Asymptotic behavior of acoustic waves scattered by very small obstacles

Abstract

<jats:p>The direct numerical simulation of the acoustic wave scattering created by very small obstacles is very expensive, especially in three dimensions and even more so in time domain. The use of asymptotic models is very efficient and the purpose of this work is to provide a rigorous justification of a new asymptotic model for low-cost numerical simulations. This model is based on asymptotic near-field and far-field developments that are then matched by a key procedure that we describe and demonstrate. We show that it is enough to focus on the regular part of the wave field to rigorously establish the complete asymptotic expansion. For that purpose, we provide an error estimate which is set in the whole space, including the transition region separating the near-field from the far-field area. The proof of convergence is established through Kondratiev's seminal work on the Laplace equation and involves the Mellin transform. Numerical experiments including multiple scattering illustrate the efficiency of the resulting numerical method by delivering some comparisons with solutions computed with a finite element software.</jats:p>

A New Class of Uniformly Stable Time-Domain Foldy–Lax Models for Scattering by Small Particles. Acoustic Sound-Soft Scattering by Circles

“…Since this is technical, we rather illustrate the timedomain counterpart with numerical experiments. Let us finally remark that a related (but more general) question was studied in [4].…”

“…It seems that there exist fewer results in the time domain (see the recent monograph by Martin [33]). Asymptotic models for acoustic and electromagnetic wave propagation were obtained in [34,4] (3D scattering by a single obstacle), [31], [19] and in [20] (transmission problems) by using matched expansion method. The integral equation approach was applied to the 3D wave scattering by multiple obstacles in the recent work [40].…”

The Mellin Transform Method: Electromagnetics, Complex Analysis, and Educational Potential [Education Corner]

“…It can thus be used for to calculate the quality factor of the TM010 mode, which is dominant when 𝑑 𝑎 ⁄ < 2, where 𝑑 is the cavity length. As such, every undergraduate electrical engineer will confront (11) in a typical microwaves course (and possibly in a more exotic one, see [18], which comes from a nuclear engineering course in MIT). The simplicity of the calculation of the electromagnetic fields inside the cavity combined with the vast applications of the device make it a popular introductory example, as can be seen from well-established textbooks in the field [19,20].…”

“…The integral in (11), which is also dealt with in more mathematical works, can be evaluated via the orthonormality relation associated with Fourier-Bessel series (e.g., see [21] and [22], which prove the aforesaid relation using Bessel-function properties). Most students, however, are not likely to be familiar with such series.…”

“…In tandem, the core element and first step of the MTM, the Mellin Transform (MT), that coined the name of the method after all, has gained a lot of attention recently in rather applied scientific regimes. For example, the MT is a popular tool in economics [8] and image processing [9] (especially under the guise of the related Fourier-Mellin Transform [10]); it has also found application in acoustics [11], wireless-channel fading studies [12], as well as in a problem concerning equilibrium of an object supported by surface tension [13,14]. All these disciplines are of immediate interest to the modern engineer.…”