Skip to main content
×
×
Home

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

  • Stefan Hein (a1), Werner Koch (a1) and Lothar Nannen (a2)
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.

Copyright
Corresponding author
Email address for correspondence: werner.koch@dlr.de
References
Hide All
1.Aguilar, J. & Combes, J. M. 1971 A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269279.
2.Akis, R. & Vasilopoulos, P. 1996 Large photonic band gaps and transmittance antiresonances in periodically modulated quasi-one-dimensional waveguides. Phys. Rev. E 53, 53695372.
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, 28052813.
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, 95949602.
5.Alster, M. 1972 Improved calculation of resonant frequencies of Helmholtz resonators. J. Sound Vib. 24 (1), 6385.
6.Aslanyan, A., Parnovski, L. & Vassiliev, D. 2000 Complex resonances in acoustic waveguides. Q. J. Mech. Appl. Maths 53, 429447.
7.Baslev, E. & Combes, J. M. 1971 Spectral properties of many body Schrödinger operators with dilation analytic interactions. Commun. Math. Phys. 22, 280294.
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, 25822585.
9.Bérenger, J. P. 1994 A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185200.
10.Cattapan, G. & Lotti, P. 2007 Fano resonances in stubbed quantum waveguides with impurities. Eur. Phys. J. B 60, 5160.
11.Cattapan, G. & Lotti, P. 2008 Bound states in the continuum in two-dimensional serial structures. Eur. Phys. J. B 66, 517523.
12.Chanaud, R. C. 1994 Effects of geometry on the resonance frequency of Helmholtz resonators. J. Sound Vib. 178 (3), 337348.
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), 599604.
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, 27122714.
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, 1095010958.
16.Den Hartog, J. P. 1947 Mechanical Vibrations. McGraw-Hill.
17.Duan, Y., Koch, W., Linton, C. M. & McIver, M. 2007 Complex resonances and trapped modes in ducted domains. J. Fluid Mech. 571, 119147.
18.East, L. F. 1966 Aerodynamically induced resonance in rectangular cavities. J. Sound Vib. 3 (3), 277287.
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.
20.Evans, D. V. & Linton, C. M. 1991 Trapped modes in open channels. J. Fluid Mech. 225, 153175.
21.Fan, S., Yanik, M. F., Wang, Z., Sandhu, S. & Povinelli, L. 2006 Advances in theory of photonic crystals. J. Lightwave Technol. 24, 44934501.
22.Fano, U. 1935 Sullo spettro di assorbimento dei gas nobili presso il limite dello spettro d’arco. Nuovo Cimento 12, 154161.
23.Fano, U. 1961 Effects of configuration interaction on intensities and phase shifts. Phys. Rev. 124 (6), 18661878.
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, 17071717.
25.Friedrich, H. & Wintgen, D. 1985 Interfering resonances and bound states in the continuum. Phys. Rev. A 32, 32313242.
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.
27.Hein, S., Hohage, T. & Koch, W. 2004 On resonances in open systems. J. Fluid Mech. 506, 255284.
28.Hein, S. & Koch, W. 2008 Acoustic resonances and trapped modes in pipes and tunnels. J. Fluid Mech. 605, 401428.
29.Hein, S., Koch, W. & Nannen, L. 2010 Fano resonances in acoustics. J. Fluid Mech. 664, 238264.
30.Hislop, P. D. & Sigal, I. M. 1996 Introduction to Spectral Theory. Springer.
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.
32.Hohage, T. & Nannen, L. 2009 Hardy space infinite elements for scattering and resonance problems. SIAM J. Numer. Anal. 47 (2), 972996.
33.Hurt, N. E. 2000 Mathematical Physics of Quantum Wires and Devices. Kluwer.
34.Ji, Z. L. 2005 Acoustic length correction of closed cylindrical side-branched tube. J. Sound Vib. 283, 11801186.
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.
36.Joe, Y. S., Satanin, M. & Kim, C. S. 2006 Classical analogy of Fano resonances. Phys. Scr. 74, 259266.
37.Jungowski, W. M., Botros, K. K. & Studzinski, W. 1989 Cylindrical side-branch as tone generator. J. Sound Vib. 131 (2), 265285.
