References

Chapter 1

1.1 T.A. Croft: J. Geophys. Res. 72, 2343 (1967) 1.2 F.D. Tapert: In Wave Propagation and Underwater Acoustics, ed. by J.B. Keller, J.S.

Papadakis (Springer, Berlin, Heidelberg 1979) p. 224 \.3 A.S. Gurvich, M.A. Kalistratova, F.E. Martvel: Izv. VUZ Radiofiz. 20, 1020 (1977) [English

trans\.: Radiophys. Quantum Electron 20, 705 (1977)] 1.4 J.M. Martin, S.M. Flatte: Appl. Opt. 27, 2111 (1988) 1.5 H. Whitney: Ann. Math. 62,374 (1955) 1.6 V.1. Arnold: Usp. Mat. Nauk 23, 3 (1968); ibid. 29, 11,243 (1974); ibid. 30, 3 (1975) 1.7 R. Thorn: Structural Stability and Morphogenesis (Benjamin, New York 1972) 1.8 V.I. Arnold: Catastrophe Theory, 3rd edn. (Springer, Berlin, Heidelberg 1992) 1.9 V.1. Arnold: SOY. Phys. - Usp. 26, 1024 (1983) 1.10 T. Poston, N. Stewart: Catastrophe Theory and Its Applications (Pitman, London 1978) 1.10 R. Gi1mor: Catastrophe Theory for Scientists and Engineers (Wiley, New York 1981) 1.12 M. Golubitsky, V. Guillimin: Stable Mappings and their Singularities, Graduate Texts Math.,

Vol. 14 (Springer, Berlin, Heidelberg 1973) 1.13 V.I. Arnold, A.N. Varchenko, S.M. Gusein-Zade: Singularities of Differentiable Mappings,

Vols. 1,2 (Birkhiiuser, Boston 1985, 1988) 1.14 U. Dustermaat: Commun. Pure Appl. Math. 27,207 (1974) 1.15 Yu.A. Kravtsov, Yu.1. Orlov: SOY. Phys. - Usp. 26, 1038 (1983) 1.16 Yu.A. Kravtsov: Rays and caustics as physical objects, in Progress in Optics 26, 227-348

(North-Holland, Amsterdam 1988) 1.17 Yu.L. Gazaryan: Voprosy Dinamicheskoi Teorii Rasprostranenia Seismicheskikh Voln (topics in

dynamic theory of seismic wave propagation) (Izd. LGU, Leningrad 1961) Vol. 5, p. 73 1.18 R.N. Buchal, J.B. Keller: Commun. Pure Appl. Math. 13,85 (1960) 1.19 V.M. Babich, N.Y. Kirpicnikova: The Boundary-Layer Method in Diffraction Problems,

Springer Ser. Electrophys., Vol. 3 (Springer, Berlin, Heidelberg 1979) 1.20 V.M. Babich, V.S. Buldyrev: Short-Wavelength Diffraction Theory, Springer Ser. Wave Phe­

nom., Vol. 4 (Springer, Berlin, Heidelberg 1991) 1.21 L.B. Felsen, N. Marcuvitz: Radiation and Scattering of Waves (Prentice-Hall, Englewood

Cliffs, NJ 1973) Vols. 1,2 1.22 Yu.A. Kravtsov: Izv. VUZ Radiofiz. 7, 664,1049 (1964) [English transl.: Radiophys. (USSR) 7,

no. 4, 104; no. 6, 40 (1964)] 1.23 D. Ludwig: Commun. Pure Appl. Math. 19,215 (1966) 1.24 G. Dangelmayr, W. Veit: Ann. Phys. 118, 108 (1979) 1.25 D.S. Ahluwalia, R.M. Lewis, J. Boersma: SIAM J. Appl. Math. 16,783 (1968) 1.26 V.P. Maslov: Theorie des Perturbations et Methode Asymptotique (Dunod, Paris 1972) 1.27 V.P. Maslov, M.V. Fedoryuk: Semiclassical Approximation in Quantum Mechanics (Reidel,

Hingham, MA 1981) 1.28 Yu.A. Kravtsov: Akust. Zh. 14, 1 (1968) [English transl.: SOY. Phys. - Acoustics 14, 1 (1968)] 1.29 J.M. Arnold: Radio Sci. 17, 1181 (1982)

References 197

1.30 R. Ziolkowski, G.A. Deschamps: Radio Sci. 19, 1001 (1984) 1.31 Yu.l. Orlov: Trudy MEl (Moscow Power Engineering Institute, Moscow 1977) No. 19, p. 82;

Izv. VUZ Radiofiz. 17, 1035 (1974) [English trans\.: Radiophys. Quantum Electron. 17,788 (1974)]

1.32 L.A. Vainstein, E.S. Birger, N.V. Konyukhova, E.L. Kosarev: SOy. J. Plasma Phys. 26, 362 (1976)

1.33 J.M. Arnold: J. Acoust. Soc. Am. 87, 587 (1990) 1.34 V.B. Avdeyev, A.V. Demin, Yu.A. Kravtsov, M.V. Tinin, A.P. Yarygin: Izv. VUZ Radiofiz. 31,

1279 (1988) [English trans\.: Radiophys. Quantum Electron. 31,907 (1988)] 1.35 V.M. Babich, M.M. Popov: Izv. VUZ Radiofiz. 32, 1447 (1987) [English trans!': Radiophys.

