Skip to main content Accessibility help

Acoustic scattering by cascades with complex boundary conditions: compliance, porosity and impedance

  • Peter J. Baddoo (a1) and Lorna J. Ayton (a1)


We present a solution for the scattered field caused by an incident wave interacting with an infinite cascade of blades with complex boundary conditions. This extends previous studies by allowing the blades to be compliant, porous or satisfy a generalised impedance condition. Beginning with the convected wave equation, we employ Fourier transforms to obtain an integral equation amenable to the Wiener–Hopf method. This Wiener–Hopf system is solved using a method that avoids the factorisation of matrix functions. The Fourier transform is inverted to obtain an expression for the acoustic potential function that is valid throughout the entire domain. We observe that the principal effect of complex boundary conditions is to perturb the zeros of the Wiener–Hopf kernel, which correspond to the duct modes in the inter-blade region. We focus efforts on understanding the role of porosity, and present a range of results on sound transmission and generation. The behaviour of the duct modes is discussed in detail in order to explain the physical mechanisms behind the associated noise reductions. In particular, we show that cut-on duct modes do not exist for arbitrary porosity coefficients. Conversely, the acoustic far-field modes are unchanged by modifications to the boundary conditions. We apply our solution to a cascade of perforated plates and see that a fractional open area of 1 % is sufficient to significantly attenuate backscattering. The solution is essentially analytic (the only numerical requirements are matrix inversion and root finding) and is therefore extremely rapid to compute.


Corresponding author

Present address: Department of Mathematics, Huxley Building, South Kensington Campus, Imperial College London, London SW7 2AZ, UK. Email address for correspondence:


Hide All
Ayton, L. J. 2016 Acoustic scattering by a finite rigid plate with a poroelastic extension. J. Fluid Mech. 791, 414438.
Ayton, L. J., Gill, J. R. & Peake, N. 2016 The importance of the unsteady Kutta condition when modelling gust–aerofoil interaction. J. Sound Vib. 378, 2837.
Baddoo, P. J. & Ayton, L. J. 2020 An analytic solution for gust–cascade interaction noise including effects of realistic aerofoil geometry. J. Fluid Mech. 886, A1.
Baddoo, P. J., Hajian, R. & Jaworski, J. W.2019 Unsteady aerodynamics of porous aerofoils. arXiv:1911.07382.
Bendali, A., Fares, M’B., Piot, E. & Tordeux, S. 2013 Mathematical justification of the Rayleigh conductivity model for perforated plates in acoustics. SIAM J. Appl. Maths 73 (1), 438459.
Bouley, S., François, B., Roger, M., Posson, H. & Moreau, S. 2017 On a two-dimensional mode-matching technique for sound generation and transmission in axial-flow outlet guide vanes. J. Sound Vib. 403, 190213.
Brambley, E. J. 2009 Fundamental problems with the model of uniform flow over acoustic linings. J. Sound Vib. 322 (4–5), 10261037.
Brandão, R. & Schnitzer, O. 2020 Acoustic impedance of a cylindrical orifice. J. Fluid Mech. 892, A7.
Cavalieri, A. V. G., Wolf, W. R. & Jaworski, J. W. 2016 Numerical solution of acoustic scattering by finite perforated elastic plates. Proc. R. Soc. Lond. A 472, 20150767.
Crighton, D. G. & Leppington, F. G. 1970 Scattering of aerodynamic noise by a semi-infinite compliant plate. J. Fluid Mech. 43 (4), 721736.
Delves, L. M. & Lyness, J. N. 1967 A numerical method for locating the zeros of an analytic function. Maths Comput. 21 (100), 543560.
Evers, I. & Peake, N. 2002 On sound generation by the interaction between turbulence and a cascade of airfoils with non-uniform mean flow. J. Fluid Mech. 463, 2552.
Fang, J. & Atassi, H. M. 1993 Compressible flows with vortical disturbances around a cascade of loaded airfoils. In Unsteady Aerodyn. Aeroacoustics, Aeroelasticity Turbomachines Propellers, pp. 149176. Springer.
Geyer, T., Sarradj, E. & Fritzsche, C. 2010 Measurement of the noise generation at the trailing edge of porous airfoils. Exp. Fluids 48 (2), 291308.
Glegg, S. A. L. 1999 The response of a swept blade row to a three-dimensional gust. J. Sound Vib. 227 (1), 2964.
Glegg, S. A. L. & Devenport, W. J. 2017 Aeroacoustics of Low Mach Number Flows: Fundamentals, Analysis, and Measurement. Academic Press.
Graham, R. R. 1934 The silent flight of owls. J. R. Aero. Soc. 38 (286), 837843.
Hall, K. C. 1997 Exact solution to category 3 problems: turbomachinery noise. In Second Comput. Aeroacoustics(CAA) Work. Benchmark Probl., pp. 4143. Springer.
Hall, K. C., Kielb, R. E. & Thomas, J. P. 2006 Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines. Kluwer Academic Publishers.
Howe, M. S. 1998 Acoustics of Fluid–structure Interactions. Cambridge University Press.
Howe, M. S., Scott, M. I. & Sipcic, S. R. 1996 The influence of tangential mean flow on the Rayleigh conductivity of an aperture. Proc. R. Soc. Lond. A 452 (1953), 23032317.
Jaworski, J. W. & Peake, N. 2013 Aerodynamic noise from a poroelastic edge with implications for the silent flight of owls. J. Fluid Mech. 723, 456479.
Jaworski, J. W. & Peake, N. 2020 Aeroacoustics of silent owl flight. Annu. Rev. Fluid Mech. 52, 395420.
Kisil, A. & Ayton, L. J. 2018 Aerodynamic noise from rigid trailing edges with finite porous extensions. J. Fluid Mech. 836, 117144.
Koch, W. 1983 Resonant acoustic frequencies of flat plate cascades. J. Sound Vib. 88 (2), 233242.
Leppington, F. G. 1977 The effective compliance of perforated screens. Mathematika 24 (02), 199215.
Lighthill, M. J. 1958 An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press.
Maierhofer, G. & Peake, N. 2020 Wave scattering by an infinite cascade of non-overlapping blades. J. Sound Vib. 481, 115418.
Myers, M. K. 1980 On the acoustic boundary condition in the presence of flow. J. Sound Vib. 71 (3), 429434.
Noble, B. 1958 Methods Based on the Wiener–Hopf Technique For the Solution of Partial Differential Equations. Pergamon Press.
Parker, R. 1967 Resonance effects in wake shedding from compressor blading. J. Sound Vib. 6 (3), 302309.
Peake, N. 1992 The interaction between a high-frequency gust and a blade row. J. Fluid Mech. 241, 261289.
Peake, N. & Parry, A. B. 2012 Modern challenges facing turbomachinery aeroacoustics. Annu. Rev. Fluid Mech. 44 (1), 227248.
Posson, H., Roger, M. & Moreau, S. 2010 On a uniformly valid analytical rectilinear cascade response function. J. Fluid Mech. 663, 2252.
Saiz, G.2008 Turbomachinery aeroelasticity using a time-linearised multi blade-row approach. PhD thesis, Imperial College London.
Strutt, J. W. 1870 On the theory of resonance. Phil. Trans. R. Soc. Lond. 161, 77118.
Trefethen, L. N. & Weideman, J. A. C. 2014 The exponentially convergent trapezoidal rule. SIAM Rev. 56 (3), 385458.
Woodley, B. M. & Peake, N. 1999 Resonant acoustic frequencies of a tandem cascade. Part 1. Zero relative motion. J. Fluid Mech. 393, 215240.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Acoustic scattering by cascades with complex boundary conditions: compliance, porosity and impedance

  • Peter J. Baddoo (a1) and Lorna J. Ayton (a1)


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.