38.Koch, W. 2005 Acoustic resonances in rectangular open cavities. AIAA J. 43 (11), 23422349.
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), 343368.
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.
41.Linton, C. M. & McIver, M. 1998 Trapped modes in cylindrical waveguides. Q. J. Mech. Appl. Maths 51, 389412.
42.Linton, C. M. & McIver, P. 2007 Embedded trapped modes in water waves and acoustics. Wave Motion 45, 1629.
43.Miroshnichenko, A. E., Flach, S. & Kivshar, Y. S. 2010 Fano resonances in nanoscale structures. Rev. Mod. Phys. 82, 22572298.
44.Moiseyev, N. 1998 Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling. Phys. Rep. 302, 211293.
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.
46.Munjal, M. L. 1987 Acoustics of Ducts and Mufflers. John Wiley & Sons.
47.Nannen, L. 2008 Hardy-Raum Methoden zur numerischen Lösung von Streu- und Resonanzproblemen auf unbeschränkten Gebieten. PhD thesis, Georg-August Universität, Göttingen.
48.Nannen, L. & Schädle, A. 2011 Hardy space infinite elements for Helmholtz-type problems with unbounded inhomogeneities. Wave Motion 48, 116129.
49.Okołowicz, J., Płoszajczak, M. & Rotter, I. 2003 Dynamics of quantum systems embedded in a continuum. Phys. Rep. 374, 271383.
50.Ordonez, G. & Kim, S. 2004 Complex collective states in a one-dimensional two-atom system. Phys. Rev. A 70, 032701.
51.Ordonez, G., Na, K. & Kim, S. 2006 Bound states in the continuum in quantum-dot pairs. Phys. Rev. A 73, 022113.
52.Ormondroyd, J. & Den Hartog, J. P. 1928 Theory of the dynamic vibration absorber. Trans. ASME 50, 925.
53.Parker, R. 1966 Resonance effects in wake shedding from parallel plates: some experimental observations. J. Sound Vib. 4 (1), 6272.
54.Persson, A., Rotter, I., Stöckmann, H.-J. & Barth, M. 2000 Observation of resonance trapping in an open microwave cavity. Phys. Rev. Lett. 85, 24782481.
55.Petrosky, T. & Subbiah, S. 2003 Electron waveguide as a model of a giant atom with a dressing field. Physica E 19, 230235.
56.Pierce, A. D. 1981 Acoustics. McGraw-Hill.
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, 233258.
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.
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.
60.Schöberl, J. 1997 NETGEN: an advancing front 2D/3D-mesh generator based on abstract rules. Comput. Vis. Sci. 1, 4152.
61.Simon, B. 1973 The theory of resonances for dilation analytic potentials and the foundations of time dependent perturbation theory. Ann. Maths 97, 247274.
62.Singh, S. 2006 Tonal noise attenuation in ducts by optimizing adaptive Helmholtz resonators. Master’s thesis, University of Adelaide.
63.Sols, F., Macucci, M., Ravaioli, U. & Hess, K. 1989 Theory for a quantum modulated transistor. J. Appl. Phys. 66, 38923906.
64.Sugimoto, N. & Imahori, H. 2006 Localized mode of sound in a waveguide with Helmholtz resonators. J. Fluid Mech. 546, 89111.
65.Tam, C. K. W. 1976 The acoustic modes of a two-dimensional rectangular cavity. J. Sound Vib. 49 (3), 353364.
66.Tang, P. K. & Sirignano, W. A. 1973 Theory of a generalized Helmholtz resonator. J. Sound Vib. 26 (2), 247262.
67.Tonon, D., Hirschberg, A., Golliard, J. & Ziada, S. 2011 Aeroacoustics of pipe systems with closed branches. Aeroacoustics 10, 201276.
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, 16861706.
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.
70.Wiersig, J. 2006 Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities. Phys. Rev. Lett. 97, 253901.
71.Xu, Y., Li, Y., Lee, R. K. & Yariv, A 2000 Scattering-theory analysis of waveguide–resonator coupling. Phys. Rev. E 62, 73897404.
72.Ziada, S. & Bühlmann, E. T. 1992 Self-excited resonances of two side-branches in close proximity. J. Fluids Struct. 6, 583601.
73.Ziada, S. & Shine, S. 1999 Strouhal numbers of flow-excited acoustic resonance of closed side branches. J. Fluids Struct. 13 (1), 127142.
74.Zworski, M. 1999 Resonances in physics and geometry. Notices Am. Math. Soc. 46 (3), 319328.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed