Journal of Engineering Mathematics

, Volume 59, Issue 4, pp 451–475 | Cite as

Acoustic scattering at a hard–soft lining transition in a flow duct

  • Sjoerd W. Rienstra
Open Access


An explicit Wiener–Hopf solution is derived to describe the scattering of sound at a hard–soft wall impedance transition at x = 0, say, in a circular duct with uniform mean flow of Mach number M. A mode, incident from the upstream hard section, scatters at x = 0 into a series of reflected modes and a series of transmitted modes. Of particular interest is the role of a possible instability along the lined wall in combination with the edge singularity. If one of the “upstream” running modes is to be interpreted as a downstream-running instability, an extra degree of freedom in the Wiener–Hopf analysis occurs that can be resolved by application of some form of Kutta condition at x = 0, for example a more stringent edge condition where wall streamline deflection \(h={\mathcal{O}}(x^{3/2})\) at the downstream side. In general, the effect of this Kutta condition is significant, but it is particularly large for the plane wave at low frequencies and should therefore be easily measurable. For small Helmholtz numbers, the reflection coefficient modulus |R 001| tends to (1 + M)/(1−M) without and to 1 with Kutta condition, while the end correction tends to ∞ without and to a finite value with Kutta condition. This is exactly the same behaviour as found for reflection at a pipe exit with flow, irrespective if this is uniform or jet flow. Although the presence of the instability in the model is hardly a question anymore since it has been confirmed numerically, a proper mathematical causality analysis is still not totally watertight. Therefore, the limit of a vortex sheet, separating zero flow from mean flow, approaching the wall has been explored. Indeed, this confirms that the Helmholtz unstable mode of the free vortex sheet transforms into the suspected mode and remains unstable. As the lined-wall vortex-sheet model predicts unstable behaviour for which experimental evidence is at best rare and indirect, the question may be raised if this model is indeed a consistent simplification of reality, doing justice to the double limit of small perturbations and a thin boundary layer. Numerical time-domain methods suffer from this instability and it is very important to decide whether the instability is at least physically genuine. Experiments based on the present problem may provide a handle to resolve this stubborn question.


