Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-25T06:15:09.589Z Has data issue: false hasContentIssue false
  • English
  • Français

Trapped modes and Fano resonances in two-dimensional acoustical duct–cavity systems

Published online by Cambridge University Press:  05 January 2012

Stefan Hein ,
Werner Koch  and
Lothar Nannen
Stefan Hein
Affiliation:
Institut für Aerodynamik und Strömungstechnik, DLR Göttingen, 37073 Göttingen, Germany
Werner Koch*
Affiliation:
Institut für Aerodynamik und Strömungstechnik, DLR Göttingen, 37073 Göttingen, Germany
Lothar Nannen
Affiliation:
Institut für Numerische und Angewandte Mathematik, Universität Göttingen, 37083 Göttingen, Germany
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Revisiting the classical acoustics problem of rectangular side-branch cavities in a two-dimensional duct of infinite length, we use the finite-element method to numerically compute the acoustic resonances as well as the sound transmission and reflection for an incoming fundamental duct mode. To satisfy the requirement of outgoing waves in the far field, we use two different forms of absorbing boundary conditions, namely the complex scaling method and the Hardy space method. In general, the resonances are damped due to radiation losses, but there also exist various types of localized trapped modes with nominally zero radiation loss. The most common type of trapped mode is antisymmetric about the duct axis and becomes quasi-trapped with very low damping if the symmetry about the duct axis is broken. In this case a Fano resonance results, with resonance and antiresonance features and drastic changes in the sound transmission and reflection coefficients. Two other types of trapped modes, termed embedded trapped modes, result from the interaction of neighbouring modes or Fabry–Pérot interference in multi-cavity systems. These embedded trapped modes occur only for very particular geometry parameters and frequencies and become highly localized quasi-trapped modes as soon as the geometry is perturbed. We show that all three types of trapped modes are possible in duct–cavity systems and that embedded trapped modes continue to exist when a cavity is moved off centre. If several cavities interact, the single-cavity trapped mode splits into several trapped supermodes, which might be useful for the design of low-frequency acoustic filters.

JFM classification

Acoustics: Acoustics Acoustics: Noise control Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 692 , 10 February 2012 , pp. 257 - 287
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Aguilar, J. & Combes, J. M. 1971 A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269–279.Google Scholar
2.Akis, R. & Vasilopoulos, P. 1996 Large photonic band gaps and transmittance antiresonances in periodically modulated quasi-one-dimensional waveguides. Phys. Rev. E 53, 5369–5372.CrossRefGoogle ScholarPubMed
3.Akis, R., Vasilopoulos, P. & Debray, P. 1995 Ballistic transport in electron stub tuners: shape and temperature dependence, tuning of the conductance output, and resonant tunneling. Phys. Rev. B 52, 2805–2813.CrossRefGoogle ScholarPubMed
4.Akis, R., Vasilopoulos, P. & Debray, P. 1997 Bound states and transmission resonances in parabolically confined cross structures: influence of weak magnetic fields. Phys. Rev. B 56, 9594–9602.Google Scholar
5.Alster, M. 1972 Improved calculation of resonant frequencies of Helmholtz resonators. J. Sound Vib. 24 (1), 63–85.Google Scholar
