Thin-layer solutions of the Helmholtz equation

Ockendon, J. R.; Tew, R. H.

doi:10.1017/s0956792520000364

Cited by 7 publications

(5 citation statements)

References 25 publications

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“…We have explored the simplest problem of diffraction on non-smooth contours, where the incidence is tangential and the contour curvature changes with jump. We hope that the developed technique will be useful in the study of similar problems with concave-convex transitions, which attract significant attention in the smooth case (see, e.g., [9,10,12,13,14,15,16,17,18]).…”

Section: Discussionmentioning

confidence: 99%

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A.V. Popov's diffraction problem revisited

Zlobina¹,

Kiselev²

2022

Preprint

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In this paper, the high-frequency diffraction of a plane wave incident along a planar boundary turning into a smooth convex contour, so that the curvature undergoes a jump, is asymptotically analysed. An approach modifying the Fock parabolic-equation method is developed. Asymptotic formulas for the wavefield in the illuminated area, shadow, and the penumbra are derived. The penumbral field is characterized by novel and previously unseen special functions that resemble Fock's integrals.

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Section: Discussionmentioning

confidence: 99%

“…There are two reasons for revisiting this problem. First, for half a century, steady interest has been seen in the description of the effects of high-frequency diffraction by contours in which the curvature is neither strictly positive nor strictly negative (see, e.g., [7,8,9,10,11,12,13,14,1,15,16,17,18,19]). A.…”

Section: Introductionmentioning

confidence: 99%

A.V. Popov's diffraction problem revisited

Zlobina¹,

Kiselev²

2022

Preprint

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“…On the convex side, there are no whispering-gallery modes; instead, there are creeping waves and rays [26][27][28][29] propagating close to the mirror while getting weaker as they shed waves tangentially into the space above. The whispering-gallery modes and the creeping waves are well understood, but the transformation of one into the other across an inflection is not [30][31][32].…”

Section: Discussionmentioning

confidence: 99%

Inflection reflection: images in mirrors whose curvature changes sign

Berry

2021

Eur. J. Phys.

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Mirrors that are convex in some places and concave in others can generate images of extended objects (such as the viewer’s face) that are curiously distorted and often topologically disrupted. Understanding these images involves the caustics of the family of rays emitted by each point of the object, and the totality of all such families constituting the rays from all points of the object. The general theory is illustrated by the simplest mirror with an inflection, whose profile is a cubic function. Simulations, and observations with a flexible plastic mirror, show how the image changes as the viewer moves relative to the mirror.

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“…Importantly the point τ = 0, corresponding to t → +∞, remains a regular point for (22) hence with a well-defined (strong) limit of Ũ0 (τ, τ 1 ) as τ → 0+. Using now the transformation (13) in reverse, determines the unitary propagator U 0 (t, t 1 ) for the unperturbed (i.e. with no boundary) problem for (2) with the above described asymptotic behaviour as t → +∞.…”

Section: Further Interpretations Of Theorem 1: Searchlight Wave Opera...mentioning

confidence: 99%

“…It is more convenient to work with an equivalently transformed via (13) problem for G(η, τ ) solving (22), which we re-write as…”

Section: Proof Of Theoremmentioning

confidence: 99%

Searchlight Asymptotics for High-Frequency Scattering by Boundary Inflection

Smyshlyaev¹,

Kamotski²

2021

Preprint

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We consider an inner problem for whispering gallery highfrequency asymptotic mode's scattering by a boundary inflection. The related boundary-value problem for a Schrödinger equation on a halfline with a potential linear in both space and time appears fundamental for describing transitions from modal to scattered asymptotic patterns, and despite having been intensively studied over several decades remains largely unsolved. We prove that the solution past the inflection point has a "searchlight" asymptotics corresponding to a beam concentrated near the limit ray, and establish certain decay and smoothness properties of the related searchlight amplitude. We also discuss further interpretations of the above result: the existence of associated generalised wave operator, and of a version of a unitary scattering operator connecting the modal and scattered asymptotic regimes.

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Thin-layer solutions of the Helmholtz equation

Cited by 7 publications

References 25 publications

A.V. Popov's diffraction problem revisited

A.V. Popov's diffraction problem revisited

Inflection reflection: images in mirrors whose curvature changes sign

Searchlight Asymptotics for High-Frequency Scattering by Boundary Inflection

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