Aircraft noise Duct acoustics Impedance models Wiener–Hopf method 


  1. 1.
    Crighton DG and Pedley TJ (1999). Michael James Lighthill (1924–1998). Notices AMS 46(1): 1226–1229 MathSciNetGoogle Scholar
  2. 2.
    Pedley TJ (2001). James Lighthill and his contributions to fluid mechanics. Annu Rev Fluid Mech 33: 1–41 CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Lighthill MJ (1952). On sound generated aerodynamically, I. General theory. Proc R Soc Lond A 211: 564–587 MATHADSMathSciNetGoogle Scholar
  4. 4.
    Crighton DG (1981). Acoustics as a branch of fluid mechanics. J Fluid Mech 106: 261–298 MATHCrossRefADSGoogle Scholar
  5. 5.
    Lighthill MJ (1954). On sound generated aerodynamically II. Turbulence as a source of sound. Proc R Soc Lond A 222: 1–32 MATHADSMathSciNetGoogle Scholar
  6. 6.
    Lighthill MJ (1962). Sound generated aerodynamically, the Bakerian Lecture 1961. Proc R Soc Lond A 267: 147–182 MATHADSCrossRefGoogle Scholar
  7. 7.
    Lighthill MJ (1993). A general introduction to aeroacoustics and atmospheric sound. In: Hardin, JC and Hussaini, MY (eds) Computational Aeroacoustics, pp. Springer-Verlag, New York Google Scholar
  8. 8.
    Stein RF (1967). Generation of acoustic and gravity waves by turbulence in an isothermal stratified atmosphere. Solar Phys 2: 385–432 CrossRefADSGoogle Scholar
  9. 9.
    Crighton DG (1975). Basic principles of aerodynamic noise generation. Prog Aerosp Sci 16: 13–96 CrossRefGoogle Scholar
  10. 10.
    Howe MS (2001). Vorticity and the theory of aerodynamic sound. J Eng Math 41(4): 367–400 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Crighton DG (1988) Aeronautical acoustics: mathematics applied to a major industrial problem. In: McKenna J, Temam R (eds) Proceedings of the first international conference on industrial and applied mathematics ICIAM’87. SIAM, Philadelphia, pp 75–89Google Scholar
  12. 12.
    Smith MJT (1989) Aircraft noise. Cambridge University PressGoogle Scholar
  13. 13.
    Lighthill MJ (1972). The propagation of sound through moving fluids, the fourth annual fairey lecture. J Sound Vibration 24: 472–492 CrossRefADSGoogle Scholar
  14. 14.
    Swinbanks MA (1975). The sound field generated by a source distribution in a long duct carrying sheared flow. J Sound Vibration 40(1): 51–76 CrossRefADSGoogle Scholar
  15. 15.
    Lighthill MJ (1965). Group velocity. J Inst Math Appl 1: 1–28 CrossRefMathSciNetGoogle Scholar
  16. 16.
    Lighthill MJ (1960). Studies on magneto-hydrodynamic waves and other anisotropic wave motions. Phil Trans R Soc Lond A 252: 397–430 CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Lighthill MJ (1978) Waves in fluids. Cambridge University PressGoogle Scholar
  18. 18.
    Rienstra SW (1999). Sound transmission in slowly varying circular and annular lined ducts with flow. J Fluid Mech 380: 279–296 MATHCrossRefADSGoogle Scholar
  19. 19.
    Rienstra SW (2003). Sound propagation in slowly varying lined flow ducts of arbitrary cross-section. J Fluid Mech 495: 157–173 MATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Rademaker ER (1990) Experimental validation of a lined-duct acoustics model including flow. Presented at ASME conference on duct acoustics, Dallas, TX, Nov. 1990Google Scholar
  21. 21.
    Ingard KU (1959). Influence of fluid motion past a plane boundary on sound reflection, absorption and transmission. J Acoust Soc Am 31(7): 1035–1036 CrossRefADSGoogle Scholar
  22. 22.
    Myers MK (1980). On the acoustic boundary condition in the presence of flow. J Sound Vibration 71(3): 429–434 MATHCrossRefADSGoogle Scholar
  23. 23.
    Eversman W and Beckemeyer RJ (1972). Transmission of sound in ducts with thin shear layers—convergence to the uniform flow case. J Acoust Soc Am 52(1): 216–220 CrossRefMATHADSGoogle Scholar
  24. 24.
    Rienstra SW (2003). A classification of duct modes based on surface waves. Wave Motion 37(2): 119–135 MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Tester BJ (1973). The propagation and attenuation of sound in ducts containing uniform or ‘Plug’ flow. J Sound Vibration 28(2): 151–203 CrossRefMATHADSGoogle Scholar
  26. 26.
    Bers A, Briggs RJ (1963) MIT Research Laboratory of Electronics Report No. 71 (unpublished)Google Scholar
  27. 27.
    Briggs RJ (1964) Electron-stream interaction with plasmas. Monograph no. 29, MIT Press, Cambridge MassachusettsGoogle Scholar
  28. 28.
    Bers A (1983) Space-time evolution of plasma instabilities—absolute and convective. In: Galeev AA, Sudan RN (eds) Handbook of plasma physics: volume 1 basic plasma physics, Chapter 3.2. North Holland Publishing Company, pp 451–517Google Scholar
  29. 29.
    Crighton DG and Leppington FG (1974). Radiation properties of the semi-infinite vortex sheet: the initial-value problem. J Fluid Mech 64(2): 393–414 MATHCrossRefADSGoogle Scholar
  30. 30.
    Jones DS and Morgan JD (1972). The instability of a vortex sheet on a subsonic stream under acoustic radiation. Proc Camb Philos Soc 72: 465–488 MATHMathSciNetGoogle Scholar
  31. 31.
    Quinn MC and Howe MS (1984). On the production and absorption of sound by lossless liners in the presence of mean flow. J Sound Vibration 97(1): 1–9 CrossRefADSGoogle Scholar
  32. 32.
    Rienstra SW (1981). Sound diffraction at a trailing edge. J Fluid Mech 108: 443–460 MATHCrossRefADSMathSciNetGoogle Scholar
  33. 33.
