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Anomalies of Acoustic Wave Scattering Near the Cut-off Points of Continuous Spectrum (A Review)

  • CLASSICAL PROBLEMS OF LINEAR ACOUSTICS AND WAVE THEORY
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Abstract

Several anomalies of wave scattering that occur in acoustic waveguides with cylindrical or corrugated rigid walls at frequencies close to the cut-off points (thresholds) of continuous spectrum are considered. The notion of threshold resonances generated by “almost standing waves”, which cause no energy transfer to infinity, is introduced. For corrugated waveguides, examples are presented to illustrate the opening of spectral gaps (wave stopping zones) and eigenvalues near their edges and common or degenerate thresholds. Weinstein’s and Wood’s anomalies are described, which occur above and below the thresholds and manifest themselves in “almost complete” reflection and transmission of waves, and in disproportionally fast variation of the diffraction pattern, respectively. Examples of complete wave transmission (“invisibility of obstacle”) are discussed along with the procedures of sharpening and smoothing of Wood’s anomalies, specifically, formation of eigenvalues embedded into the continuous spectrum and corresponding trapped waves. The Sommerfeld, Umov–Mandelshtam, and limiting absorption principles are compared along with the specific features of their application at thresholds.

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Notes

  1. The term is conventionally used for waveguides with soft walls generating Dirichlet boundary conditions.

  2. In this paper, elastic waveguides are understood as waveguides described by a two- or three-dimensional system of equations of elasticity theory.

  3. It differs from the continuous spectrum by the set of EVs of infinite multiplicity, which are absent in cylindrical waveguides.

  4. Point \(\eta = - \pi \) corresponds to the same Floquet wave as point \(\eta = \pi \) does, and, therefore, the first of them is omitted.

  5. Sequence (3) of EF for the problem of an isolated container.

  6. Criterion of wave trapping was proposed in [74] for quantum waveguides. It may be adjusted to acoustic waveguides, but requires a large set of new notation.

  7. One more case \({{J}_{m}}(H) = 0\) allows many possibilities, including the retention of TR; they are considered in detail in [83] and omitted below.

  8. Its construction will be described in the journal named Sbornik Mathematics.

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Funding

This work was supported by the Russian Scientific Foundation (project no. 17-11-01003).

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  1. St. Petersburg State University, 199034, St. Petersburg, Russia

    S. A. Nazarov

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Translated by E. Golyamina

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Nazarov, S.A. Anomalies of Acoustic Wave Scattering Near the Cut-off Points of Continuous Spectrum (A Review). Acoust. Phys. 66, 477–494 (2020). https://doi.org/10.1134/S1063771020050115

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