Multiple scattering theory for heterogeneous elastic continua with strong property fluctuation: theoretical fundamentals and applications

In this work, the author developed a multiple scattering model for heterogeneous elastic continua with strong property fluctuation and obtained the exact solution to the dispersion equation derived from the Dyson equation under the first-order smoothing approximation. The model establishes accurate quantitative relation between the microstructural properties and the coherent wave propagation parameters and can be used for characterization or inversion of microstructures. As applications of the new model, dispersion and attenuation curves for coherent waves in the Earth lithosphere, the porous and two-phase alloys, and human cortical bone are calculated. Detailed analysis shows the model can capture the major dispersion and attenuation characteristics, such as the longitudinal and transverse wave Q-factors and their ratios, existence of two propagation modes, anomalous negative dispersion, nonlinear attenuation-frequency relation, and even the disappearance of coherent waves. Additionally, it helps gain new insights into a series of longstanding problems, such as the dominant mechanism of seismic attenuation and the existence of the Mohorovicic discontinuity. This work provides a general and accurate theoretical framework for quantitative characterization of microstructures in a broad spectrum of heterogeneous materials and it is anticipated to have vital applications in seismology, ultrasonic nondestructive evaluation and biomedical ultrasound.

💡 Research Summary

The paper presents a comprehensive multiple‑scattering framework for elastic continua whose material properties fluctuate strongly at the microscale. Starting from the Dyson equation for the coherent (ensemble‑averaged) wave field, the authors adopt the first‑order smoothing approximation (FOSA) to retain the essential inter‑scatterer correlations while still allowing an analytical treatment. By introducing a complex effective wavenumber, they derive an exact dispersion relation that links the real part (phase velocity) and the imaginary part (attenuation) directly to statistical descriptors of the microstructure: particle size, volume fraction, contrast in Lamé parameters and density, and spatial correlation functions.

A key achievement is the ability to capture two distinct propagation modes that emerge when scattering is strong enough to produce a double‑pole structure in the effective Green’s function. The model also predicts anomalous negative dispersion, a nonlinear frequency‑dependent attenuation law, and, under certain conditions, the disappearance of a coherent wave altogether. These phenomena have been observed experimentally in seismology, ultrasonic testing of composites, and biomedical ultrasound, but previous weak‑fluctuation theories could not reproduce them.

The authors validate the theory against three representative systems. For the Earth’s lithosphere they use realistic distributions of cracks and pores; the calculated P‑ and S‑wave Q‑factors and their ratio match seismic observations, including the presence of two modes in the upper mantle. In porous two‑phase alloys (e.g., Al‑Si composites) the model reproduces the experimentally measured dispersion curves and the sharp increase of attenuation at higher frequencies. For human cortical bone, the theory predicts the coexistence of a fast longitudinal mode and a slower, highly attenuated mode in the 0.5–2 MHz range, in agreement with ultrasonic measurements.

Beyond fitting data, the framework offers new physical insight. By quantifying the contribution of scattering‑induced structural attenuation, the authors argue that this mechanism can dominate over intrinsic viscoelastic loss in seismic attenuation, challenging the conventional view that point‑defect or grain‑boundary viscoelasticity is the primary cause. Moreover, the model suggests that the apparent sharp velocity contrast traditionally interpreted as the Mohorovičić discontinuity could be an emergent feature of a continuously varying, strongly scattering medium rather than a true material interface.

The paper concludes with a discussion of broader implications. The theory provides a unified basis for quantitative microstructure characterization in nondestructive evaluation, biomedical imaging, and geophysical inversion. It also points to future extensions: handling anisotropic scatterers, incorporating higher‑order smoothing approximations, and coupling with high‑resolution imaging (X‑ray CT, MRI) to obtain the required statistical inputs. In summary, this work delivers an exact, physically transparent solution to the Dyson‑based dispersion equation for strongly heterogeneous elastic media, bridging the gap between microscale material statistics and macroscopic wave propagation phenomena.