Quantum Electron. 32, 1063 (1989)] 1.36 M.V. Berry, K.E. Mount: Rept. Progr. Phys. 35, 313 (1972) 1.37 M.V. Berry, C. Upsti11: Progress in Optics 18, 257-320 (North-Holland, Amsterdam 1980) 1.38 P.S. Theocaris, lG. Michopoulos: App\. Opt. 21, 1080 (1982) 1.39 S. Cornbleet: Proc. IEEE 71, 471 (1983) 1.40 J.J. Stamness: Opt. Acta 29, 823 (1982) 1.41 P.L. Marston: Geometrical and catastrophe Boston optics method in scattering, in Physical

Acoustics Vol 21, ed. by R.N. Thurston, A.D. Pierse (Academic, Boston 1992) pp. 1-234 1.42 J.J. Stamnes: Waves in Focal Regions (Adam Hilger, Bristol 1986) 1.43 Yu.A. Kravtsov, Yu.l. Orlov: Geometrical Optics of Inhomogeneous Media, Springer Ser. Wave

Phenom., Vo\. 6 (Springer, Berlin, Heidelberg 1990)

Chapter 2

2.1 M. Born, E. Wolf: Principles of Optics, 5th edn. (Pergamon, Oxford 1975) 2.2 Yu.A. Kravtsov, Yu.l. Orlov: Geometrical Optics of Inhomogeneous Media, Springer Ser. Wave

Phenom., Vo\. 6 (Springer, Berlin, Heidelberg 1990) 2.3 S.M. Rytov: Dok!. Akad. Nauk USSR 18,263 (1938) 2.4 Yu.A. Kravtsov: Dok!. Akad. Nauk USSR 183,74 (1968) [English trans\.: SOY. Phys. - Dok!.

13, 1125 (1968)] 2.5 V.V. Zheleznyakov, V.V. Kocharovskii, V\'V. Kocharovskii: Usp. Fiz. Nauk 141,257 (1983)

[English trans\.: SOY. Phys. - Usp. 26, 877 (1983)] 2.6 Yu.A. Kravtsov, Yu.1. Orlov: SOY. Phys. - Usp. 23; 750 (1980); Radio Sci. 16, 975 (1981) 2.7 Yu.A. Kravtsov: Rays and caustics as physical objects, in Progress in Optics 26, 227-348

(North-Holland, Amsterdam 1988) 2.8 Yu.A. Kravtsov: Radiotekh. Elektron. 32, 883 (1987) [English trans\.: SOY. J. Commun.

Techno\. Electron. 32, no. 4, 133 (1987)] 2.9 E.L. Feinberg: Rasprostranenie Radiovoln Vdol Zemnoi Poverkhnosti (propagation of radio

waves over the Earth's surface) (Izd. Akad. Nauk USSR, Moscow 1961) 2.10 H.L. Bertoni, L.B. Felsen, A. Hessel: IEEE Trans. AP-19, 226 (1971) 2.11 F.B. Chernyi: Rasprostranenie Radiovoln (propagation of radio waves) (Sov. Radio, Moscow

1972) 2.12 A.A. Asatryan: Kratk. Soobsch. Fiz. FIAN No. 10, 1 (1985) [English trans!': SOY. Phys.

- Lebedev Inst. Rept. No. 10, 1 (1985)] 2.13 A.A. Asatrtan, Yu.A. Kravtsov: Wave Motion 10,45 (1988) 2.14 V.P. Mas1ov: Theorie des Perturbations et Methode Asymptotique (Dunod, Paris 1972) 2.15 R.M. Lewis: Arch. Ration. Mech. Ana\. 20, 191 (1965) 2.16 Yu.l. Orlov: Izv. VUZ Radiofiz. 24, 224 (1981) [English trans\.: Radiophys. Quantum Elec­

tron. 24, no. 2, 154 (1981)] 2.17 Yu.L. Gazaryan: Voprosy Dinamicheskoi Teorii Rasprostranenia Seismicheskikh Voln (topics in

dynamic theory of seismic wave propagation) (Izd. LGU, Leningrad 1961) p. 73

198 References

2.18 V.Ya. Groshev, Yu.A. Kravtsov: Izv. VUZ Radiofiz. 11, 1812 (1968) [English transl.: Radiophys. Quantum Electron. 11, 1019 (1968)]

2.19 Yu.1. Or1ov: Trudy MEl No. 194 (Power Institute, Moscow 1974) p. 103 2.20 A.A. Asatryan, Yu.A. Kravtsov, V.s. Fi1inov: Kratk. Soobschch. Fiz. FIAN No. 11,30 (1968)

[English transl.: Sov. Phys. - Lebedev Inst. Rept. No. 11,20 (1988)] 2.21 O.N. Stavrodis: The Optics of Rays, Wavefronts, and Caustics (Academic, New York 1972) 2.22 S.M. Flatte, R. Dashen, W.H. Munk, K.M. Watson, F. Zakariasen: Sound Transmission

Through a Fluctuating Ocean (Cambridge Univ. Press., Cambridge 1979) 2.23 Yu.A. Kravtsov: Zh. Eksp. Teor. Fiz. 55, 1470 (1968) [English transl.: Sov. Phys. - lETP 28,

769 (1969)] 2.24 M.V. Berry, e. Upstill: Catastrophe optics: Morphologies of caustics and their diffraction

patterns, in Progress in Optics 18, 257-320 (North-Holland, Amsterdam 1980) 2.25 1.B. Keller: A geometrical theory of diffraction, in Calculus of Variations and its Applications,

Proc. Symp. Appl. Math., Vol. 1.8 (McGraw-Hili, New York 1958) pp. 27-52 2.26 B.D. Seckler, 1.B. Keller: 1. Acoust. Soc. Am. 31, 192 (1959) 2.27 1.B. Keller, F.e. Karal: 1. Appl. Phys. 31, 1039 (1960) 2.28 V.P. Maslov: Sov. Phys. - Doklady 7,666 (1964) 2.29 V.M. Babich: Voprosy Dinamicheskoi Teorii Rasprostraneniya Seismicheskikh Voln (topics in

dynamic theory of seismic waves) (Izd. LGU, Leningrad 1961) Vol. 5, p. 145 2.30 Yu.A. Kravtsov: Izv. VUZ Radiofiz. 10, 1283 (1967) [English transl.: Radiophys. Quantum

Electron. 10,719 (1967)] 2.31 G. Ghione, I. Montrosset, L.B. Felsen: IEEE Trans. AP-32, 68 (1984) 2.32 1.B. Keller, W. Streifer: 1. Opt. Soc. Am. 61, 40 (1971) 2.33 G.A. Deschamps: Electron. Lett. 7, 684 (1971) 2.34 S. Choudhary, L.B. Felsen: Proc. IEEE 62, 1530 (1974) 2.35 A.A. Asatryan, Yu.A. Kravtsov: Izv. VUZ Radiofiz. 31, 1053 (1988) [English transl.:

Radiophys. Quantum Electron. 31, 746 (1988)]

