Skip to main content Accessibility help

Cones of silence, complex rays and catastrophes: high-frequency flow–acoustic interaction effects

  • J. T. Stone (a1) (a2), R. H. Self (a1) and C. J. Howls (a2)


In this paper we develop a novel ray solver for the time-harmonic linearized Euler equations used to predict high-frequency flow–acoustic interaction effects from point sources in subsonic mean jet flows. The solver incorporates solutions to three generic ray problems found in free-space flows: the multiplicity of rays at a receiver point, propagation of complex rays and unphysical divergences at caustics. We show that these respective problems can be overcome by an appropriate boundary value reformulation of the nonlinear ray equations, a bifurcation-theory-inspired complex continuation, and an appeal to the uniform functions of catastrophe theory. The effectiveness of the solver is demonstrated for sources embedded in isothermal parallel and spreading jets, with the fields generated containing a wide variety of caustic structures. Solutions are presented across a large range of receiver angles in the far field, both downstream, where evanescent complex rays generate the cone of silence, and upstream, where multiple real rays are organized about a newly observed cusp caustic. The stability of the caustics is verified for both jets by their persistence under parametric changes of the flow and source. We show the continuation of these caustics as surfaces into the near field is complicated due to a dense caustic network, featuring a chain of locally hyperbolic umbilic caustics, generated by the tangency of rays as they are channelled upstream within the jet.


