High frequency analysis of Helmholtz equations: case of two point sources
Elise Fouassier
TL;DR
This work analyzes the high-frequency (ε→0) limit of the Helmholtz equation with a source term that is a sum of two point sources. By employing Wigner measures, the authors show that the limiting energy distribution μ satisfies a Liouville transport equation with a source term equal to the sum of the single-source contributions, effectively decoupling the two sources in the limit. The analysis hinges on uniform bounds in Besov-like norms and a refined radiation condition, enabling a precise description of μ on the energy shell |ξ|^2 = 1 and establishing additivity of the source terms. The results extend the one-source case and justify the intuition that two closely spaced point sources behave independently in the high-frequency regime, with the energy radiated from each source contributing separately to μ.
Abstract
We derive the high frequency limit of the Helmholtz equation with source term when the source is the sum of two point sources. We study it in terms of Wigner measures (quadratic observables). We prove that the Wigner measure associated with the solution satisfies a Liouville equation with, as source term, the sum of the source terms that would be created by each of the two point sources taken separately. The first step, and main difficulty, in our study is the obtention of uniform estimates on the solution. Then, from these bounds, we derive the source term in the Liouville equation together with the radiation condition at infinity satisfied by the Wigner measure.