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), 1652–1664.

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), 363–381.

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), 193–208.

Berry, M. V. & Upstill, C.
1980
Catastrophe optics: morphologies of caustics and their diffraction patterns. Prog. Opt.
18, 257–346.

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), 465–493.

Červený, V., Popov, M. M. & Pšenčík, I.
1982
Computation of wave fields in inhomogeneous media – Gaussian beam approach. Geophys. J. Intl
70 (1), 109–128.

Chapman, C. J.
1999
Caustics in cylindrical ducts. Proc. R. Soc. Lond. A
455, 2529–2548.

Chapman, S. J., Lawry, J. M. H., Ockendon, J. R. & Tew, R. H.
1999
On the theory of complex rays. SIAM Rev.
41 (3), 417–509.

Connor, J. N. L. & Curtis, P. R.
1984
Differential equations for the cuspoid canonical integrals. J. Math. Phys.
25, 2895–2902.

Crighton, D. G. & Gaster, M.
1976
Stability of slowly diverging jet flow. J. Fluid Mech.
77 (02), 397–413.

Durbin, P. A.
1983a
High frequency Green function for aerodynamic noise in moving media. Part I. General theory. J. Sound Vib.
91 (4), 519–525.

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), 527–538.

Freund, J. B. & Fleischman, T. G.
2002
Ray traces through unsteady turbulence. AIAA J.
1 (1), 83–96.

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), 499–522.

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, 315–333.

Grikurov, V. E.
1980
Caustic overlap in a surface waveguide and a generalization of the ray method. Radiophys. Quantum Electron.
23 (9), 690–695.

Hanyga, A.
1996
Point-to-curve ray tracing. Pure Appl. Geophys.
148 (3–4), 387–420.

Hanyga, A.
1997
Canonical functions of asymptotic diffraction theory associated with symplectic singularities. In Symplectic Singularities and Geometry of Gauge Fields, vol. 36, pp. 57–71. Banach Center Publications.

Hayes, W. D.
1970
Kinematic wave theory. Proc. R. Soc. Lond. A
320 (1541), 209–226.

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), 1203–1213.

Jones, D. S.
1976
The mathematical theory of noise shielding. Prog. Aerosp. Sci.
17 (3), 149–229.

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), 981–992.

Lighthill, M. J.
1952
On sound generated aerodynamically. I. General theory. Proc. R. Soc. Lond. A
211 (1107), 564–587.

Lighthill, M. J.
1954
On sound generated aerodynamically. II. Turbulence as a source of sound. Proc. R. Soc. Lond. A
222 (1148), 1–32.

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), 2005–2010.

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 http://dlmf.nist.gov/.
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), 79–99.

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, 157–168.

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), 456–464.

Tester, B. J. & Morfey, C. L.
1976
Developments in jet noise modelling – theoretical predictions and comparisons with measured data. J. Sound Vib.
46 (1), 79–103.

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), 793–830.