6.Aslanyan, A., Parnovski, L. & Vassiliev, D. 2000 Complex resonances in acoustic waveguides. Q. J. Mech. Appl. Maths 53, 429–447.Google Scholar
7.Baslev, E. & Combes, J. M. 1971 Spectral properties of many body Schrödinger operators with dilation analytic interactions. Commun. Math. Phys. 22, 280–294.Google Scholar
8.Bayer, M., Gutbrod, T., Reithmaier, J. P., Forchel, A., Reinecke, T. L., Knipp, P. A., Dremin, A. A. & Kulakovskii, V. D. 1998 Optical modes in photonic molecules. Phys. Rev. Lett. 81, 2582–2585.CrossRefGoogle Scholar
9.Bérenger, J. P. 1994 A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200.Google Scholar
10.Cattapan, G. & Lotti, P. 2007 Fano resonances in stubbed quantum waveguides with impurities. Eur. Phys. J. B 60, 51–60.Google Scholar
11.Cattapan, G. & Lotti, P. 2008 Bound states in the continuum in two-dimensional serial structures. Eur. Phys. J. B 66, 517–523.CrossRefGoogle Scholar
12.Chanaud, R. C. 1994 Effects of geometry on the resonance frequency of Helmholtz resonators. J. Sound Vib. 178 (3), 337–348.Google Scholar
13.Chew, W. C. & Weedon, W. H. 1994 A 3-D perfectly matched medium from modified Maxwell’s equation with stretched coordinates. Microwave Opt. Technol. Lett. 7 (13), 599–604.CrossRefGoogle Scholar
14.Danglot, J., Carbonell, J., Fernandez, M., Vanbesien, O. & Lippens, D. 1998 Modal analysis of guiding structures patterned in a metallic phononic crystal. Appl. Phys. Lett. 73, 2712–2714.Google Scholar
15.Debray, P., Raichev, O. E., Vasilopoulos, P., Rahman, M., Perrin, R. & Mitchell, W. C. 2000 Ballistic electron transport in stubbed quantum waveguides: experiment and theory. Phys. Rev. B 61, 10950–10958.Google Scholar
16.Den Hartog, J. P. 1947 Mechanical Vibrations. McGraw-Hill.Google Scholar
17.Duan, Y., Koch, W., Linton, C. M. & McIver, M. 2007 Complex resonances and trapped modes in ducted domains. J. Fluid Mech. 571, 119–147.CrossRefGoogle Scholar
18.East, L. F. 1966 Aerodynamically induced resonance in rectangular cavities. J. Sound Vib. 3 (3), 277–287.CrossRefGoogle Scholar
19.El Boudouti, E. H., Mrabti, T., Al-Wahsh, H., Djafari-Rouhani, B., Akjouj, A. & Dobrzynski, L. 2008 Transmission gaps and Fano resonances in an acoustic waveguide: analytical model. J. Phys.: Condens. Matter 20, 255212.Google Scholar
20.Evans, D. V. & Linton, C. M. 1991 Trapped modes in open channels. J. Fluid Mech. 225, 153–175.Google Scholar
21.Fan, S., Yanik, M. F., Wang, Z., Sandhu, S. & Povinelli, L. 2006 Advances in theory of photonic crystals. J. Lightwave Technol. 24, 4493–4501.Google Scholar
22.Fano, U. 1935 Sullo spettro di assorbimento dei gas nobili presso il limite dello spettro d’arco. Nuovo Cimento 12, 154–161.CrossRefGoogle Scholar
23.Fano, U. 1961 Effects of configuration interaction on intensities and phase shifts. Phys. Rev. 124 (6), 1866–1878.CrossRefGoogle Scholar
24.Fellay, A., Gagel, F., Maschke, K., Virlouvet, A. & Khater, A. 1997 Scattering of vibrational waves in perturbed quasi-one-dimensional multichannel waveguides. Phys. Rev. B 55, 1707–1717.Google Scholar
25.Friedrich, H. & Wintgen, D. 1985 Interfering resonances and bound states in the continuum. Phys. Rev. A 32, 3231–3242.Google Scholar
26.González, J. W., Pacheco, M., Rosales, L. & Orellana, P. A. 2010 Bound states in the continuum in graphene quantum dot structures. EPL 91, 66001.Google Scholar
27.Hein, S., Hohage, T. & Koch, W. 2004 On resonances in open systems. J. Fluid Mech. 506, 255–284.Google Scholar
