Some limitations for the applicability of the ray method in elastodynamics

Abstract

The ray method in the frequency domain describes the asymptotic behavior of the wave field for large frequencies which is not, in general, uniform with respect to the distance between the source and the point of observation. This implies that one can face situations in which the results obtained by the ray method may not be reliable for large distances from the source, if the dominant frequency is not high enough. In this paper we present two examples of such a type of problem for rather simple elastodynamic problems: a homogeneous medium; and a constant gradient velocity medium without interfaces. For both cases, the sources are assumed to be represented by two successive terms of the ray series, and therefore there are no point sources problems in our study. To this end we employ the following criterion of validity of the ray method: the absolute value of the ratio of the second term of the ray series to the first one must be less than unity. This criterion turns out to be sensitive to the radius of curvature of the initial wave front and to the initial distribution of the amplitude (or energy) along it. In the worst case of an initially planar wave front with nonuniform distribution of the amplitude, the second term of the ray series increases very fast. This gives rise to strong limitations to the use of the ray method with respect to distance if the frequency is fixed, or with respect to frequency for large distances. The depolarization phenomenon in both cases is discussed as well.