences, and, finally, (4) the development of different auxiliary ideas underlying the quantitative evaluation of the fields under consideration. Further, we intended to pay due attention to the application of methods proposed for the investigation and evaluation of fields in quantitatively studying wave fields arising in specific problems on wave propagation in media comprising one monowave thin layer.In accordance with this intention, initially paper [1] included one more section (consisting of three subsections) that presented the results of certain quantitative studies and evaluations of the monowave interference fields represented in [1] by formulas (8.1), (8.3), and (8.6) and also by formulas (2.83) and (2.88).In that section, first, in relevant layered media we formulated problems on the propagation of waves excited by a given point source with a variable shape function of its impulse, and we chose certain schemes for observing fields excited in media, which yield the interference wave representations indicated above of the latter fields in the form of repeated integrals of the Lamb method. Further, real pararaeters of the medium were specified, and the elements of interference wave fields (to which it was possible to assign a clear physical meaning) were computed for several reference points of observation profiles of the field, and next the total wave field was evaluated. The results of computations were presented in the form of seismograms, which were considered in the classical correlation manner of studying experimental seismic data. The phase and group velocities of propagation of the chosen wave groups, the character of their attenuation with distance depending on the frequency from the spectral band of the signal impulse, and other characteristics were evaluated both theoretically (numerically) and experimentally.In the ease of the field from (2.83) in [1], special emphasis was places on studying sufficiently lowfrequency interference waves of head type excited in a liquid layer, namely, on the longitudinal (p) and the transverse (8) wave in the underlying elastic half-space. Examining such waves is of interest from different points of view, in particular, because on the second sheet of the complex (~) plane the behavior (as the frequency parameter ~ varies) of the roots of the dispersion equation of the problem that are located in a neighborhood of the cut corresponding to the p-head wave in the liquid layer is crucial for evaluating the spectral function of the field. Irregularities in the behavior of such roots lead to a manifest...

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On problems to be studied in the immediate future

Petrashen¹

1997

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Algorithms for evaluating basic components of spectral functions of interference wave fields

Kiselev¹

1997

Simple algorithms for finding a numerical solution on a complex plane of dispersion equations dependent on a real parameter are described. Algorithms for numerical integration along stationary contours on a complez plane in the case where the integrand possesses singularities of the pole and branch-point type are also presented. These algorithms are applied to compute the spectral functions of wave fields propagating in layered media. Bibliography: 3 titles.In the present paper, an interference wave field u(t) (in the case of chosen models of media and points of observation) is represented (see w167 3 in [1]) by the iterated integrals On the original Lamb integration contour A, the integrand in (2) is always highly oscillating, which makes the direct numerical evaluation of the integral quite complicated. Because of this, it turns out to be more advantageous to evaluate the spectral function by using methods of contour integration. These methods start by carrying out a deformation of the original contour A to the stationary lines 10 of steepest ascent of the phase function f(() that are defined by the equations which must be compensated for by adding (with appropriate signs) the residues of the integrand from (2) at all roots that are intersected by the contour A (see w Secs. 4, 5 from [1]) to the integral under transformation. In addition, in the process of deformation the contour A may come across the branch points r = 7 of the factor at the exponent, which is conventionally referred to in what follows as "the slowly varying part" of the integrand from (2). In such a case, the cut (7, oo) is preliminarily deformed, starting with the point ( = 7, into the half-plane ~ < 0 up to its coincidence with the stationary line F~ of steepest ascent of the phase function with origin at the point ~ = 7. The original contour )~ bypasses the cut F~ along a loop that can be reduced, by continuous deformation, to one of its boards.

The classical and generalized love problems in a domain of low-frequency interference waves of signal type. I

Смирнова

Surkov

1998

Low-frequency interference waves propagating through a thin elastic layer, which is in rigid contact with an elastic half-space, are considered. A new method for computing the fields of low-frequency waves suggested by G. I. Pc,rashes allows us to compute the fields of these waves and to draw some conclusions concerning their behavior. In particular, we may conclude that the thin upper layer significantly distorts a signal sent deep into a medium. Bibliography: 8 titles.