×

zbMATH — the first resource for mathematics

The mathematical physics of rainbows and glories. (English) Zbl 0974.78008
Summary: A detailed qualitative summary of the optical rainbow is provided at several complementary levels of description, including geometrical optics (ray theory), the Airy approximation, Mie scattering theory, the complex angular momentum (CAM) method, and catastrophe theory. The phenomenon known commonly as the glory is also discussed from both physical and mathematical points of view: backward glories, rainbow-glories and forward glories. While both rainbows and glories result from scattering of the incident radiation, the primary rainbow arises from scattering at about \(138°\) from the forward direction, whereas the (backward) glory is associated with scattering very close to the backward direction. In fact, it is a more complex phenomenon physically than the rainbow, involving a variety of different effects (including surface waves) associated with the scattering droplet. Both sets of optical phenomena – rainbows and glories – have their counterparts in atomic, molecular and nuclear scattering, and these are addressed also.
The conceptual foundations for understanding rainbows, glories and their associated features range from classical geometrical optics, through quantum mechanics (in particular scattering from a square well potential; the associated Regge poles and scattering amplitude functions) to diffraction catastrophes. Both the scalar and the electromagnetic scattering problems are reviewed, the latter providing details about the polarization of the rainbow that the scalar problem cannot address. The basis for the complex angular momentum (CAM) theory (used in both types of scattering problem) is a modification of the Watson transform, developed by Watson in the early part of this century in the study of radio wave diffraction around the Earth. This modified Watson transform enables a valuable and accurate approximation to be made to the Mie partial-wave series, which while exact, converges very slowly at high frequencies.
The theory and many applications of the CAM method were developed in a fundamental series of papers by Nussenzveig and co-workers (including an important interpretation based on the concept of tunneling), but many other contributions have been made to the understanding of these beautiful phenomena, including descriptions in terms of so-called diffraction catastrophes. The rainbow is a fine example of an observable event which may be described at many levels of mathematical sophistication using distinct mathematical approaches, and in so doing the connections between several seemingly unrelated areas within physics become evident.
Reviewer: Reviewer (Berlin)

MSC:
78A45 Diffraction, scattering
78A05 Geometric optics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Sassen, K., J. opt. soc. am., 69, 1083, (1979)
[2] Nussenzveig, H.M., Sci. am., 236, 116, (1977)
[3] Berry, M.V., Sci. prog. oxf., 57, 43, (1969)
[4] Greenler, R., Rainbows, halos and glories, (1980), Cambridge University Press Cambridge
[5] Nussenzveig, H.M., J. math. phys., 10, 82, (1969)
[6] Nussenzveig, H.M., J. math. phys., 10, 125, (1969)
[7] van de Hulst, H.C., Light scattering by small particles, (1981), Dover New York
[8] Tricker, R.A.R., Introduction to meteorological optics, (1970), Elsevier New York
[9] Bohren, C.F.; Huffman, D.R., Absorption and scattering of light by small particles, (1983), Wiley New York
[10] Nussenzveig, H.M., Diffraction effects in semiclassical scattering, (1992), Cambridge University Press Cambridge
[11] Boyer, C.B., The rainbow, from myth to mathematics, (1987), Princeton University Press Princeton, NJ
[12] Logan, N.A., Proc. IEEE, 53, 773, (1965)
[13] Pedgely, D.E., Weather, 41, 401, (1986)
[14] Walker, J.D., Am. J. phys., 44, 421, (1976)
[15] Lynch, D.K.; Livingston, W., Color and light in nature, (1995), Cambridge University Press New York
[16] J.D. Austin, F.B. Dunning, Math. Teacher (1988) 484 (September issue); see also S. Janke, UMAP Module 724, COMAP, Inc., Lexington, MA, 1992.