    Koch W and Möhring W (1983). Eigensolutions for liners in uniform mean flow ducts. AIAA J 21: 200–213 MATHADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Daniels PG (1985). On the unsteady Kutta condition. Quar J Mech Appl Math 31: 49–75 CrossRefMathSciNetGoogle Scholar
  35. 35.
    Goldstein ME (1981). The coupling between flow instabilities and incident disturbances at a leading edge. J Fluid Mech 104: 217–246 MATHCrossRefADSMathSciNetGoogle Scholar
  36. 36.
    Crighton DG, Innes D (1981) Analytical models for shear-layer feed-back cycles. AIAA81-0061, AIAA Aerospace Sciences Meeting, 19th, St. Louis, MO, 12–15 Jan. 1981Google Scholar
  37. 37.
    Brandes M, Ronneberger D (1995) Sound amplification in flow ducts lined with a periodic sequence of resonators. AIAA paper 95–126, 1st AIAA/CEAS Aeroacoustics Conference, Munich, Germany, 12–15 June 1995Google Scholar
  38. 38.
    Aurégan Y, Leroux M, Pagneux V (2005) Abnormal behavior of an acoustical liner with flow. Forum Acusticum 2005, BudapestGoogle Scholar
  39. 39.
    Munt RM (1977). The interaction of sound with a subsonic jet issuing from a semi-infinite cylindrical pipe. J Fluid Mech 83(4): 609–640 CrossRefADSMathSciNetGoogle Scholar
  40. 40.
    Munt RM (1990). Acoustic radiation properties of a jet pipe with subsonic jet flow: I. The cold jet reflection coefficient. J Sound Vibration 142(3): 413–436 ADSMathSciNetGoogle Scholar
  41. 41.
    Morgan JD (1974). The interaction of sound with a semi-infinite vortex sheet. Quart J Mech Appl Math 27: 465–487 MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Bechert DW (1980). Sound absorption caused by vorticity shedding, demonstrated with a jet flow. J Sound Vibration 70: 389–405 CrossRefADSGoogle Scholar
  43. 43.
    Bechert DW (1988). Excitation of instability waves in free shear layers. Part 1. Theory. J Fluid Mech 186(186): 47–62 MATHGoogle Scholar
  44. 44.
    Howe MS (1979). Attenuation of sound in a low Mach number nozzle flow. J Fluid Mech 91: 209–229 MATHCrossRefADSGoogle Scholar
  45. 45.
    Cargill AM (1982). Low-frequency sound radiation and generation due to the interaction of unsteady flow with a jet pipe. J Fluid Mech 121: 59–105 MATHCrossRefADSMathSciNetGoogle Scholar
  46. 46.
    Cargill AM (1982). Low frequency acoustic radiation from a jet pipe—a second order theory. J Sound Vibration 83: 339–354 MATHCrossRefADSGoogle Scholar
  47. 47.
    Rienstra SW (1983). A small Strouhal number analysis for acoustic wave-jet flow-pipe interaction. J Sound Vibration 86: 539–556 MATHCrossRefADSGoogle Scholar
  48. 48.
    Rienstra SW (1984). Acoustic radiation from a semi-infinite annular duct in a uniform subsonic mean flow. J Sound Vibration 94(2): 267–288 ADSGoogle Scholar
  49. 49.
    Crighton DG (1985). The Kutta condition in unsteady flow. Ann Rev Fluid Mech 17: 411–445 CrossRefADSGoogle Scholar
  50. 50.
    Peters MCAM, Hirschberg A, Reijnen AJ and Wijnands APJ (1993). Damping and reflection coefficient measurements for an open pipe at low Mach and low Helmholtz numbers. J Fluid Mech 256: 499–534 CrossRefADSGoogle Scholar
  51. 51.
    Cummings A (1983). Acoustic nonlinearities and power losses at orifices. AIAA J 22: 786–792 ADSMathSciNetGoogle Scholar
  52. 52.
    Allam S and Åbom M (2006). Investigation of damping and radiation using full plane wave decomposition in ducts. J Sound Vibration 292: 519–534 CrossRefADSGoogle Scholar
  53. 53.
    Michalke A (1965). On spatially growing disturbances in an inviscid shear layer. J Fluid Mech 23(3): 521–544 CrossRefADSMathSciNetGoogle Scholar
  54. 54.
    Jones DS and Morgan JD (1974). A linear model of a finite Helmholtz instability. Proc R Soc Lond A 344: 341–362 Google Scholar
  55. 55.
    Noble B (1958). Methods based on the Wiener–Hopf technique. Pergamon Press, London MATHGoogle Scholar
  56. 56.
    Heins AE and Feshbach H (1947). The coupling of two acoustical ducts. J Math Phys 26: 143–155 MATHMathSciNetGoogle Scholar
  57. 57.
    Levine H and Schwinger J (1948). On the radiation of sound from an unflanged circular pipe. Phys Rev (APS) 73(4): 383–406 MATHCrossRefADSMathSciNetGoogle Scholar
  58. 58.
    Rienstra SW (1986) Hydrodynamic instabilities and surface waves in a flow over an impedance wall. In: Comte-Bellot G, Ffowcs Williams JE (eds) Proceedings IUTAM symposium ‘aero- and hydro-acoustics’ 1985 Lyon. Springer-Verlag, Heidelberg, pp 483–490Google Scholar
  59. 59.
    Abramowitz M and Stegun IA (1964). Handbook of mathematical functions. National Bureau of Standards, Dover Publications Inc., New York MATHGoogle Scholar
  60. 60.
    Rienstra SW (2006) Impedance models in time domain, including the extended Helmholtz resonator model. AIAA Paper 2006-2686, 12th AIAA/CEAS Aeroacoustics Conference, 8–10 May 2006, Cambridge, MA, USAGoogle Scholar
  61. 61.
    Miles JW (1957). On the reflection of sound at an interface of relative motion. J Acoust Soc Am 29(2): 226–228 CrossRefMathSciNetADSGoogle Scholar
  62. 62.
    Kelvin L (1871). Hydrokinetic solutions and observations. Philos Mag 4(42): 362–377 Google Scholar
  63. 63.
    von Helmholtz H (1868). On discontinuous movement of fluids. Philos Mag 4(36): 337–346 Google Scholar
  64. 64.
    Chevaugeon N, Remacle J-F, Gallez X (2006) Discontinuous Galerkin implementation of the extended Helmholtz resonator impedance model in time domain. AIAA paper 2006-2569, 12th AIAA/CEAS Aeroacoustics Conference, Cambridge, MA, 8–10 May 2006Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations