Functional-Analytic Justification of the Time-Domain Foldy-Lax Approximation for Dispersive Acoustic Media: A Feynman-Diagram Viewpoint
Arpan Mukherjee, Mourad Sini
TL;DR
The paper develops a rigorous time-domain Foldy-Lax framework for dispersive acoustic media composed of bubble clusters, incorporating Minnaert resonance to model dispersion. It builds a functional-analytic foundation using Hardy-Sobolev and Zen spaces, proving unique solvability of a delayed-coupled hyperbolic system for bubble amplitudes and representing the solution as a convergent Neumann series of convolutions. The authors derive geometric truncation-error bounds and quantify how higher-order multi-scattering terms scale with bubble size and inter-bubble spacing, showing meaningful contributions up to N < 1/(1-p) near resonance. A novel Feynman-diagram mapping links multi-scattering paths to diagrammatic vertices and propagators, providing an interpretable and computationally tractable view of higher-order interactions. The results yield explicit error and scaling estimates with practical implications for cavitation therapy, seismic imaging, and metamaterial design, and are illustrated via an air-bubbles-in-water example that demonstrates the framework’s quantitative predictions.
Abstract
This work provides a rigorous functional-analytic justification for a time-domain Foldy-Lax framework that describes multiple acoustic scattering by a cluster of dispersive resonators (modeling gas-filled bubbles), explicitly incorporating dispersion via the Minnaert resonance. The model is formulated as a delayed-coupled hyperbolic system for bubble amplitude interactions. We combine time-domain integral equations, Laplace transforms, and Hardy-Sobolev space techniques to analyze this system, establishing its unique solvability in anisotropic Hilbert spaces, with solutions expressed as convergent Neumann series of convolution operators. We derive geometric decay of truncation errors for resonant incident waves and quantify the contribution of $N$-th order multi-scattering, showing it scales with \(\varepsilon^{N(1-p)+1}\) (relating bubble radius \(\varepsilon\) and inter-bubble distance scaling as $\varepsilon^p$, $p<1$). This dominates the measurement errors, which are of order $\varepsilon^2$, thereby allowing us to capture fields generated by inter-bubble interactions of order $N<\frac{1}{1-p}$. This provides a quantitative relation between the spectra band width of the source field, the closeness distance between the bubbles and the order $N$ of the relevant interactions between the bubbles. Furthermore, a novel connection to Feynman diagrams maps multi-scattering paths to diagrammatic vertices and propagators, simplifying the interpretation of higher-order interactions and kinematic constraints. This framework advances accurate transient wave prediction in dispersive media, with implications for cavitation therapy, seismic imaging, and metamaterial engineering.