28.Hein, S. & Koch, W. 2008 Acoustic resonances and trapped modes in pipes and tunnels. J. Fluid Mech. 605, 401–428.CrossRefGoogle Scholar
29.Hein, S., Koch, W. & Nannen, L. 2010 Fano resonances in acoustics. J. Fluid Mech. 664, 238–264.Google Scholar
30.Hislop, P. D. & Sigal, I. M. 1996 Introduction to Spectral Theory. Springer.CrossRefGoogle Scholar
31.Hladky-Hennion, A.-C., Vasseur, J., Djafari-Rouhani, B. & de Billy, M. 2008 Sonic band gaps in one-dimensional phononic crystals with a symmetric stub. Phys. Rev. B 77, 104304.CrossRefGoogle Scholar
32.Hohage, T. & Nannen, L. 2009 Hardy space infinite elements for scattering and resonance problems. SIAM J. Numer. Anal. 47 (2), 972–996.CrossRefGoogle Scholar
33.Hurt, N. E. 2000 Mathematical Physics of Quantum Wires and Devices. Kluwer.CrossRefGoogle Scholar
34.Ji, Z. L. 2005 Acoustic length correction of closed cylindrical side-branched tube. J. Sound Vib. 283, 1180–1186.CrossRefGoogle Scholar
35.Jing, X., Wang, X. & Sun, X. 2007 Broadband acoustic liner based on the mechanism of multiple cavity resonance. Presented at 13th AIAA/CEAS Aeroacoustics Conference, Rome, Italy. AIAA Paper 2007-3537.Google Scholar
36.Joe, Y. S., Satanin, M. & Kim, C. S. 2006 Classical analogy of Fano resonances. Phys. Scr. 74, 259–266.Google Scholar
37.Jungowski, W. M., Botros, K. K. & Studzinski, W. 1989 Cylindrical side-branch as tone generator. J. Sound Vib. 131 (2), 265–285.Google Scholar
38.Koch, W. 2005 Acoustic resonances in rectangular open cavities. AIAA J. 43 (11), 2342–2349.Google Scholar
39.Kriesels, P. C., Peters, M. C. A. M., Hirschberg, A., Wijnands, A. P. J., Iafrati, A., Riccardi, G., Piva, R. & Bruggeman, J. C. 1995 High amplitude vortex-induced pulsations in a gas transport system. J. Sound Vib. 184 (2), 343–368.Google Scholar
40.Ladrón de Guevara, M. L., Claro, F. & Orellana, P. A. 2003 Ghost Fano resonance in a double quantum dot molecule attached to leads. Phys. Rev. B 67, 195335.CrossRefGoogle Scholar
41.Linton, C. M. & McIver, M. 1998 Trapped modes in cylindrical waveguides. Q. J. Mech. Appl. Maths 51, 389–412.Google Scholar
42.Linton, C. M. & McIver, P. 2007 Embedded trapped modes in water waves and acoustics. Wave Motion 45, 16–29.Google Scholar
43.Miroshnichenko, A. E., Flach, S. & Kivshar, Y. S. 2010 Fano resonances in nanoscale structures. Rev. Mod. Phys. 82, 2257–2298.Google Scholar
44.Moiseyev, N. 1998 Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling. Phys. Rep. 302, 211–293.CrossRefGoogle Scholar
45.Moiseyev, N. 2009 Suppression of Feshbach resonance widths in two-dimensional waveguides and quantum dots: a lower bound for the number of bound states in the continuum. Phys. Rev. Lett. 102, 167404.Google Scholar
46.Munjal, M. L. 1987 Acoustics of Ducts and Mufflers. John Wiley & Sons.Google Scholar
47.Nannen, L. 2008 Hardy-Raum Methoden zur numerischen Lösung von Streu- und Resonanzproblemen auf unbeschränkten Gebieten. PhD thesis, Georg-August Universität, Göttingen.Google Scholar
48.Nannen, L. & Schädle, A. 2011 Hardy space infinite elements for Helmholtz-type problems with unbounded inhomogeneities. Wave Motion 48, 116–129.CrossRefGoogle Scholar
49.Okołowicz, J., Płoszajczak, M. & Rotter, I. 2003 Dynamics of quantum systems embedded in a continuum. Phys. Rep. 374, 271–383.Google Scholar
50.Ordonez, G. & Kim, S. 2004 Complex collective states in a one-dimensional two-atom system. Phys. Rev. A 70, 032701.Google Scholar
51.Ordonez, G., Na, K. & Kim, S. 2006 Bound states in the continuum in quantum-dot pairs. Phys. Rev. A 73, 022113.Google Scholar
