Abstract
We study the propagation of waves in a waveguide which is the union of two semi-infinite cylinders connected by a thin rectangular ligament. It is shown that almost complete or even complete transmission of the piston mode at a prescribed frequency can be achieved via fine tuning of the plate sizes, although, of course, almost complete wave reflection occurs in the generic case. The result is obtained by applying an asymptotic analysis of the scattering coefficients of the acoustic wave, in particular, by using the dimension reduction procedure on the thin ligament. Possible generalizations of the problem formulation and related open questions are discussed.
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REFERENCES
R. Mittra and S. W. Lee, Analytical Techniques in the Theory of Guided Waves (Macmillan, New York, 1971).
S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries (Walter de Gruyter, Berlin, 1994).
L. A. Weinstein, The Theory of Diffraction and the Factorization Method (Sovetskoe Radio, Moscow, 1966; Golem, Boulder, Colo., 1969).
A. V. Shanin, “Weinstein’s diffraction problem: Embedding formula and spectral equation in parabolic approximation,” SIAM J. Appl. Math. 70, 1201–1218 (2009).
S. A. Nazarov, “Scattering anomalies in a resonator above the thresholds of the continuous spectrum,” Sb. Math. 206 (6), 782–813 (2015).
A. I. Korolkov, S. A. Nazarov, and A. V. Shanin, “Stabilizing solutions at thresholds of the continuous spectrum and anomalous transmission of waves,” Z. Angew. Math. Mech. 96 (10), 1245–1260 (2016).
A. V. Shanin and A. I. Korolkov, “Diffraction of a mode close to its cut-off by a transversal screen in a planar waveguide,” Wave Motion 68, 218–241 (2017).
A. I. Korolkov and A. V. Shanin, “Diffraction by a grating consisting of absorbing screens of different height: New equations,” J. Math. Sci. 206, 270–287 (2015).
S. A. Nazarov, “Transmission of waves through a small aperture in the cross-wall in an acoustic waveguide,” Sib. Math. J. 59 (1), 85–101 (2018).
S. Molchanov and B. Vainberg, “Scattering solutions in networks of thin fibers: Small diameter asymptotics,” Commun. Math. Phys. 273 (2), 533–559 (2007).
D. Grieser, “Spectra of graph neighborhoods and scattering,” Proc. London Math. Soc. 97, 718–752 (2008).
K. Pankrashkin, “Eigenvalue inequalities and absence of threshold resonances for waveguide junctions,” J. Math. Anal. Appl. 449 (1), 907–925 (2017).
U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124 (6), 1866–1878 (1961).
Y. Duan, W. Koch, C. M. Linton, and M. McIver, “Complex resonances and trapped modes in ducted domains,” J. Fluid Mech. 571, 119–147 (2007).
G. Cattapan and P. Lotti, “Fano resonances in stubbed quantum waveguides with impurities,” Eur. Phys. J. B 60 (1), 51–60 (2007).
E. H. El Boudouti, T. Mrabti, H. Al-Wahsh, B. Djafari-Rouhani, A. Akjouj, and L. Dobrzynski, “Transmission gaps and Fano resonances in an acoustic waveguide: Analytical model,” J. Phys. Condens. Matter 20 (25), 255212 (2008).
T. Hohage and L. Nannen, “Hardy space infinite elements for scattering and resonance problems,” SIAM J. Numer. Anal. 47 (2), 972–996 (2009).
S. Hein, W. Koch, and L. Nannen, “Trapped modes and Fano resonances in two-dimensional acoustical duct-cavity systems,” J. Fluid Mech. 692, 257–287 (2012).
S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71 (2), 026611 (2005).
S. P. Shipman and H. Tu, “Total resonant transmission and reflection by periodic structures,” SIAM J. Appl. Math. 72 (1), 216–239 (2012).
S. P. Shipman and A. T. Welters, “Resonant electromagnetic scattering in anisotropic layered media,” J. Math. Phys. 54 (10), 103511 (2013).
G. S. Abeynanda and S. P. Shipman, “Dynamic resonance in the high-Q and near-monochromatic regime,” MMET, IEEE, 10.1109, MMET, 7544100 (2016).
L. Chesnel and S. A. Nazarov, “Non reflection and perfect reflection via Fano resonance in waveguides,” Commun. Math. Sci. 16 (7), 1779–1800 (2018).
G. A. Kriegsmann, “Complete transmission through a two-dimensional diffraction grating,” SIAM J. Appl. Math. 65 (1), 24–42 (2004).
É. Bonnetier and F. Triki, “Asymptotic of the Green function for the diffraction by a perfectly conducting plane perturbed by a sub-wavelength rectangular cavity,” Math. Methods Appl. Sci. 33 (6), 772–798 (2010).