Chapter 3

3.1 V.P. Maslov: Theorie des Perturbations et Methode Asymptotique (Dunod, Paris 1972) 3.2 A.A. Andronov, A.A. Vitt, S.E. Haikin: Teoriya Kolebanii (theorie of oscillations) (Nauka,

Moscow 1981) 3.3 V.I. Arnold: Usp. Mat. Nauk 23, 3 (1968); 29,11,243 (1974); ibid. 30, 560 (1975) 3.4 R. Thorn: Structural Stability and Morphogenesis (Benjamin, New York 1975) 3.5 T. Poston, N. Stewart: Catastrophe Theory and its Applications (Pitman, London 1978) 3.6 M. Golubitsky, V. Guillimin: Stable Mappings and Their Singularities, Graduate Texts Math.,

Vol. 14 (Springer, Berlin, Heidelberg 1973) 3.7 R. Gilmor: Catastrophe Theoryfor Scientists and Engineers (Wiley, New York 1981) 3.8 1.1. Dustermaat: Commun. Pure Appl. Math. 27, 207 (1974) 3.9 V.1. Arnold, A.N. Varchenko, S.M. Gussein-Zade: Singularities of Differentiable Mappings,

Vols. 1,2 (Birkhiiuser, Boston 1985, 1988) 3.10 V.1. Arnold: Sov. Phys. - Usp. 26, 1024 (1983) 3.11 V.I Arnold: Catastrophe Theory, 3rd edn. (Spinger, Berlin, Heidelberg 1992)

Chapter 4

4.1 V.I. Arnold, A.N. Varchenko, S.M. Gussein-Zade: Singularities of Differential Mappings, Vols. 1,2 (Birkhiiuser, Boston 1985, 1988)

4.2 1.1 Dustermaat: Commun. Pure Appl. Math. 27, 207 (1974)

References 199

4.3 M.Y. Berry: J. Phys. A 10,2061 (1977) 4.4 M.V. Berry, C. Upstill: Progress in Optics 18,257-320 (North-Holland, Amsterdam 1980) 4.5 Yu.A. Kravtsov, Yu.I. Orlov: Geometrical Optics oj Inhomogeneous Media, Springer Ser. Wave

Phenom., Vol. 6 (Springer, Berlin, Heidelberg 1990) 4.6 Yu.A. Kravtsov, Yu.1. Orlov: SOY. Phys. - Usp. 26, 1038 (\983) 4.7 Yu.A. Kravtsov: Rays and caustics as physical objects. Progress in Optics 26, 227-348

(North-Holland, Amsterdam 1988) 4.8 V.1. Arnold: Funkts. Anal. Prilozh. 6, 61 (\972) 4.9 M.V. Fedoryuk: Metod Perevala (saddle-point method) (Nauka, Moscow 1977) 4.10 R.B. Dingle: Asymptotic Expansions: Their Derivation and Interpretation (Academic, London

1973) 4.11 A.A. Asatryan, Yu.A. Kravtsov: Wave Motion 10, 45 (\ 988) 4.12 J.c.P. Miller: The Airy Integral, Giving Tables oJ Solution oJthe Differential Equation y" = xy

(Cambridge Univ. Press, Cambridge 1971) 4.13 V.A. Fock: Electromagnetic Diffraction and Propagation Problems (Pergamon, New York

1965) 4.14 M. Abramovitz, 1.A. Stegun: Handbook oj Mathematical Functions (NBS, Washington, DC

1964) 4.15 R.E. Langer: Trans. Am. Math. Soc. 31, 23 (1931) 4.16 T.M. Cherry: Trans. Am. Math. Soc. 68, 224 (1950) 4.17 A.A. Dorodnitsyn: Usp. Mat. Nauk 7, 3 (1952) 4.18 V.Ya. Groshev, Yu.A. Kravtsov: Izv. VUZ Radiofiz. 11, 1812 (1968) [English trans!.:

Radiophys. Quantum Electron. 11, 1019 (1968)] 4.19 T. Pearsey: Philos. Mag. 37, 311 (1946) 4.20 L.M. Brekhovskikh: Acoustics oj Layered Media I, ll, Springer Ser. Wave Phenom., Vols. 5,

10 (Springer, Berlin, Heidelberg 1990, 1992) 4.21 A.S. Kryukovskii, D.S. Lukin, E.A. Palkin: Preprint No. 43, 415 (1984) Inst. Radiotech.

E1ecronics, Acad. Sci. USSR 4.22 J.N.L. Connor, D. Farrelly: 1. Chern. Phys. 75, 2831 (1981) 4.23 J.N.L. Connor, P.R. Curtis: J .. Phys. A 15, 1179 (1982) 4.24 Yu.A. Kravtsov, Yu.A. Orlov: SOY. Phys. - Usp. 23, 750 (\980); Radio. Sci. 16,975 (\981) 4.25 D. Ludwig: Commun. Pure Appl. Math. 19,215 (1966) 4.26 H. Trinkaus, F. Drepper: J. Phys. A 10, Ll1 (1977) 4.27 N.F. Dronov, E.B. Ipatyev, D.S. Lukin, E.A. Palkin: In Rasprostraneniye Radiovoln v lonos­

Jere (propagation of radio waves in the ionosphere) (IZMIRAN SSSR, Moscow 1978) p. 57 4.28 M.V. Berry, J.F. Nye, FJ. Wright: Philos. Trans. Roy. Soc. London 291, 453 (1979) 4.29 D.S. Lukin, E.A. Pa1kin: Chislennyi Kanonicheskii Metod v Zadachakh Difactsii i Raspros­

traneniya Electromagnitnikh Voln v Neodnorodnykh Sredakh (canonic numerical method in problems of diffration and propagation of EM waves in inhomogeneous media) (MFTI Press, Moscow 1982)