[17] Whitaker, R.J., Phys. teacher, 12, 283, (1974)
[18] Humphreys, W.J., Physics of the air, (1964), Dover New York
[19] Ford, K.W.; Wheeler, J.A., Ann. phys., 7, 259, (1959)
[20] Hundhausen, E.; Pauly, H., Z. physik, 187, 305, (1965)
[21] Airy, G.B., Trans. camb. phil. soc., 6, 379, (1838)
[22] Fowles, G.R., Introduction to modern optics, (1975), Dover New York
[23] Born, M.; Wolf, E., Principles of optics, (1965), Pergamon Oxford
[24] Mie, G., Ann. physik, 25, 377, (1908)
[25] Watson, G.N., Proc. R. soc. A, 95, 83, (1918)
[26] Adam, J., Phys. rep., 142, 263, (1986)
[27] Adam, J., J. math. phys., 30, 744, (1989)
[28] Adam, J., Wave motion, 12, 385, (1990)
[29] Boyer, C.B., Am. J. phys., 18, 360, (1950)
[30] Van der Pol, B.; Bremmer, H., Phil. mag., 24, 141, (1937)
[31] B. Van der Pol, H. Bremmer, Phil. Mag. 24 (1937) 825. (See also 25 (1938) 817.)
[32] A. Sommerfeld, Partial Differential Equations in Physics, Academic Press, New York, 1964 (Appendix II).
[33] Nussenzveig, H.M., Ann. phys., 34, 23, (1965)
[34] Nussenzveig, H.M., (), 203
[35] Jackson, J.D., Phys. rep., 320, 27, (1999)
[36] Jackson, J.D., Classical electrodynamics, (1998), Wiley New York · Zbl 0114.42903
[37] Macdonald, J.E., Am. J. phys., 31, 282, (1963)
[38] Pernter, J.M.; Exner, F.M., Meteorologische optik, (1910), W. Braumüller Vienna
[39] Buchwald, E., Ann. phys., 43, 488, (1948)
[40] Malkus, W.V.R.; Bishop, R.H.; Briggs, R.O., NACA technical notes, 1622, (1948)
[41] Penndorf, R., J. opt. soc. am., 52, 402, (1962)
[42] Wang, R.T.; van de Hulst, H.C., Appl. opt., 30, 106, (1991)
[43] Ungut, A.; Grehan, G.; Gouesbet, G., Appl. opt., 20, 2911, (1981)
[44] Querfeld, C.W., J. opt. soc. am., 55, 105, (1965)
[45] Patitsas, A.J., Can. J. phys., 50, 3172, (1972)
[46] Wiscombe, W.J., Appl. opt., 19, 1505, (1980)
[47] Khare, V.; Nussenzveig, H.M., ()
[48] H.C. Bryant, N. Jarmie, Ann. Phys. (NY) 47 (1968) 127. See also Chapter 8 in Light from the Sky (readings from Scientific American), Freeman, San Francisco, 1980.
[49] Können, G.P., Polarized light in nature, (1985), Cambridge University Press Cambridge
[50] van de Hulst, H.C., J. opt. soc. am., 37, 16, (1947)
[51] See M. Born, E. Wolf, Principles of Optics, Pergamon, Oxford, 1965 and references therein.
[52] Naik, Y.G.; Joshi, R.M., J. opt. soc. am., 45, 733, (1955)
[53] V. Khare, in: A.D. Boardman (Ed.), Electromagnetic Surface Modes, Wiley, New York, 1982 (Chapter 11).
