Multiscale analysis of the acoustic scattering by many scatterers of impedance type

Challa, Durga Prasad; Sini, Mourad

doi:10.1007/s00033-016-0652-0

Cited by 15 publications

(26 citation statements)

References 39 publications

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“…We call the upper bounds of the Lipschitz character of B m 's, M m a x , d m i n , d m a x , and κ m a x the set of the apriori bounds. In , we have shown that there exist a positive constant a 0 , λ − , and λ + depending only on the set of the apriori bounds and on n m a x such that if

a \leq a_{0}, | λ_{m, 0} | \leq λ_{+}, | R (λ_{m, 0}) | \geq λ_{-}, β < 1, s \leq 2 - β, \frac{s}{3} \leq t

then the far‐field pattern

U^{\infty} (\overset{x}{true}, θ)

has the following asymptotic expansion

U^{\infty} (\overset{x}{true}, θ) = V_{n}^{\infty} (\overset{x}{true}, θ) + \sum_{m = 1}^{M} V_{n}^{t} (- \overset{x}{true}, z_{m}) {double-struckQ}_{m} + O ((), a^{3 - s - 2 β}),

uniformly in

\overset{x}{true}

and θ in

{double-struckS}^{2}

. The constant appearing in the estimate O (.)…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…Hence, the total field U t := U i + U s satisfies the following exterior impedance problem of the acoustic waves

(normalΔ + κ^{2} n^{2} (x)) U^{t} = 0 in {double-struckR}^{3} ∖ ((), \cup_{m = 1}^{M} {\overset{D}{true}}_{m}),

{(|, \frac{\partial U^{t}}{\partial ν_{m}} + λ_{m} U^{t})}_{\partial D_{m}} = 0, 1 \leq m \leq M,

\frac{\partial U^{s}}{\partial | x |} - iκ U^{s} = o ((), \frac{1}{| x |}), | x | \to \infty,

Again, the scattering problem – is well‐posed in the Hölder or Sobolev spaces (see in the case I λ m >0). As we said for –, this last condition can be relaxed to allow I λ m to be negative . Applying Green's formula to U s , we can show that the scattered field U s ( x , θ ) has the following asymptotic expansion:

U^{s} (x, θ) = \frac{e^{iκ | x |}}{4 π | x |} U^{\infty} (\overset{x}{true}, θ) + O (| x |^{- 2}), | x | \to \infty,

where the function

U^{\infty} (\overset{x}{true}, θ)

for

(\overset{x}{true}, θ) \in {double-struckS}^{2} ...

…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…Because we have the freedom in choosing the three functions K , P 0 , and λ 0 , then we can generate a large family of indices of refraction. In particular if we choose the surface impedance to have negative imaginary parts, which is mathematically possible , then we show that the equivalent medium will be passive only if the index of refraction is negative, that is, it behaves as a metamaterial. we can choose the coefficients K , P 0 , and λ 0 so that the equivalent index of refraction is educed to the unity. This means that the domain Ω modeled by the original index of refraction n is cloaked. …”

Section: Introductionmentioning

confidence: 94%

“…A rigorous approximation of the scattered fields, with error estimates in terms of the number M of the small scatterers, their minimum distance d , and the maximum radius a , was derived in using the integral equations approach. Based on these last estimates, we can characterize the equivalent medium with explicit error estimates in terms of the parameter a , a << 1, in appropriate regimes described by the other parameters M and d in terms of a , namely M := M ( a ):= O ( a − s ) and d := d ( a )≈ a t , as a << 1, with non‐negative parameters s and t .…”

Section: Introductionmentioning

confidence: 99%

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A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index

Alsaedi

Ahmad

Challa

et al. 2015

Math Methods in App Sciences

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We deal with the scattering of an acoustic medium modeled by an index of refraction n varying in a bounded region Ω of double-struckR3 and equal to unity outside Ω. This region is perforated with an extremely large number of small holes Dm's of maximum radius a, a << 1, modeled by surface impedance functions. Precisely, we are in the regime described by the number of holes of the order M:=O(aβ − 2), the minimum distance between the holes is d ∼ at, and the surface impedance functions of the form λm∼λm,0a−β with β > 0 and λm,0 being constants and eventually complex numbers. Under some natural conditions on the parameters β,t, and λm,0, we characterize the equivalent medium generating approximately the same scattered waves as the original perforated acoustic medium. We give an explicit error estimate between the scattered waves generated by the perforated medium and the equivalent one, respectively, as a→0. As applications of these results, we discuss the following findings: If we choose negative‐valued imaginary surface impedance functions, attached to each surface of the holes, then the equivalent medium behaves as a passive acoustic medium only if it is an acoustic metamaterial with index of refraction $\tilde {n}(x)=‐n(x),\; x \in \upOmega $ñ(x)=−n(x),x∈Ω and $\tilde {n}(x)=1,\; x \in \mathbb {R}^{3}\setminus {\overline {\upOmega }}$ñ(x)=1,x∈R3∖Ω¯. This means that with this process, we can switch the sign of the index of the refraction from positive to negative values. We can choose the surface impedance functions attached to each surface of the holes so that the equivalent index of refraction $\tilde {n}$ñ is $\tilde {n}(x)=1,\; x \in \mathbb {R}^{3}$ñ(x)=1,x∈R3. This means that the region Ω modeled by the original index of refraction n is approximately cloaked. Copyright © 2015 John Wiley & Sons, Ltd.