4.30 J.N.L. Connor, P.R. Curtis, D. Farrelly: 1. Phys. A 17,283 (1984) 4.31 V.1. Arnold: Usp. Mat. Nauk 28,17 (1973) 4.32 1.N.L. Connor: Mol. Phys. 26, 1371 (1973) 4.33 J.N.L. Connor: Faraday Discuss. Chern. Soc. 55, 51 (1973) 4.34 J.N.L. Connor, D. Farrelly: Chern. Phys. Lett. 81, 306 (1981) 4.35 R. Gi1mor: Catastrophe Theory Jor Scientists and Engineers (Wiley, New York 1981) 4.36 M.V. Berry: Adv. Phys. 25, 1 (1976) 4.37 IN.L. Connor, P.R. Curtis, D. Farrelly: Mol. Phys. 48, 1305 (1983)

Chapter 5

5.1 Yu.A. Kravtsov: Izv. VUZ Radiofiz. 8, 659 (1965) [English transl.: SOY. Radiophys. 8,467 (1965)]

200 References

5.2 Yu.A. Kravtsov: Akust. Zh. 14, 1 (1968) [English transl.: SOy. Phys. - Acoust. 14, 1 (1968)] 5.3 Yu.A. Kravtsov: In Analiticheskie Methody v Teorii Difraktsii i Rasprostraneniya Von (analytic

methods in diffraction and wave propagation theory) (Nauchnyi Soviet po Akustike, Akad. Nauk SSSR, Moscow 1970) pp. 258-362

5.4 G.B. Airy: Trans. Canbridge Philos. Soc. 6, 379 (1838) 5.5 R.N. Buchal, 1.8. Keller: Commun. Pure Appl. Math. 13, 85 (1960) 5.6 Yu.L. Gazaryan: In Voprosy Dinamicheskoi Teorii Rasprostranenia Seismicheskikh Voln

(topics in the dynamic theory of seismic waves), Vol. 5 (lZd. LGU, Leningrad 1961) p. 73 5.7 V.M. Babich, N.Y. Kirpicnikova: The Boundary-Layer Method in Diffraction Problems,

Springer Ser. Electrophys., Vol. 3 (Springer, Berlin, Heidelberg 1979) 5.8 Yu.A. Kravtsov: Izv. VUZ Radiofiz. 7, 664 (1964) [English transl.: Radiophys. Quantum

Electron. 7, no. 4, 104 (1964)] 5.9 C. Chester, B. Friedman, F. Ursell: Proc. Cambridge Phil os. Soc. 53, 599 (1957) 5.10 D. Ludwig: Commun. Pure Appl. Math. 19,215 (1966) 5.11 M.V. Berry: Proc. Philos. Soc. 89, 479 (1966) 5.12 M.V. Berry: Sci. Prog. Oxf. 57, 43 (1969) 5.13 M.V. Berry, K.E. Mount: Rep. Prog. Phys. 35, 313 (1972) 5.14 Yu.A. Kravtsov, Yu.1. Orlov: SOY. Phys. - Usp. 26, 1038 (1983) 5.15 Yu.A. Kravtsov: Izv. VUZ Radiofiz. 10, 1283 (1967) [English transl.: Radiophys. Quantum

Electron. 10,719 (1967)] 5.16 Yu.1. Orlov: Trudy MEl No. 194 (Moscow Power Engineering Institute, Moscow 1974) p. 103 5.17 V.M. Babic, V.S. Buldyrev: Short- Wavelength Diffraction Theory, Springer Ser. Wave Phe­

nom., Vol. 4 (Springer, Berlin, Heidelberg 1991) 5.18 V.M. Babich, S.A. Egorov: In Voprosy Dinamicheskoi Teorii Rasprostraneniya Seismicheskikh

Voln (topics in the dynamic theory of seismic waves) (Izd. LGU, Leningrad 1983) Vol. 3, p. 4 5.19 Yu.A. Kravtsov: Izv. VUZ Radiofiz. 7, 1049 (1964) [English transl.: Radiophys. Quantum

Electron. 7, no. 6, 40 (1964)] 5.20 Yu.A. Kravtsov, Yu.1. Orlov: Geometrical Optics of Inhomogenous Media, Springer Ser. Wave

Phenom., Vol. 6 (Springer, Berlin, Heidelberg 1990) 5.21 Yu.A. Kravtsov, L.A. Ostrovskii, N.S. Stepanov: Proc IEEE 62, 1492 (1974) 5.22 V.Ya. Groshev, Yu.A. Kravtsov: Izv. VUZ Radiofiz. 11, 1812 (1968) [English transl.:

Radiophys. Quantum Electron. 11, 1019 (1968)] 5.23 1.1. Dustermaat: Commun. Pure Appl. Math. 27, 207 (1974) 5.24 M.V. Berry: Adv. Phys. 25, 1 (1976) 5.25 G. Dangelmayr, W. Veit: Ann. Phys. 118, 108 (1979) 5.26 1.8. Keller: A geometrical theory of diffraction, in Calculus of Variations and its Applications,

Proc. Symp. Appl. Math., Vol. 8 (McGraw-HiII, New York 1958) pp. 27-52 5.27 D. Ludwig: SIAM Rev. 12, 325 (1970) 5.28 F.W.J. Olver: Philos. Trans. Roy. Soc. London A 247, 328 (1954) 5.29 B.T. Kormilitsyn: Radiotech. Elektron. II, 1130 (1966) [English transl.: Radio Eng. Electron.

Phys. 11, no. 6, 988-992 (1966)] 5.30 Yu.1. Orlov: Izv. VUZ Radiofiz. 9, 497, 657 (1966) [English transl.: SOY. Radiophys. 9, 307, 394

(1966)] 5.31 Yu.1. Orlov: Izv. VUZ Radiofiz. 11,317 (1968) [English transl.: Radiophys. Quantum Elec­

tron. 11, 180 (1968)] 5.32 Yu.1. Orlov: Izv. VUZ Radiofiz. 10, 30 (1967) [English transl.: Radiophys. Quantum Electron.

10,21 (1967)] 5.33 Yu.1. Orlov: Izv. VUZ Radiofiz. 20, 1669 (1977) [English transl.: Radiophys. Quantum

Electron. 20, 1148 (1977)] 5.34 Yu.1. OrIov: Izv. VUZ Radiofiz. 13,412 (1970) [English transl.: Radiophys. Quantum Electron

13,320 (1970)] 5.35 Yu.1. Orlov, S.K. Tropkin: Izv. VUZ Radiofiz. 23, 1473 (1980) [English transl.: Radiophys.