[54] Bryant, H.C.; Cox, A.J., J. opt. soc. am., 56, 1529, (1966)
[55] Fahlen, T.S.; Bryant, H.C., J. opt. soc. am., 58, 304, (1968)
[56] Maecker, H., Ann. phys., 4, 409, (1948)
[57] Woodward, B.W.; Bryant, H.C., J. opt. soc. am., 57, 430, (1967)
[58] Langley, D.S.; Morrell, M.J., Appl. opt., 30, 3459, (1991)
[59] Khare, V.; Nussenzveig, H.M., Phys. rev. lett., 33, 976, (1974)
[60] Probert-Jones, J.R., J. opt. soc. am. A, 1, 822, (1984)
[61] Nussenzveig, H.M., J. opt. soc. am., 69, 1068, (1979)
[62] Langley, D.S.; Marston, P.L., Phys. rev. lett., 47, 913, (1981)
[63] Marston, P.L.; Langley, D.S., J. opt. soc. am., 72, 456, (1982)
[64] Nussenzveig, H.M.; Wiscombe, W.J., Optics lett., 5, 455, (1980)
[65] Schiff, L.I., Quantum mechanics, (1968), McGraw-Hill New York · Zbl 0068.40202
[66] Ford, K.W.; Wheeler, J.A., Ann. phys., 7, 287, (1959)
[67] Bremmer, H., Terrestrial radio waves, (1949), Elsevier Amsterdam · Zbl 0033.32202
[68] Berry, M.V., Proc. phys. soc., 89, 479, (1966)
[69] Mason, E.A.; Monchick, L.J., J. chem. phys., 41, 2221, (1964)
[70] Mason, E.A.; Munn, R.J.; Smith, F.J., Endeavour, 30, 91, (1971)
[71] Buck, U., Rev. mod. phys., 46, 369, (1974)
[72] Chester, C.; Friedman, B.; Ursell, F., Proc. camb. phil. soc., 53, 599, (1957)
[73] Lighthill, M.J., Waves in fluids, (1987), Cambridge University Press Cambridge · Zbl 0153.30201
[74] Langer, R.E., Phys. rev., 51, 669, (1937)
[75] Jorna, S., Proc. R. soc. A, 281, 99, (1964)
[76] Rubinow, S.I., Ann. phys., 14, 305, (1961)
[77] Pekeris, C.L., Proc. symp. appl. math., 2, 71, (1950)
[78] Copson, E.T., Asymptotic expansions, (1965), Cambridge University Press Cambridge · Zbl 0123.26001
[79] Mott, N.F.; Massey, H.S.W., The theory of atomic collisions, (1949), Clarendon Press Oxford · Zbl 0039.22401
[80] Remler, E.A., Phys. rev. A, 3, 1949, (1971)
[81] Rich, W.G.; Bobbio, S.M.; Champion, R.L.; Doverspike, L.D., Phys. rev. A, 4, 2253, (1971)
[82] Goldberg, D.A.; Smith, S.M., Phys. rev. lett., 33, 715, (1974)
[83] Friedman, W.A.; McVoy, K.W.; Shuy, G.W.T., Phys. rev. lett., 33, 308, (1974)
[84] Miller, W.H., J. chem. phys., 48, 464, (1968)
[85] Miller, W.H., J. chem. phys., 51, 3631, (1969)
[86] Boyle, J.F., Mol. phys., 22, 993, (1971)
[87] Buck, U.; Kick, M.; Pauly, H., J. chem. phys., 56, 3391, (1972)
[88] Guimarães, L.G.; Nussenzveig, H.M., Opt. commun., 89, 363, (1992)
[89] Guimarães, L.G.; Nussenzveig, H.M., J. mod. opt., 41, 625, (1994)
[90] Nussenzveig, H.M., Nucl. phys., 11, 499, (1959)
[91] Eisberg, R.M., Fundamentals of modern physics, (1961), Wiley New York · Zbl 0097.39402
[92] Nussenzveig, H.M., Causality and dispersion relations, (1972), Academic Press New York · Zbl 0093.44202
[93] Wiscombe, W.J., Appl. opt., 19, 1505, (1980)
[94] Berry, M.V., Proc. phys. soc., 88, 285, (1966)
[95] Bertocchi, L.; Fubini, S.; Furlan, G., Nuovo cimento, 35, 596, (1965)
[96] Bollini, C.J.; Giambiagi, J.J., Nuovo cimento, 26, 619, (1962)
[97] Bollini, C.J.; Giambiagi, J.J., Nuovo cimento, 28, 341, (1963)
[98] Barut, A.O.; Calogero, F., Phys. rev., 128, 1383, (1962)
[99] Patashinskii, A.Z.; Pokrovskii, V.L.; Khalatnikov, I.M., Jetp, 17, 1387, (1963)
[100] Newton, R.G., The complex j-plane, (1964), Benjamin New York · Zbl 0119.44004
[101] Debye, P.J., Ann. phys. ser. 4, 30, 57, (1909)
[102] V. Khare, Ph.D. Thesis, University of Rochester, NY, 1975.
