Foldy–Lax approximation on multiple scattering by many small scatterers

Abstract

The multiple scattering of time harmonic wave emitted by a localized source through a medium with many scatterers can be approximated by an Foldy–Lax self-consistent system when the relative radius of each scatterer is small and the distribution of scatterers is sparse. The scattering amplitude in the Foldy–Lax self-consistent system will be specified in terms of scatterer volume and scattering strength. By neglecting the self-interaction effect, the difference from the exciting field in the Foldy–Lax formula to the analytic wave field given implicitly by the Lippmann–Schwinger integral equation is compared. An upper bound of the difference is obtained in terms of scaled radius and sparsity of the distribution of the scatterers.

Keywords:

• multiple scattering
• Foldy–Lax self-consistent system
• Lippmann–Schwinger integral equation

AMS Subject Classifications::

• 31A10
• 35J05
• 74J20

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Erratum

Acknowledgements

This work was suggested by Professor George Papanicolaou, who devoted much time and energy on this research. The author is grateful to his hospitality and generous help during his visit to Stanford University. He thanks Professor P. A. Martin for many valuable suggestions. Many helpful comments from anonymous referee are gracefully acknowledged. The research is partially supported by the National Natural Science Foundation of China (Nos. 11126082, 11171211 and 11171212).