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The Foldy–Lax approximation is valid for nearly resonating frequencies

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Abstract

The waves (including acoustic, electromagnetic and elastic ones) propagating in the presence of a cluster of inhomogeneities undergo multiple interactions between them. When these inhomogeneities have sub-wavelength sizes, the dominating field due to the these multiple interactions is the Foldy–Lax field. This field models the multiple interactions between the equivalent point-like scatterers, located at the centers of the small inhomogeneities, with scattering coefficients related to geometrical/material properties of each inhomogeneities, as polarization coefficients. One of the related questions left open for a long time is whether we can reconstruct this Foldy–Lax field from the scattered field measured far away from the cluster of the small inhomogeneities. This would allow us to see (or detect) the multiple interactions between them. This is the Foldy–Lax approximation (or Foldy–Lax paradigm). In this work, we show that this approximation is indeed valid as soon as the inhomogeneities enjoy critical scales between their sizes and contrasts. These critical scales allow them to generate resonances which can be characterized and computed. The main result here is that exciting the cluster by incident frequencies which are close to the real parts of these resonances allows us to reconstruct the Foldy–Lax field from the scattered waves collected far away from the cluster itself (as the farfields). In short, we show that the Foldy–Lax approximation is valid using nearly resonating incident frequencies. This result is demonstrated by using, as small inhomogeneities, dielectric nanoparticles for the 2D TM model of electromagnetic waves and bubbles for the 3D acoustic waves.

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Notes

  1. The notation p.v means the Cauchy principal value.

  2. Recall that in 3D, \(\lambda _n=\delta ^2 \tilde{\lambda }_n\) where \(\tilde{\lambda }_n\) are the eigenvalues of \(\widetilde{A_0}\)

  3. Where \(\varvec{\delta }\) is the Kronecker symbol.

  4. From (1.22), we see that \(\gamma \sim \delta ^{-2} \vert \log (\delta )\vert ^{-1}\). Therefore and the second part of Hypotheses 1, we deduce that \(k_n =O(1)\) for \(\delta \ll 1\)

  5. To avoid more complicated notations and simplify the exposition, we assume that \(\tau _{j} {{:}{=}} b - b_{0}(z_{j}) = \tau , \,\, j=1,\ldots ,M.\) The final results stay valid without this assumption.

  6. As in 3D case, to simplify the computations, we write the detailed proof with \(\tau \), but of course we can do it with \(\left\{ \tau _{j} \right\} _{j=1}^{M}\).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Kuwait University, P.O. Box 5969, 13060, Safat, Kuwait

    Abdulaziz Alsenafi

  2. RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040, Linz, Austria

    Ahcene Ghandriche & Mourad Sini

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  1. Abdulaziz Alsenafi
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  2. Ahcene Ghandriche
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Correspondence to Abdulaziz Alsenafi.

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Alsenafi, A., Ghandriche, A. & Sini, M. The Foldy–Lax approximation is valid for nearly resonating frequencies. Z. Angew. Math. Phys. 74, 11 (2023). https://doi.org/10.1007/s00033-022-01896-5

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  • DOI: https://doi.org/10.1007/s00033-022-01896-5

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