Small defects reconstruction in waveguides from multifrequency one-side scattering data

Abstract

<p style='text-indent:20px;'>Localization and reconstruction of small defects in acoustic or electromagnetic waveguides is of crucial interest in nondestructive evaluation of structures. The aim of this work is to present a new multi-frequency inversion method to reconstruct small defects in a 2D waveguide. Given one-side multi-frequency wave field measurements of propagating modes, we use a Born approximation to provide a <inline-formula><tex-math id="M1">\begin{document}$ \text{L}^2 $\end{document}</tex-math></inline-formula>-stable reconstruction of three types of defects: a local perturbation inside the waveguide, a bending of the waveguide, and a localized defect in the geometry of the waveguide. This method is based on a mode-by-mode spacial Fourier inversion from the available partial data in the Fourier domain. Indeed, in the available data, some high and low spatial frequency information on the defect are missing. We overcome this issue using both a compact support hypothesis and a minimal smoothness hypothesis on the defects. We also provide a suitable numerical method for efficient reconstruction of such defects and we discuss its applications and limits.</p>

The Helmholtz problem in slowly varying waveguides at locally resonant frequencies

“…If Ω is a regular waveguide, we find the same expression for the wave field as in [17]. We also see that the behavior of propagative and evanescent modes in a perturbed waveguide is similar to that in a regular waveguide.…”

“…Source f . This proof is an adaptation of the proof presented in Appendix B of [17]. Using the results on the modal decomposition presented in Appendix A of [17], we know that the equation ( 50) is equivalent to Here, the relative L 2 error between u and u app is 7.73%.…”

“…Using these two propositions, we are now able to justify the approximation of (H) by (H ) which, as in [19] and [17], is a Born approximation. However, here we show that this approximation remains valid near resonance frequencies.…”

“…In this section, we recall some classical results about modal decompositions, the proofs of which can be found in [17,18]. Definition 1.…”

“…Using the wavenumber function, one can adapt the classic Sommerfeld (or outgoing) condition, defined in [17] for regular waveguides, to general varying waveguides Ω. This condition will be used later to guarantee uniqueness for the source problem (2).…”