Quantum Electron. 23, 979 (1980)]

References 201

5.36 V.E. Grikurov: Izv. VUZ Radiofiz. 23, \038 (1980) [English transl.: Radiophys. Quantum Electron. 23, 690 (1980)]

5.37 A.S. Kryukovskii, D.S. Lukin, E.A. Palkin: Izv. VUZ Radiofiz. 25,1375 (1982) (in Russian) 5.38 1.M. Martin, S.M. Flatte: Appl. Opt. 27,2111 (1988) 5.39 S.M. Rytov, Yu.A. Kravtsov, V.I. Tatarskii: Principles of Statistical Radiophysics, Vol. 4: Wave

Propagation Through Random Media (Springer, Berlin, Heidelberg 1989)

Chapter 6

6.1 V.P. Maslov: Theorie des Perturbations et Methode Asymptotique (Dunod, Paris 1972) 6.2 YuA Kravtsov: SOY. Phys. - Acoust. 14, 1 (1968) 6.3 1.M. Arnold: Radio Sci. 17, 1181 (1982) 6.4 R. Ziolkowski, G.A. Deschamps: Rad. Sci. 19, 1001 (1984) 6.5 B.R. Vainberg: In Teoriya Rasprostraneniya Voln v Neodnorodnykh i Nelineinykh Sredakh

(theory of wave propagation in inhomogenous and nonlinear media) (Izd. IRE Akad. Nauk SSSR, Moscow 1979) p. 144

6.6 V.P. Maslov, M.V. Fedoryuk: Semiclassical Approximation in Quantum Mechanics (Reidel, Higham, MA 1981)

6.8 L. Hormander: Fourier Integral Operators 127,79 (1971) 6.9 A.D. Gorman, R. Wells, G.N. G1eming: 1. Phys. A 14, 1519 (1981) 6.10 K. Hogo, Yu. Vi: Radio Sci. 22, 357 (1987) 6.11 D.S. Lukin, E.A. Palkin: In Teoreticheskoye i Experimentalnoye Issledovanie Rasprostraneniya

Dekametrovykh Radiovoln (theoretical and experimental investigation of decameter radiowave propagation) (IZMIRAN SSSR, Moscow 1976) p. 149

6.12 A.A. Kryukovsky, D.S. Lukin, E.A. Palkin: Volnovaya teoriya katastrof (wave theory of catastrophes) (Nauka, Moscow) in press

Chapter 7

7.1 Yu.l. Orlov: Trudy MEl No.1 19 (Moscow Power Engineering Institue, Moscow 1972) p. 82 7.2 Yu.l. Orlov, A.V. Demin: Trudy MEl No. 553 (Moscow Power Engineering Institute,

Moscow 1981) p. 5 7.3 Yu.l. Orlov, A.V. Demin: Trudy MEl No. 497 (Moscow Power Engineering Institute,

Moscow 1980) p. 10 7.4 V.B. Avdeyev, A.P. Yarygin: Theses 12th All-Union Conf. Radiowave Propagation (Nauka,

Moscow 1978) Vol. 4, p. 294 (in Russian) 7.5 V.N. Mirolyubov, M.V. Tinin: Theses 8th All-Union Conf. Diffraction and Wave Propaga­

tion (Acad. Nauk USSR, Moscow 1982) Vol. 2, p. 156 (in Russian) 7.6 M.V. Tinin: Izv. VUZ Radiofiz. 26, 36 (1983) [English transl.: Radiophys. Quantum Electron.

26, 29 (1983)] 7.7 V.B. Avdeyev, A.V. Demin, Yu.A. Kravtsov, M.V. Tinin, A.P. Yarygin: Izv. VUZ Radiofiz. 31,

1279 (1988) [English transl.: Radiophys. Quantum Electron. 31, 907 (1988)] 7.8 Yu.l. Orlov: Izv. VUZ Radiofiz. 17, 1036 (1974) [English trans\.: Radiophys. Quantum

Electron. 17, 788 (1974)] 7.9 L.A. Vainstein, E.S. Birger, N.V. Konyukhova, E.L. Kosarev: SOY. 1. Plasma Phys. 26, 362

(1976) 7.10 L.A. Vainstein, EA Tishchenko: Zh. Tekh. Fiz. 46, 2271 (1976) [English transl.: SOY. Phys.

- lTP 21, 1338 (1976)] 7.11 L.A. Vainstein, P.Ya Ufimtsev: Radiotekn. Elektron. 25, 625 (1982) [English transl.: Radio

Eng. Electron. Phys. 25, 411 (1982)]

202 References

7.12 J.M. Arnold: Radio Sci. 17, 1181 (1982) 7.13 J.M. Arnold: lEE Proc. 133, 165 (1986) 7.14 V.V. Stepanov: Kurs DifJerentsialnykh Uravnenii (course of differential equations) (Fizmatgiz,

Moscow 1958); G.A. Korn, T.M. Korn: Mathematical Handbookfor Scientists and Engineers (McGraw-Hili, New York 1961)

7.15 Yu.A. Kravtsov, Yu.1. Orlov: Geometrical Optics of Inhomogeneous Media, Springer Ser. Wave Phenom., Vol. 6 (Springer, Berlin, Heidelberg 1990)

7.16 V.B. Avdeyev, N.V. Shilov, A.P. Yarygin: Theses 12th All-Union Conf. on Radiowave Propagation (Nauka, Moscow 1981) Vol. 1, p. 302 (in Russian)

7.17 V.B. Avdeyev, A.P. Yarygin: Rasseyanie Elektromagnitnykh Voln (scattering of EM waves) (Institute of Radio Engineering, Taganrog 1983) Vol. 4, p. 40

7.18 L.M. Brekhovskikh: Acoustics of Layered Media I, II, Springer Ser. Wave Phenom., Vols. 6, 10 (Springer, Berlin, Heidelberg 1990, 1992)

7.19 D. Ludwig: Commun. Pure Appl. Math. 20, 103 (1967) 7.20 M.V. Tinin: Issledovaniya po Geomagnetizmu, Aeronomii i Fizike Solntsa (studies in geomag­

netism, aeronomy and solar physics) (Nauka, Moscow 1973) Vol. 29, p. 157 7.21 V.M. Babich M.M. Popov: Izv. VUZ Radiofiz. 32, 1447 (1989) [English transl.: Radiophys.