Corresponding author

Email addresses for correspondence:,


Hide All
Abrahams, I. D., Kriegsmann, G. A. & Reiss, E. L. 1989 On the development of caustics in shear flows over rigid walls. SIAM J. Appl. Maths 49 (6), 16521664.
Allgower, E. L. & Georg, K. 1990 Numerical Continuation Methods. Springer.
Amodei, D., Keers, H., Vasco, D. & Johnson, L. 2006 Computation of uniform wave forms using complex rays. Phys. Rev. E 73 (3), 036704.
Avila, G. S. S. & Keller, J. B. 1963 The high-frequency asymptotic field of a point source in an inhomogeneous medium. Commun. Pure Appl. Maths 16 (4), 363381.
Balsa, T. F. 1976 The far field of high frequency convected singularities in sheared flows, with an application to jet-noise prediction. J. Fluid Mech. 74 (02), 193208.
Berry, M. V. & Upstill, C. 1980 Catastrophe optics: morphologies of caustics and their diffraction patterns. Prog. Opt. 18, 257346.
Bleistein, N. & Handelsman, R. A. 1986 Asymptotic Expansions of Integrals. Dover.
Candel, S. M. 1977 Numerical solution of conservation equations arising in linear wave theory: application to aeroacoustics. J. Fluid Mech. 83 (03), 465493.
Červený, V., Popov, M. M. & Pšenčík, I. 1982 Computation of wave fields in inhomogeneous media – Gaussian beam approach. Geophys. J. Intl 70 (1), 109128.
Chapman, C. J. 1999 Caustics in cylindrical ducts. Proc. R. Soc. Lond. A 455, 25292548.
Chapman, S. J., Lawry, J. M. H., Ockendon, J. R. & Tew, R. H. 1999 On the theory of complex rays. SIAM Rev. 41 (3), 417509.
Connor, J. N. L. & Curtis, P. R. 1984 Differential equations for the cuspoid canonical integrals. J. Math. Phys. 25, 28952902.
Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77 (02), 397413.
Durbin, P. A. 1983a High frequency Green function for aerodynamic noise in moving media. Part I. General theory. J. Sound Vib. 91 (4), 519525.
Durbin, P. A. 1983b High frequency Green function for aerodynamic noise in moving media. Part II. Noise from a spreading jet. J. Sound Vib. 91 (4), 527538.
Freund, J. B. & Fleischman, T. G. 2002 Ray traces through unsteady turbulence. AIAA J. 1 (1), 8396.
Goldstein, M. E. 1976 Aeroacoustics. McGraw-Hill.
Goldstein, M. E. 1982 High frequency sound emission from moving point multipole sources embedded in arbitrary transversely sheared mean flows. J. Sound Vib. 80 (4), 499522.
Goldstein, M. E.1991 Noise from turbulent shear flows. In Aeroacoustics of Flight Vehicles: Theory and Practice. Volume 1. Noise Sources (ed. H. H. Hubbard), pp. 291–310. DTIC Document.
Goldstein, M. E. 2003 A generalized acoustic analogy. J. Fluid Mech. 488, 315333.
Grikurov, V. E. 1980 Caustic overlap in a surface waveguide and a generalization of the ray method. Radiophys. Quantum Electron. 23 (9), 690695.
Hanyga, A. 1996 Point-to-curve ray tracing. Pure Appl. Geophys. 148 (3–4), 387420.
Hanyga, A. 1997 Canonical functions of asymptotic diffraction theory associated with symplectic singularities. In Symplectic Singularities and Geometry of Gauge Fields, vol. 36, pp. 5771. Banach Center Publications.
Hayes, W. D. 1970 Kinematic wave theory. Proc. R. Soc. Lond. A 320 (1541), 209226.
Ilário, C. R., Azarpeyvand, M., Rosa, V., Self, R. H. & Meneghini, J. R. 2017 Prediction of jet mixing noise with Lighthill’s acoustic analogy and geometrical acoustics. J. Acoust. Soc. Am. 141 (2), 12031213.
Jones, D. S. 1976 The mathematical theory of noise shielding. Prog. Aerosp. Sci. 17 (3), 149229.
Keller, H. B. 1992 Numerical Methods for Two-point Boundary-value Problems. Dover.
Keller, H. B. & Perozzi, D. J. 1983 Fast seismic ray tracing. SIAM J. Appl. Maths 43 (4), 981992.
Lighthill, M. J. 1952 On sound generated aerodynamically. I. General theory. Proc. R. Soc. Lond. A 211 (1107), 564587.
Lighthill, M. J. 1954 On sound generated aerodynamically. II. Turbulence as a source of sound. Proc. R. Soc. Lond. A 222 (1148), 132.
Lilley, G. M. 1958 On noise from air jets. Aero. Res. Counc. 20, 376-N40-FM 2724.
Mani, R., Gliebe, P. R. & Balsa, T. F.1978 High velocity jet noise source location and reduction. Task 2, Fed. Aviation Admin. Rep., FAA-RD-76-II.
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. McGraw-Hill.
Norris, A. N. 1986 Complex point-source representation of real point sources and the Gaussian beam summation method. J. Opt. Soc. Am. A 3 (12), 20052010.
Olver, F. W. J., Olde Daalhuis, A. B., Lozier, D. W., Schneider, B. I., Boisvert, R. F., Clark, C. W., Miller, B. R. & Saunders, B. V.2016 NIST digital library of mathematical functions. Release 1.0.13 of 2016-09-16. Available from
Pereyra, V., Lee, W. H. K. & Keller, H. B. 1980 Solving two-point seismic-ray tracing problems in a heterogeneous medium. Part 1. A general adaptive finite difference method. Bull. Seismol. Soc. Am. 70 (1), 7999.
Pierce, A. D. 1981 Acoustics: An Introduction to its Physical Principles and Applications. McGraw-Hill.
Poston, T. & Stewart, I. 2014 Catastrophe Theory and its Applications. Dover.
Ribner, H. S.1995 An extension of the Lighthill theory of jet noise to encompass refraction and shielding. NASA Tech. Mem. 110163.
Sambridge, M. S. & Kennett, B. L. N. 1990 Boundary value ray tracing in a heterogeneous medium: a simple and versatile algorithm. Geophys. J. Intl 101, 157168.
Sneddon, I. N. 2006 Elements of Partial Differential Equations. Dover.
Stone, J. T., Self, R. H. & Howls, C. J. 2017 Aeroacoustic catastrophes: upstream cusp beaming in Lilley’s equation. Proc. R. Soc. Lond. A 473, 20160880.
Stone, J. T., Self, R. H. & Howls, C. J.2014 A complex ray-tracing tool for high-frequency mean-field flow interaction effects in jets. AIAA Paper 2014-2757.
Tam, C. K. W. & Pastouchenko, N. N. 2002 Noise from fine-scale turbulence of nonaxisymmetric jets. AIAA J. 40 (3), 456464.
Tester, B. J. & Morfey, C. L. 1976 Developments in jet noise modelling – theoretical predictions and comparisons with measured data. J. Sound Vib. 46 (1), 79103.
Thom, R. 1989 Structural Stability and Morphogenesis, Advanced Books Classic Series. Addison-Wesley.
Wundrow, D. W. & Khavaran, A. 2004 On the applicability of high-frequency approximations to Lilley’s equation. J. Sound. Vib. 272 (3), 793830.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


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