SEISMIC WAVES IN ANELASTIC NON‐LINEAR MEDIA—A THEORETICAL CONTRIBUTION *

A bstract Numerical solutions of the wave equation for a particular type of non‐linear “constant Q ” medium were carried out. These solutions were obtained after the transformation of the space derivatives in finite differences; power series of the time are used to express the solutions. The medium is characterized by a not single valued stress‐strain relation; the stresses are greater when the absolute values of strain are increasing (passive work), and are less when they are decreasing (active work). A loss of energy follows which is constant for every cycle and independent of frequency. This model represents the simplest type of medium in agreement with the laboratory data on rock samples. For a similar medium the stress’values do not depend only on the instantaneous value of the strain, but also on the previous strain values, i.e. the history of the medium. All these characteristics are not compatible with linearity and require particular types of computing procedures similar to the one shown in this paper. The results of calculations here shown refer both to the propagation of an isolated wave and to the generation of a wave in a spherical hole by change of pressure. They refer particularly to the shape, the amplitude and the width of the isolated wave along the propagation path. The most important results for this type of medium are the following: a ) for a plane single isolated wave, the displacement amplitude wave does not change along the propagated distance. The width increases linearly as function of the distance; b ) the corresponding particle velocity decreases in amplitude; c ) for single isolated spherical waves the displacement amplitude decreases with propagated distance only due to the geometric factor, i.e. inversely proportional to the propagated distance; its width increases in the same way as for plane waves. The comparison between these theoretical results with the field and seismological data show a sufficiently good agreement as far as the value interval of wave width and frequencies is concerned. Less satisfactory is the comparison regarding a linear dependence of the wave width on the distance. This fact happens probably because in the field often long trains of waves and not isolated waves occur. In effect, for trains of waves the behaviour is different than that of an isolated wave; particularly, for the former the frequency variations along the travelled path is less and the displacement variations greater. However, it seems likely that a further similar theoretical research for trains of waves propagating in this type of non‐linear medium might be carried out to complete the present research.