Multiple scattering effects due to a random array of obstacles are considered. Employing a ``configurational averaging'' procedure, a criterion is obtained for the validity of approximate integral equations describing the various field quantities of interest. The extinction theorem is obtained and shown to give rise to the forward‐amplitude theorem of multiple scattering. In the limit of vanishing correlations in position, the complex propagation constant κ of the scattering medium is obtained. Under appropriate restrictions, the expression for κ is shown to include both the square‐root law of isotropic scatterers and the additive rule for cross sections valid for sufficiently low densities of anisotropic obstacles. Some specific examples from acoustics and electromagnetic theory then indicate that at least in the simplest cases the results remain valid for physically allowable densities of obstacles.

Multiple scattering of waves

Elastic wave scattering by a general elastic heterogeneity having slightly different density and elastic constants from the surrounding medium is formulated using the equivalent source method and Born approximation In the low‐frequency range (Rayleigh scattering) the scattered field by an arbitrary heterogeneity having an arbitrary variation of density and elastic constants can be equated to a radiation field from a point source composed of a unidirectional force proportional to the density contrast between the heterogeneity and the medium, and a force moment tensor proportional to the contrasts of elastic constant It is also shown that the scattered field can be decomposed into an “impedance‐type” field, which has a main lobe in the backscattering direction and no scattering in the exact forward direction, and a “velocity type” scattered field, which has a main lobe in the forward scattering direction and no scattering in the exact backward direction For Mie scattering we show that the scattered far f

Scattering characteristics of elastic waves by an elastic heterogeneity

The physical properties which affect the propagation of cracks in specimens fractured by thermal shock are discussed. The driving force for crack propagation is provided by the elastic energy stored at fracture. The mechanism of energy dissipation which will tend to arrest the propagating cracks is provided by the “effective surface energy” required to produce the newly formed crack surfaces. An expression is derived applicable to a body of spherical shape for the mean area traversed by cracks nucleated by thermal shock. Three numerical examples are given for materials with widely different physical properties, and their fracture behavior is predicted. Good agreement with experiment was obtained. Thermal shock damage resistance parameters suitable for the relative comparison of the “degree of damage” to be expected in materials fractured by thermal shock are proposed. The criteria for a low degree of damage are high values of Young's modulus of elasticity, Poisson's ratio, and effective surface energy and low values of strength. Recommendations are made for the selection of materials for severe thermal shock, where the best materials available are known to fail.

Elastic Energy at Fracture and Surface Energy as Design Criteria for Thermal Shock

Three single‐scattering approximations for coefficients in Biot’s equations of poroelasticity are considered: the average T‐matrix approximation (ATA), the coherent potential approximation (CPA), and the differential effective medium (DEM). The scattering coefficients used here are exact results obtained previously for scattering from a spherical inclusion of one Biot material imbedded in another otherwise homogeneous Biot material. The CPA has been shown previously to guarantee that, if the coefficients for the scattering materials satisfy Gassmann’s equation, then the effective coefficients for the composite medium satisfy Brown and Korringa’s generalization of Gassmann’s equation. A collection of similar results is obtained here showing that the coefficients derived from ATA, CPA, or DEM all satisfy the required conditions for consistency. It is also shown that Gassmann’s equation will result from any of these single‐scattering approximations if the collection of scatterers includes only spheres of fluid and of a single type of elastic solid.

Single‐scattering approximations for coefficients in Biot’s equations of poroelasticity

A unified model is proposed for explaining the frequency dependence of Q−1 and the formation of velocity seismograms by a single scattering process. Adopting Birch's law and a linear correlation between P and S wave velocities, we statistically describe the inhomogeneous medium by one random fluctuation, which is spatially characterized by the exponential autocorrelation function. We calculate Q−1 for P and S waves on the basis of the Born approximation, supposing seismic waves attenuate due to scattering largely by rapidly fluctuating random structure. Resulting Qs−1 explains the observed frequency dependence well, when the mean square fractional fluctuation and the correlation distance are chosen to be 0.01 and 2 km, respectively. Nonspherical radiation due to a point shear dislocation and frequency dependent nonisotropic scattering including conversion between P and S waves complicate the seismograms. Taking those complexities into account, we synthesize the “envelopes” of three-component velocity amplitudes of P and S coda waves by summing up energy singly scattered by distributed inhomogeneities. From numerical calculations, we found that the radial component of the P coda wave is excited even in the nodal direction of the P wave radiation mostly due to SP scattering occurring near the hypocenter. The P coda waves appearing on the transverse components are a complex mixture of PP, PS, and SP scattering. The S coda waves are excited mostly due to SS scattering in all directions, and appear in all three components; their polarizations are controlled by the radiation pattern even in the later part of S coda waves.

Attenuation and envelope formation of three-component seismograms of small local earthquakes in randomly inhomogeneous lithosphere

Scattering by a spherical obstacle of a plane longitudinal wave propagating in an isotropically elastic solid is computed. Expressions for the scattered wave and the total scattered energy are given. Three special types of obstacle—an isotropically elastic sphere, a spherical cavity, and a rigid sphere—are discussed in detail, especially for Rayleigh scattering. The result for the isotropically elastic sphere is compared with the well‐known result of scattering of a plane wave propagating in an ideal fluid by a sphere of another ideal fluid.

Chung F Ying

Papers

Scattering of a Plane Longitudinal Wave by a Spherical Obstacle in an Isotropically Elastic Solid