J. Lin and H. Zhang, “Scattering and field enhancement of a perfect conducting narrow slit,” SIAM J. Appl. Math. 77 (3), 951–976 (2017).
J. Lin and H. Zhang, “Scattering by a periodic array of subwavelength slits I: Field enhancement in the diffraction regime,” Multiscale Model. Simul. 16 (2), 922–953 (2018).
J. Lin, S. Shipman, and H. Zhang, “A mathematical theory for Fano resonance in a periodic array of narrow slits,” arXiv:1904.11019 (2019).
J. T. Beale, “Scattering frequencies of resonators,” Commun. Pure Appl. Math. 26 (4), 549–563 (1973).
A. A. Arsen’ev, “The existence of resonance poles and scattering resonances in the case of boundary conditions of the second and third kind,” USSR Comput. Math. Math. Phys. 16 (3), 171–177 (1976).
R. R. Gadyl’shin, “Characteristic frequencies of bodies with thin spikes: I. Convergence and estimates,” Math. Notes 54 (6), 1192–1199 (1993).
V. A. Kozlov, V. G. Maz’ya, and A. B. Movchan, “Asymptotic analysis of a mixed boundary value problem in a multi-structure,” Asymptotic Anal. 8 (2), 105–143 (1994).
S. A. Nazarov, “Junctions of singularly degenerating domains with different limit dimensions 1,” J. Math. Sci. 80, 1989–2034 (1996).
S. A. Nazarov, “Junctions of singularly degenerating domains with different limit dimensions 2,” J. Math. Sci. 97, 155–195 (1999).
S. A. Nazarov, “Asymptotic analysis and modeling of the junction of a massive body with thin rods,” J. Math. Sci. 127, 2172–2263 (2003).
R. R. Gadyl’shin, “On the eigenvalues of a ‘dumb-bell with a thin handle’”, Izv. Math. 69 (2), 265–329 (2005).
P. Joly and S. Tordeux, “Matching of asymptotic expansions for wave propagation in media with thin slots: I. The asymptotic expansion,” SIAM Multiscale Model. Simul. 5 (1), 304–336 (2006).
F. L. Bakharev and S. A. Nazarov, “Gaps in the spectrum of a waveguide composed of domains with different limiting dimensions,” Sib. Math. J. 56 (4), 575–592 (2015).
A.-S. Bonnet-Ben Dhia, L. Chesnel, and S. A. Nazarov, “Perfect transmission invisibility for waveguides with sound hard walls,” J. Math. Pures Appl. 111, 79–105 (2018).
L. Chesnel, S. A. Nazarov, and J. Taskinen, “Surface waves in a channel with thin tunnels at the bottom: Non-reflecting underwater topography,” Asymptotic Anal. 118 (1), 81–122 (2020).
M. Van Dyke, Perturbation Methods in Fluid Mechanics (Academic, New York, 1964).
A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Nauka, Moscow, 1989; Am. Math. Soc., RI, Providence, 1992).
V. Maz’ya, S. Nasarov, and B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains (Birkhäuser, Basel, 2000), Vols. 1, 2.
V. S. Vladimirov, Generalized Functions in Mathematical Physics (Nauka, Moscow, 1979; Mir, Moscow, 1979).
O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973; Springer-Verlag, New York, 1985).
V. A. Kondrat’ev, “Boundary value problems for elliptic equations in domains with conical or corner points,” Tr. Mosk. Mat. O–va 16, 219–292 (1963).
V. A. Kozlov, V. G. Maz’ya, and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities (Am. Math. Soc., Providence, 1997).
S. A. Nazarov, “The Neumann problem in angular domains with periodic boundaries and parabolic perturbations of the boundaries,” Trans. Moscow Math. Soc. 67, 153–208 (2007).
S. A. Nazarov, “Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide,” Theor. Math. Phys. 167 (2), 606–627 (2011).
S. A. Nazarov, “Enforced stability of a simple eigenvalue in the continuous spectrum of a waveguide,” Funct. Anal. Appl. 47 (3), 195–209 (2013).
L. C hesnel, S. A. Nazarov, and V. Pagneux, “Invisibility and perfect reflectivity in waveguides with finite length branches,” SIAM J. Appl. Math. 78 (4), 2176–2199 (2018).
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This work was supported by the Russian Science Foundation, project no. 17-11-01003.
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Translated by I. Ruzanova
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Nazarov, S.A., Chesnel, L. Anomalies of Acoustic Wave Propagation in Two Semi-Infinite Cylinders Connected by a Flattened Ligament. Comput. Math. and Math. Phys. 61, 646–663 (2021). https://doi.org/10.1134/S0965542521040096
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DOI: https://doi.org/10.1134/S0965542521040096