Scattering ofP andS waves by a spherically symmetric inclusion

Scattering of an arbitrary elastic wave incident upon a spherically symmetric inclusion is considered and solutions are developed in terms of the spherical vector system of Petrashen, which produces results in terms of displacements rather than displacement potentials and in a form suitable for accurate numerical computations. Analytical expressions for canonical scattering coefficients are obtained for both the cases of incidentP waves and incidentS waves. Calculations of energy flux in the scattered waves lead to elastic optical theorems for bothP andS waves, which relate the scattering cross sections to the amplitude of the scattered fields in the forward direction. The properties of the solutions for a homogeneous elastic sphere, a sphere filled by fluid, and a spherical cavity are illustrated with scattering cross sections that demonstrate important differences between these types of obstacles. A general result is that the frequency dependence of the scattering is defined by the wavelength of the scattered wave rather than the wavelength of the incident wave. This is consistent with the finding that the intensity of theP→S scattering is generally much stronger than theS→P scattering. When averaged over all scattering angles, the mean intensity of theP→S converted waves is2Vp2/Vs4times the mean intensity of theS→P converted waves, and this ratio is independent of frequency. The exact solutions reduce to simple and easily used expressions in the case of the low frequency (Rayleigh) approximation and the low contrast (Rayleigh-Born) approximation. The case of energy absorbing inclusions can also be obtained by assigning complex values to the elastic parameters, which leads to the result that an increase in attenuation within the inclusion causes an increased scattering cross section with a marked preference for scatteredS waves. The complete generality of the results is demonstrated by showing waves scattered by the earth's core in the time domain, an example of high-frequency scattering that reveals a very complex relationship between geometrical arrivals and diffracted waves.

[1]  C. Chapman,et al.  Diffracted Seismic Signals and Their Numerical Solution , 1972 .

[2]  K. Aki.  Scattering and attenuation of shear waves in the lithosphere , 1980 .

[3]  P. Waterman,et al.  Matrix theory of elastic wave scattering , 1976 .

[4]  C. Chapman.  The Turning Point of Elastodynamic Waves , 1974 .

[5]  J. Gubernatis,et al.  Interpretation of elastic‐wave scattering theory for analysis and design of flaw‐characterization experiments: The long‐wavelength limit , 1979 .

[6]  K. Aki.  Analysis of the seismic coda of local earthquakes as scattered waves , 1969 .

[7]  H. Nussenzveig.  High‐Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission , 1969 .

[8]  H. V. Hulst.  Light Scattering by Small Particles , 1957 .

[9]  M. Hinders.  Plane-elastic-wave scattering from an elastic sphere , 1991 .

[10]  C. H. Chapman,et al.  Diffraction of P Waves by the Core and an Inhomogeneous Mantle , 1970 .

[11]  Charles Elbaum,et al.  Ultrasonic Methods in Solid State Physics , 1969 .

[12]  G. Mie.  Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen , 1908 .

[13]  J. Rial,et al.  Seismic waves at the epicenter's antipode , 1980 .

[14]  Eytan Domany,et al.  Formal aspects of the theory of the scattering of ultrasound by flaws in elastic materials , 1977 .

[15]  J. Cleary,et al.  Evidence for scattering of seismic PKP waves near the mantle-core boundary , 1974 .

[16]  P. Richards.  On the adequacy of plane-wave reflection/transmission coefficients in the analysis of seismic body waves , 1976 .

[17]  Rohn Truell,et al.  Scattering of a Plane Longitudinal Wave by a Spherical Obstacle in an Isotropically Elastic Solid , 1956 .

[18]  Ru-Shan Wu,et al.  Scattering characteristics of elastic waves by an elastic heterogeneity , 1985 .

[19]  L. Johnson,et al.  Scattering of elastic waves by a spherical inclusion–II. Limitations of asymptotic solutions , 1993 .

[20]  J. Ansell.  On the scattering of SH waves from a point source by a sphere , 1978 .

[21]  Eytan Domany,et al.  The Born approximation in the theory of the scattering of elastic waves by flaws , 1977 .

[22]  Keiiti Aki,et al.  Scattering of P waves under the Montana Lasa , 1973 .

[23]  K. Aki.  Scattering conversions P to S versus S to P , 1992, Bulletin of the Seismological Society of America.

[24]  D. Ludwig.  Diffraction by a Circular Cavity , 1970 .

[25]  K. Aki,et al.  Elastic wave scattering by a random medium and the small‐scale inhomogeneities in the lithosphere , 1985 .

[26]  L. Johnson,et al.  Scattering of elastic waves by a spherical inclusion—I. Theory and numerical results , 1993 .

[27]  V. Cormier.  Time‐domain modelling of PKIKP precursors for constraints on the heterogeneity in the lowermost mantle , 1995 .

[28]  Robert A. Phinney,et al.  P wave diffraction theory and the structure of the core‐mantle boundary , 1966 .