52.Ormondroyd, J. & Den Hartog, J. P. 1928 Theory of the dynamic vibration absorber. Trans. ASME 50, 9–25.Google Scholar
53.Parker, R. 1966 Resonance effects in wake shedding from parallel plates: some experimental observations. J. Sound Vib. 4 (1), 62–72.Google Scholar
54.Persson, A., Rotter, I., Stöckmann, H.-J. & Barth, M. 2000 Observation of resonance trapping in an open microwave cavity. Phys. Rev. Lett. 85, 2478–2481.Google Scholar
55.Petrosky, T. & Subbiah, S. 2003 Electron waveguide as a model of a giant atom with a dressing field. Physica E 19, 230–235.CrossRefGoogle Scholar
56.Pierce, A. D. 1981 Acoustics. McGraw-Hill.Google Scholar
57.Porter, R. & Evans, D. V. 1999 Rayleigh–Bloch surface waves along periodic gratings and their connection with trapped modes in waveguides. J. Fluid Mech. 386, 233–258.CrossRefGoogle Scholar
58.Rotter, S., Libisch, F., Burgdörfer, J., Kuhl, U. & Stöckmann, H.-J. 2004 Tunable Fano resonances in transport through microwave billiards. Phys. Rev. E 69, 046208.CrossRefGoogle ScholarPubMed
59.Sadreev, A. F., Bulgakov, E. N. & Rotter, I. 2006 Bound states in the continuum in open quantum billiards with a variable shape. Phys. Rev. B 73, 235342.Google Scholar
60.Schöberl, J. 1997 NETGEN: an advancing front 2D/3D-mesh generator based on abstract rules. Comput. Vis. Sci. 1, 41–52.Google Scholar
61.Simon, B. 1973 The theory of resonances for dilation analytic potentials and the foundations of time dependent perturbation theory. Ann. Maths 97, 247–274.Google Scholar
62.Singh, S. 2006 Tonal noise attenuation in ducts by optimizing adaptive Helmholtz resonators. Master’s thesis, University of Adelaide.Google Scholar
63.Sols, F., Macucci, M., Ravaioli, U. & Hess, K. 1989 Theory for a quantum modulated transistor. J. Appl. Phys. 66, 3892–3906.CrossRefGoogle Scholar
64.Sugimoto, N. & Imahori, H. 2006 Localized mode of sound in a waveguide with Helmholtz resonators. J. Fluid Mech. 546, 89–111.Google Scholar
65.Tam, C. K. W. 1976 The acoustic modes of a two-dimensional rectangular cavity. J. Sound Vib. 49 (3), 353–364.Google Scholar
66.Tang, P. K. & Sirignano, W. A. 1973 Theory of a generalized Helmholtz resonator. J. Sound Vib. 26 (2), 247–262.Google Scholar
67.Tonon, D., Hirschberg, A., Golliard, J. & Ziada, S. 2011 Aeroacoustics of pipe systems with closed branches. Aeroacoustics 10, 201–276.Google Scholar
68.Venakides, S., Haider, M. A. & Papanicolaou, V. 2000 Boundary integral calculation of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures. SIAM J. Appl. Maths 60, 1686–1706.Google Scholar
69.Wang, X. F., Kushwaha, M. S. & Vasilopoulos, P. 2001 Tunability of acoustic spectral gaps and transmission in periodically stubbed waveguides. Phys. Rev. B 65, 035107.Google Scholar
70.Wiersig, J. 2006 Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities. Phys. Rev. Lett. 97, 253901.Google Scholar
71.Xu, Y., Li, Y., Lee, R. K. & Yariv, A 2000 Scattering-theory analysis of waveguide–resonator coupling. Phys. Rev. E 62, 7389–7404.Google Scholar
72.Ziada, S. & Bühlmann, E. T. 1992 Self-excited resonances of two side-branches in close proximity. J. Fluids Struct. 6, 583–601.Google Scholar
73.Ziada, S. & Shine, S. 1999 Strouhal numbers of flow-excited acoustic resonance of closed side branches. J. Fluids Struct. 13 (1), 127–142.Google Scholar
74.Zworski, M. 1999 Resonances in physics and geometry. Notices Am. Math. Soc. 46 (3), 319–328.Google Scholar