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a \leq a_{0}, | λ_{m, 0} | \leq λ_{+}, | R (λ_{m, 0}) | \geq λ_{-}, β < 1, s \leq 2 - β, \frac{s}{3} \leq t

then the far‐field pattern

U^{\infty} (\overset{x}{true}, θ)

has the following asymptotic expansion

U^{\infty} (\overset{x}{true}, θ) = V_{n}^{\infty} (\overset{x}{true}, θ) + \sum_{m = 1}^{M} V_{n}^{t} (- \overset{x}{true}, z_{m}) {double-struckQ}_{m} + O ((), a^{3 - s - 2 β}),

uniformly in

\overset{x}{true}

and θ in

{double-struckS}^{2}

. The constant appearing in the estimate O (.)…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…Hence, the total field U t := U i + U s satisfies the following exterior impedance problem of the acoustic waves

(normalΔ + κ^{2} n^{2} (x)) U^{t} = 0 in {double-struckR}^{3} ∖ ((), \cup_{m = 1}^{M} {\overset{D}{true}}_{m}),

{(|, \frac{\partial U^{t}}{\partial ν_{m}} + λ_{m} U^{t})}_{\partial D_{m}} = 0, 1 \leq m \leq M,

\frac{\partial U^{s}}{\partial | x |} - iκ U^{s} = o ((), \frac{1}{| x |}), | x | \to \infty,

U^{s} (x, θ) = \frac{e^{iκ | x |}}{4 π | x |} U^{\infty} (\overset{x}{true}, θ) + O (| x |^{- 2}), | x | \to \infty,

where the function

U^{\infty} (\overset{x}{true}, θ)

for

(\overset{x}{true}, θ) \in {double-struckS}^{2} ...

…”

Section: Statement Of the Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index

Alsaedi

Ahmad

Challa

et al. 2015

Math Methods in App Sciences

Self Cite

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show abstract

“…We have two cases as follows: Under the conditions, on κ , that

τ : = \min_{i \neq j} \cos false(κ false| z_{i} - z_{j} false| false) > 0

and if C m ( κ 0 , λ m ) is positive with

\frac{L_{m} false(κ_{0}, λ_{m} false)}{d}

uniformly bounded, in terms of a , from above by a certain constant, then the above algebraic system is invertible and hence κ is not a quasi‐resonance (or even a resonance), see Challa and Sini …”

Section: Proof Of Theorem 14mentioning

confidence: 99%

Estimation of a class of quasi‐resonances generated by multiple small particles with high surface impedances

Challa

Mouffouk

Sini

2019

Math Methods in App Sciences

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We deal with the stationary acoustic waves propagating in a cluster of small particles enjoying high contrasts. Such contrasts allow the appearance of (complex valued) resonances that are close to the real line as the size of the particles becomes small. For single (but not necessarily small) particles, we derive the characteristic equation that generates a class of these resonances (the ones for which the corresponding eigenfunctions are uniformly constant). For multiple and small particles, we provide sufficient conditions on the contrasts that generates quasi‐resonances for which the corresponding eigenfunctions are uniformly constant. Precisely, we show that, if we distribute the particles on a uniform line, then the existence of such quasi‐resonances is related to the eigenvalues of the Harary matrix. To show these results, we take, as the small contrasted particles, small obstacles with high surface impedances λ of the form λ: = βa−1 − αβa−1 + h where a is the maximum radi of the particles, with a < <1, and β is a universal and positive constant depending only on the shape of the particles (but not on their size). In this case, if the relative constant α is an eigenvalue of the Harary matrix, then the used frequency is a quasi resonance of the cluster of the small particles where the error of approximation is of the order maxfalse(ah,3.0235pta1−hfalse) for h ∈ (0,1) as a < <1.

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Mesoscale Approximation of the Electromagnetic Fields

Bouzekri

Sini

2021

Ann. Henri Poincaré

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Multiscale analysis of the acoustic scattering by many scatterers of impedance type

Cited by 15 publications

References 39 publications

A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index

A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index

Estimation of a class of quasi‐resonances generated by multiple small particles with high surface impedances

Mesoscale Approximation of the Electromagnetic Fields

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