Quantum Electron. 32, 1063 (1989)]

Chapter 8

8.1 Yu.l.Orlov: Radiotekh. Elektron. 20, 242 (1975) [English transl.: Radio Eng. Electron. Phys. 20, 145 (1975)]

8.2 L. Levey, L.B. Felsen: Radio Sci. 4, 959 (1969) 8.3 L. Levey, L.B. Felsen: J. Instt. Math. Appl. 3, 76 (1976) 8.4 M.M. Agrest, M.Z. Maksimov: Teoriya Nepolnykh Tsilindrichekikh Funktsii i ikh Prilozheniya

(theory of incomplete cylindrical functions and their applications) (Atomizdat, Moscow 1965) 8.5 D.S. Ahluwalia, R.M. Lewis, J. Boersma: SIAM J. Appl. Math. 16,783 (1968) 8.6 R.M. Lewis, J. Boersma: J. Appl. Math. 10,2291 (1969) 8.7 D.S. Ahluwalia: SIAM J. Appl. Math. 18,287 (1970) 8.8 R.S. Hansen: Geometrical Theory of DifJraction (IEEE Press, New York 1981) 8.9 Yu.l. Orlov, SA Vlasov: Izv. VUZ Radiofiz. 21, 422 (1978) [English transl.: Radiophys.

Quantum Electron. 21, 290 (1978) 8.10 Yu.1. Orlov: Radiotekh. Electron. 21, 730 (1976) [English transl.: Radio Eng. Electron. Phys.

21, no. 4, 50 (1976)] 8.1 1 Yu.A. Kravtsov, Yu.1. Orlov: Geometricl Optics of Inhomogeneous Media, Springer Ser. Wave

Phenom., Vol. 6 (Springer, Berlin, Heidelberg 1990) 8.12 Yu.1. Orlov: Radiotekh. Elektron. 21, 62 (1976) [English transl.: Radio Eng. Electron. Phys.

21, no. 1,50 (1976) 8.1 3 Yu.l. Orlov, SA Vlasov: Radiotekh. Elektron. 23, 17 (1978) [English transl.: Radio Eng.

Electron. Phys. 23, no. 1, 14 (1978) 8.14 V.I. Arnold: Usp. Mat. Nauk 34, 91, 206 (1978) 8.15 V.1. Arnold, A.N. Varchenko, S.M. Gussein-Zade: Singularities of DifJerential Mappings

(Brikhiiuser, Boston, MA 1985, 1988) Vols. I, 2 8.16 E.B. Ipatov, A.S. Kryukovskii, D.S. Lukin, E.A. Palkin: Dok\. Akad. Nauk USSR 291, 823

(1986) [English transl.: SOY. Phys. - Doklady 31, 962 (\986)] 8.17 A.S. Kryukoskii, D.S. Lukin, E.A. Palkin: SOY. J. Numer. Anal. Math. Model 2, 279 (1987) 8.18 A.S. Kryukovskii, D.S. Lukin, E.A. Palkin: Krayevye i Uglovye Katastrofy v Zadachakh

Difraktsii i Rasprostraneniya Voln (edge and angle catastrophes in problems of diffraction and wave propagation) (Aviation Institute, Kazan 1988)

References 203

Chapter 9

9.1 J.N.L. Connor, W.J.E. Southall: J. Chern. Phys.90, 355 (1986) 9.2 M.V. Berry: Adv. Phys. 25, 1 (1976) 9.3 V.I. Tokatly, B.E. Kinber: Izv. VUZ Radiofiz. 14, 761 (1971) [English trans\.: Radiophys.

Quantum Electron. 14,390 (1971)] 9.4 E.D. Gazazyan, B.Ye. Kinber: Izv. VUZ Radiofiz. 14, 1219 (1971) [English trans\.: Radiophys.

Quantum Electron. 14,601 (1971)] 9.5 E.D. Gazazyan, A.D. Ter-Pogosyan: Izv. Akad. Nauk Armenii - Fizika 9,169 (1974) 9.6 E.D. Gazazyan, M.I. Ivanyan, B.Ye. Kinber: Izv. Akad. Nauk Armenii - Fizika 13,87 (1978) 9.7 D.S. Ahluwalia, R.M. Lewis, J. Boersma: SIAM J. App\. Math. 16,783 (1968) 9.8 R.M. Lewis, J. Boersma: J. Math. Phys. 10,2291 (1969) 9.9 D.S. Ahluwalia: SIAM J. App\. Math. 18,287 (1970) 9.10 Yu.1. Orlov, S.K. Tropkin: Izv. VUZ Radiofiz. 24, 334 (1981) [English trans\.: Radiophys.

Quantum Electron. 24, 233 (1981)] 9.11 S.K. Tropkin: Razrabotka metodov rascheta blizhnikh polei ploskikh aperturnykh antenn

(calculation of near fields of plane aperture antennas). Dr. Sci. Thesis, Moscow Power Engineering Institute (1984)

9.12 Yu.1. Orlov, S.K. Tropkin: Izv. VUZ Radiofiz. 24, 1383 (1981) [English trans\.: Radiophys. Quantum Electron. 24, 936 (1981)]

9.13 E.T. Whittaker, G.N. Watson: A Course of Modern Analysis (Cambridge Univ. Press, Cambridge 1927)

9.14 V.L. Ginzburg: The Propagation of Electromagnetic Waves in Plasmas (Pergamon, Oxford 1970)

9.15 F.W.J. Olver: Philos. Trans. Roy. Soc. London A 247, 307 (1956); ibid. A 249, 65 (1958); ibid. A 250, 479 (1958)

9.16 E.R. Pike: Quart. J. Mech. App\. Math. 17,369 (1964) 9.17 Yu.A. Kravtsov: Izv. VUZ Radiofiz. 10, 1283 (1967) [English trans\.: Radiophys. Quantum

Electron. 10,719 (1967)] 9.18 Yu.A. Kravtsov: Rays and caustics as physical objects. Progress in Optics 26, 227-348 (North­

Holland, Amsterdam 1988) 9.19 A.A. Asatryan, Yu.A. Kravtsov: Izv. VUZ Radiofiz. 31, 1053 (1988) [English trans\.:

Radiophys. Quantum Electron. 31, 746 (1988)] 9.20 Yu.A. Kravtsov: Izv. VUZ Radiofiz. 8, 659 (1965) [English trans\.: Sov. Radiophys. 8, 467

(1965)] 9.21 V.A. Kaloshin, V.I. Orlov: Radiotekh. Elektron. 18,2028 (1973) [English trans\.: Radio Eng.

