Journal of Engineering Mathematics

, Volume 65, Issue 4, pp 345–354 | Cite as

Low-frequency acoustic reflection at a hard–soft lining transition in a cylindrical duct with uniform flow

  • E. J. Brambley


The low-frequency limit of the reflection coefficient for downstream-propagating sound in a cylindrical duct with uniform mean flow at a sudden hard–soft wall impedance transition is considered. The scattering at such a transition for arbitrary frequency was analysed by Rienstra (2007, J Eng Maths 59:451–475), who, having derived an exact analytic solution, also considered the plane-wave reflection coefficient, R 011, in the low-frequency limit, and it is this result that is reconsidered here. This reflection coefficient was shown to be significantly different with or without the application of a Kutta-like condition and the corresponding inclusion or exclusion of an instability wave over the impedance wall, assuming an impedance independent of frequency. This analysis is here rederived for a frequency-dependent locally-reacting impedance, and a dramatic difference is seen. In particular, the Kutta condition is shown to have no effect on R 011 in the low-frequency limit for impedances with Z(ω) ~ −ib/ω for some b > 0 as ω → 0, which includes the mass–spring–damper and Helmholtz resonator impedances, although, interestingly, not the enhanced Helmholtz resonator model. This casts doubt on the usefulness of the low-frequency plane-wave reflection coefficient as an experimental test for the presence of instability waves over the surface of impedance linings. The plane-wave reflection coefficient is also derived in the low-frequency limit for a thin shell boundary, based on the scattering analysis of Brambley and Peake (2008, J Fluid Mech 602:403–426), who suggested the model as a well-posed regularization of the mass–spring–damper impedance. The result might be interpretable as evidence for the nonexistence of an instability over an acoustic lining.


Acoustic lining Impedance boundary Low-frequency asymptotics Myers’ boundary condition Scattering 


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  1. 1.
    Rienstra WS (2007) Acoustic scattering at a hard–soft lining transition in a flow duct. J Eng Math 59: 451–475MATHCrossRefGoogle Scholar
  2. 2.
    Eversman W, Beckemeyer RJ (1972) Transmission of sound in ducts with thin shear layers—Convergence to the uniform flow case. J Acoust Soc Am 2: 216–220CrossRefGoogle Scholar
  3. 3.
    Myers MK (1980) On the acoustic boundary condition in the presence of flow. J Sound Vib 71: 429–434MATHCrossRefADSGoogle Scholar
  4. 4.
    Rienstra WS (2003) A classification of duct modes based on surface waves. Wave Motion 37: 119–135MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Rienstra SW (2006) Impedance models in time domain, including the extended Helmholtz resonator model. AIAA 2006-2686Google Scholar
  6. 6.
    Brambley EJ, Peake N (2006) Classification of aeroacoustically relevant surface modes in cylindrical lined ducts. Wave Motion 43: 301–310CrossRefMathSciNetGoogle Scholar
  7. 7.
    Dickey NS, Selamet A, Ciray MS (2001) An experimental study of the impedance of perforated plates with grazing flow. J Acoust Soc Am 110: 2360–2370CrossRefADSGoogle Scholar
  8. 8.
    Aurégan Y, Leroux M (2003) Failures in the discrete models for flow duct with perforations: an experimental investigation. J Sound Vib 265: 109–121CrossRefADSGoogle Scholar
  9. 9.
    Jones MG, Watson WR, Parrott TL (2005) Benchmark data for evaluation of aeroacoustic propagation codes with grazing flow. AIAA 2005-2853Google Scholar
  10. 10.
    Noble B (1958) Methods based on the Wiener–Hopf technique for the solution of partial differential equations. Pergamon Press, New YorkMATHGoogle Scholar
  11. 11.
    Koiter TW (1954) Approximate solution of Wiener–Hopf type integral equations with applications. I. General theory. Proc K Ned Akad Wet B 57: 558–564MathSciNetGoogle Scholar
  12. 12.
    Abramowitz M, Stegun IA (1964) Handbook of mathematical functions, 9th edn. Dover, NYMATHGoogle Scholar
  13. 13.
    Brambley EJ, Peake N (2008) Stability and acoustic scattering in a cylindrical thin shell containing compressible mean flow. J Fluid Mech 602: 403–426MATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Crighton DG (2001) Asymptotic factorization of Wiener–Hopf kernels. Wave Motion 33: 51–65MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK

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