Full low‐frequency asymptotic expansion for second‐order elliptic equations in two dimensions

Kleinman, R. E.; Vainberg, Boris

doi:10.1002/mma.1670171207

Cited by 20 publications

(12 citation statements)

References 15 publications

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“…The problem of scattering by a small sound-soft cylinder is a problem of low-frequency asymptotics. Using the general theory of Kleinman and Vainberg [14], we find that (2.5) where the origin is inside the cylinder's cross section S and…”

Section: Foldy's Methodmentioning

confidence: 99%

Semi-Infinite Arrays of Isotropic Point Scatterers. A Unified Approach

Linton¹,

Martin²

2004

SIAM J. Appl. Math.

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Abstract. We solve the two-dimensional problem of acoustic scattering by a semi-infinite periodic array of identical isotropic point scatterers, i.e., objects whose size is negligible compared to the incident wavelength and which are assumed to scatter incident waves uniformly in all directions. This model is appropriate for scatterers on which Dirichlet boundary conditions are applied in the limit as the ratio of wavelength to body size tends to infinity. The problem is also relevant to the scattering of an E-polarized electromagnetic wave by an array of highly conducting wires. The actual geometry of each scatterer is characterized by a single parameter in the equations, related to the single-body scattering problem and determined from a harmonic boundary-value problem. Using a mixture of analytical and numerical techniques, we confirm that a number of phenomena reported for specific geometries are in fact present in the general case (such as the presence of shadow boundaries in the far field and the vanishing of the circular wave scattered by the end of the array in certain specific directions). We show that the semi-infinite array problem is equivalent to that of inverting an infinite Toeplitz matrix, which in turn can be formulated as a discrete Wiener-Hopf problem. Numerical results are presented which compare the amplitude of the wave diffracted by the end of the array for scatterers having different shapes.

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Section: Foldy's Methodmentioning

confidence: 99%

Semi-Infinite Arrays of Isotropic Point Scatterers. A Unified Approach

Linton¹,

Martin²

2004

SIAM J. Appl. Math.

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“…(29) is clearly suitable for a perturbative calculation of T ± (p) and hence the scattering amplitude (30). Next, we recall that according to (15) and (16),…”

Section: Transfer-matrix In Two Dimensionsmentioning

confidence: 99%

Exact solution of the two-dimensional scattering problem for a class of δ-function potentials supported on subsets of a line

Loran¹,

Mostafazadeh²

2018

J. Phys. A: Math. Theor.

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We use the transfer matrix formulation of scattering theory in two-dimensions to treat the scattering problem for a potential of the form v(x, y) = ζ δ(ax + by)g(bx − ay) where ζ, a, and b are constants, δ(x) is the Dirac δ function, and g is a real-or complex-valued function. We map this problem to that of v(x, y) = ζ δ(x)g(y) and give its exact and analytic solution for the following choices of g(y): i) A linear combination of δ-functions, in which case v(x, y) is a finite linear array of two-dimensional δ-functions; ii) A linear combination of e iαny with α n real; iii) A general periodic function that has the form of a complex Fourier series. In particular we solve the scattering problem for a potential consisting of an infinite linear periodic array of two-dimensional δ-functions. We also prove a general theorem that gives a sufficient condition for different choices of g(y) to produce the same scattering amplitude within specific ranges of values of the wavelength λ. For example, we show that for arbitrary real and complex parameters, a and z, the potentials z ∞ n=−∞ δ(x)δ(y − an) and a −1 zδ(x)[1 + 2 cos(2πy/a)] have the same scattering amplitude for a < λ ≤ 2a.

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“…But the integers a and t' and the polynomial P of Theorem 1.1(9) was not known, even for equations of second order. Recently, Kieinmann and Vainberg [4] obtained the complete asymptotic expansion in the case that A(x,a) coincides with the Laplace operator in some neighbourhood of infinity. We apply the idea of [4] to the system case.…”

Section: B(xa)u = Bn(xa)u = C Vi(x)a"(x)aju I J = Lmentioning

confidence: 99%

On the Low-frequency Asymptotic Expansion for some Second-order Elliptic Systems in a Two-dimensional Exterior Domain

Dan

1996

Math. Meth. Appl. Sci.

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Full low‐frequency asymptotic expansion for second‐order elliptic equations in two dimensions

Cited by 20 publications

References 15 publications

Semi-Infinite Arrays of Isotropic Point Scatterers. A Unified Approach

Semi-Infinite Arrays of Isotropic Point Scatterers. A Unified Approach

Exact solution of the two-dimensional scattering problem for a class of δ-function potentials supported on subsets of a line

On the Low-frequency Asymptotic Expansion for some Second-order Elliptic Systems in a Two-dimensional Exterior Domain

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