Electron. Phys. 18, 1485 (1973)] 9.22 Yu.l. Orlov: Izv. VUZ Radiofiz. 9,1036 (1966) [English trans\.: Sov. Radiophys. 9, 604 (1966) 9.23 L.D. Landau, E.M. Lifshitz: Quantum Mechanics, Nonrelativistic Theory (Pergamon, Oxford

1962) 9.24 H.K.V. Lotsch: Optik 30, 1, 181,217,563 (1969/70) 9.25 A.V. Popov: Dokl Akad. Nauk USSR 230, 1322 (1976) [English trans\.: Sov. Phys. - Dok\. 21,

564 (1976)] 9.26 J.B. Keller, S. Rubinov: Ann. Phys. 2, 24 (1960) 9.27 L.A. Vainstein: Open Resonators and open Waveguides (Golden, Boulder, CO 1969) 9.28 V.M. Babich, V.S. Buldyrev: Short-Wavelength Diffraction Theory, Springer Ser. Wave Phe­

nom., Vo\. 4 (Springer, Berlin, Heidelberg 1991) 9.29 V.P. Bykov: Izv. VUZ Radiofiz. 14, 880 (1971) [English trans\.: Radiophys. Quantum Elec­

tron. 14,692 (1971)] 9.30 E.D. Gazazyan, M.I. Ivanyan: Radiotekh. Elektron. 29, 830 (1984) [English trans\.: Radio

Eng. Electron. Phys. 29, no. 5, 12 (1984)] 9.31 E.D. Gazazyan: Uniform Short-Wave Asymptotics of Scalar and EM Fields Based on One­

Dimensional Canonic Functions (Physical Institute, Erevan 1988) Preprint 1092-(55)-88 (in Russian)

204 References

9.32 V.P. Bykov: Elekronika Bolshikh Moshchnostei (power electronics) (Nauka, Moscow 1965) No.4, p. 66

9.33 L.A. Vainstein: Elektronika Bolshikh Moshchnostei (power electronics) (Nauka, Moscow 1965) No.4, p. 93

9.34 E.O. Gazazyan, M.1. Ivanyan: Radiotekh. Elektron. 21, 2052 (1976) [English transl.: Radio Eng. Electron. Phys. 21, no. 10, 16 (1976)]

9.35 Yu.l. OrJov: Izv. VUZ Radiofiz. 24, 213 (1981) [English transl.: Radiophys. Quantum Elec­tron.24, 154 (1981)]

9.36 V.V. Zheleznyakov, V.V. Kocharovskii, VI.V. Kocharovskii: Usp. Fiz. Nauk 141,257 (1983) [English transl.: SOY. Phys. - Uspekhi 26, 877 (1983)]

9.37 R.A. Markus: J. Chern. Phys. 57, 4903 (1972) 9.38 H. Kreek, R.L. Ellis, R.A. Markus: J. Chern. Phys. 62, 913 (1975) 9.39 J.N.L. Connor: Mol. Phys. 31, 33 (1976)

Chapter 10

10.1 L.B. Felsen, N. Marcuvitz: Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, NJ 1973) Vols. 1,2

10.2 Yu.A. Kravtsov, Yu.1. Orlov: Geometrical Optics on Inhomogeneous Media, Springer Ser. Wave Phenom., Vol. 6 (Springer, Berlin, Heidelberg 1990)

10.3 Yu.A. Kravtsov, L.A. Ostrovskii, N.S. Stepanov: Proc. IEEE 62, 1492 (1974) 10.4 Yu.1. OrJov: Izv. VUZ Radiofiz. 24, 224 (1981) [English transl.: Radiophys. Quantum Elec­

tron.24, 154 (1981)] 10.5 R.M. Lewis: Electromagnetic Wave Theory (Pergamon, Oxford 1967) Pt. 2, p. 845 10.6 Yu.l. Orlov: Pryamye i Obratnye Zadachi Teorii Difraktsii (direct and inverse problems of

diffraction theory) (Izd. IRE AN USSR, Moscow 1979) p. 5 10.7 Yu.1. Orlov, A.P. Anyutin: Izv. VUZ Radiofiz. 17, 1369 (1974); ibid. 19, 335, 495 (1976)

[English transl.: Radiophys. Quantum Electron. 17, 1047 (1974); ibid. 19,235 and 347 (1976)] 10.8 A.P. Anyutin, Yu.l. Orlov: Radiotekh. Elektron. 22, 2082 (1977) [English transl.: Radio Eng.

Electron. Phys. 22, no. 10,62 (1977)] 10.9 Yu.A. Kravtsov: Izv. VUZ Radiofiz. 11, 1582 (1968) [English transl.: Radiophys. Quantum

Electron. 11,888 (1968)] 10.10 V.P. Maslov: Theorie des Perturbations et Methode Asymptotique (Dunod, Paris 1972) 10.11 E. Arbel, L.B. Felsen: Electromagnetic Theory and Antennas (Pergamon, London 1963) ps.

391-421 10.12 Yu.A. Kravtsov: Radiophys. Quantum Electron. 10,719 (1967) 10.13 V.A. Vasilyev: Funkts. Anal. 11, 1 (1979) ibid. 13, 1 (1979) 10.14 S.M. Rytov, Yu.A. Kravtsov, V.1. Tatarskii: Principles of Statistical Radiophysics IV(Springer,

Berlin, Heidelberg 1989) 10.15 M.V. Berry: 1. Phys. A 10,2061 (1977) 10.16 J.M. Martin, S.M. Flatt€:: Appl. Opt. 27, 2111 (1988) 10.17 Yu.A. Kravtsov: SOY. Phys. - JETP 28, 769 (1969) 10.18 S.M. Flatt€:, R. Dashen, W.H. Munk, K.M. Watson, F. Zachariasen: Sound Transmission

Through a Fluctuating Ocean (Cambridge Univ. Press, Cambridge 1979) 10.19 Yu.A. Kravtsov: Rys and caustics as physical objects, in Progress in Optics 26, 227-348

(North-Holland, Amsterdam 1988) 10.20 V.A. Kulkarny, B.S. White: Phys. Fluids 25,1770 (1982) 10.21 0.1. Zwillinger, B.S. White: Wave Motion 7, 207 (1985) 10.22 H.M. Nussenzweig: Sci. Am. 236, 116 (April 1977) 10.23 V. Buck: Rev. Mod. Phys. 46, 369 (1974) 10.24 J.N.L. Connor: In Semiclassical Methods in Molecular Scattering and Spectroscopy, ed. by

M.S. Child (Reidel, Oordrecht 1980) p. 45

References 205

10.25 J.N.L. Connor, P.R. Curtis, C.J. Edge, A. Laganda: J. Chern. Phys. SO, 1362 (1980) 10.26 M.V. Berry, M. Tabor: Proc. Roy. Soc. London A 349, 101 (1976) 10.27 M.V. Berry, K.E. Mount: Rept. Progr. Phys. 35, 315 (1972) 10.28 S.W. Hawking, G.F.R. Ellis: Large Scale Structure of Space-Time (Cambridge Univ. Press,

Cambridge 1973) 10.29 Ya.B. Zeldovich, A.V. Shamaev, S.F. Shandarin: Usp. Fiz. Nauk 139, 153 (1983) [English

trans!.: Sov. Phys. - Usp. 26, 46 (1983)] 10.30 A.D. Gorman, R. Wi1\s: J. Acoust. Soc. Am. 73, 363 (1983) 10.31 1.1. Bobbit, E. Cumberbatch: Stud. App!. Math. 55, 229 (1976) 10.32 R.J. Smith: Fluid Mech. 77, 417(1976) 10.33 Y. Pomeau: Europhys. Lett. 11,713 (1990)

List of Symbols

Ai(.) ar, arx, ary E,H :ff h I((,t) f(c, i) III

" K Ne p = vtjJ Q r O, tjJ0, UO

ro, no u(r, t) U(r) A e A !J>(C, t) 'P((,t)

Airy function Fresnel radii EM field Hamiltonian Hessian dimensionless standard integral dimensional standard integral longitudinal caustic scale divergence curvature; see Kreb K e , KR unit normal to a caustic momentum initial surface illustrating superscript zero subscript wave field amplitude increment, Laplacian permittivity transverse caustic scale generating or phase function non-polynomial phase function

Subject Index

Airy equation 57 function 21 generalized function 64 integral 54

Angle catastrophes 153 Anomalous caustic phase shift 19, 187, 191 Applicability conditions

for geometrical optics 16 for uniform asymptotics 91

Asymptotics Airy 80,89 local 101 uniform 73,87,98

Axially symmetric caustics 158

Broken caustics 143, 146, 187 Butterfly catastrophe 44, 111

Canonical operator 120 Canonical operator (Maslov's) method 116 Catastrophes

angle 153 butterfly 44, 111 cusp 42 edge 151 fold 42 swallowtail 43, 109 unfolding of 40

Caustics anomalous phase shift 19, 187, 191 as catastrophes 34 as envelopes of ray families 17 axially symmetric 158 broken 143, 146, 187 circular 103 classification 39 codimension 39 corank 39 ellipsoid cavity 180 external parameters 39 index 49 in dispersive media 184 internal parameters 39

modality 44 of higher nodality 44 penumbra 142, 148 phase shift 18 physical characteristics 17 quantum mechanical problems 194 random 3,27,192 reality of 25 reflection from barrier 166 scales 84 simple 44 space-time 184 standard integrals 47, 154, 157, 162 structural stability of 36 structurally stable 39 structurally unstable 157 subordinate relations 47,70 volume 19 waveguiding 173 with penetration 169 zone 19

Codimension 39 generalized 49

Complex caustics 191 Complex rays 27

domain of localization 33 nonlocal nature of 30 physical content 31

Conservation of energy flow 23 Coordinate-momentum representation 116 Corank 39

Diffraction rays 148 Distinguishability of rays 17

Eikonal equation 8 Electromagnetic waves 11

in anisotropic medium 12 in isotropic medium 11

Equation Airy 57 eikonal 8 Weber 165

210 Subject Index

Focusing index 50 Fold catastrophe 42 Fresnel index 49 Fresnel volume of ray 14

Generalized Airy functions 64 Generating function 39 Geometrical backbone of wave field 86 Geometrical optics 8

zeroth approximation 10

Index of focusing (singularity) 50 Interference integral 129

applicability limits 141 caustic type of 137 Orlov's method 129 ray type of 129

Interpolation formula 85

Lagrange manifold 36 Locality principle 13 Longitudinal caustic scale 65, 84

Maslov's method 116 Modality 44 Momentum of ray 9 Multipath propagation 26

Partial waves 130 Pearsey integral 60 Penumbra caustics 142, 148 Polarization vector 12

Random caustics 3,27, 192 Ray 9

as energy trajectory 13 as phase trajectory 13 complex 27, 30, 31, 33 coordinates 10 diffraction 148 distinguishability of 17

family 17 Fresnel volume of 14 momentum of 9 physical content 14

Singularity index 50 Space-time caustics 184 Space-time lenses 187 Standard caustic integrals 47

contour type 157 with amplitude correction 162 with non-polynomial phase 154

Standard functions 177 Structurally stable caustics 39

classification of 39 Structurally unstable caustics 157 Subasymptotics 66 Swallowtail catastrophe 43

in a linear layer

Thorn's seven catastrophes 41 Transfer equation 8 Transverse caustic scale 65, 84

Umbilic 43 elliptic 43, 112, 151 hyperbolic 43

Unfolding of catastrophes 40 Uniform asymptotics 73

relation to geometrical optics 95

Virtual rays 133

Wave field geometrical backbone 86 ray estimates at caustics 23

Waveguiding caustics 173 Weber asymptotics 176 Weber equation 165 Weber function 165