[103] de Bruijn, N.G., Asymptotic methods in analysis, (1958), North-Holland Amsterdam · Zbl 0082.04202
[104] Olver, F.W.J., Asymptotics and special functions, (1974), Academic Press New York · Zbl 0303.41035
[105] Watson, G.N., A treatise on the theory of Bessel functions, (1966), Cambridge University Press Cambridge · Zbl 0174.36202
[106] Keller, J.B., ()
[107] Szegö, G., Proc. London math. soc., 36, 2, 427, (1934)
[108] Khare, V.; Nussenzveig, H.M., Phys. rev. lett., 38, 1279, (1977)
[109] J.M. Peterson, Phys. Rev. 125 (1962) 955; K.W. McVoy, L. Heller, M. Bolsterli, Rev. Mod. Phys. 39 (1967) 245; K.W. McVoy, Ann. Phys. 43 (1967) 91.
[110] Wyatt, P.J., Phys. rev., 127, 1837, (1962)
[111] Rösch, S., Appl. opt., 7, 233, (1968)
[112] Fock, V.A., Electromagnetic diffraction and propagation problems, (1965), Pergamon Press Oxford
[113] Leontovich, M.A., Bull. acad. sci. USSR, 8, 16, (1944)
[114] Nussenzveig, H.M., Comm. at. mol. phys., 23, 175, (1989)
[115] Keller, J.B., J. opt. soc. am., 52, 116, (1962)
[116] Berry, M.V., J. phys. B, 2, 381, (1969)
[117] Nussenzveig, H.M., J. phys. A, 21, 81, (1988)
[118] Nussenzveig, H.M.; Wiscombe, W.J., Phys. rev. lett., 59, 1667, (1987)
[119] Nussenzveig, H.M.; Wiscombe, W.J., Phys. rev. A, 43, 2093, (1991)
[120] Berry, M.V.; Mount, K.E., Rep. prog. phys., 35, 315, (1972)
[121] Nussenzveig, H.M.; Wiscombe, W.J., Phys. rev. lett., 45, 1490, (1980)
[122] H.M. Nussenzveig, in: S. Haroche, J.C. Gay, G. Grynberg (Eds.), Atomic Physics 11, World Scientific Press, Singapore, 1989, p. 421 see also [114].
[123] Johnson, B.R., J. opt. soc. am., 10, 343, (1993)
[124] Labianca, F.M., J. acoust. soc. am., 53, 1137, (1973)
[125] J. Adam, Phys. Rep. 142 (1986) 263 (see Appendix 5).
[126] G.P. Können, J.H. de Boer, 18 (1979) 1961.
[127] Bricard, J., Ann. phys., 14, 148, (1940)
[128] J.R. Meyer-Arendt, Introduction to Classical and Modern Optics, Prentice-Hall, Englewood Cliffs, NJ, 1972 (Chapter 3.1).
[129] Jeffreys, H.; Jeffreys, B.S., Methods of mathematical physics, (1966), Cambridge University Press Cambridge · Zbl 0037.31704
[130] Pruppacher, H.R.; Beard, K., Q.J.R. meteorol. soc., 96, 247, (1970)
[131] Fraser, A.B., J. atmos. sci., 29, 211, (1972)
[132] Walker, J.D., Sci. am., 237, 138, (July 1977)
[133] Walker, J.D., Sci. am., 242, 174, (June 1980)
[134] Walker, D., Weather, 5, 324, (1950)
[135] Dave, J.V., Appl. opt., 8, 155, (1969)
[136] Mobbs, S.D., J. opt. soc. am., 69, 1089, (1979)
[137] Lynch, D.K.; Schwartz, P., Appl. opt., 30, 3415, (1991)
[138] Malkus, W.V.R., Weather, 10, 331, (1955)
[139] Voltz, F.E., Physics of precipitation, (1960), American Geophysical Union Washington DC
[140] Lock, J.A., J. opt. soc. am. A, 6, 1924, (1989)
[141] Fraser, A.B., J. opt. soc. am., 73, 1626, (1983)
[142] T. Young, Phil. Trans. Roy. Soc. xcii (1802) 12387.
[143] Spilhaus, A.F., J. meteorol., 5, 108, (1948)
[144] Möbius, W., Abh. kgl. saechs. ges. wiss. math. phys. kl., 30, 108, (1907)
[145] Fraser, A.B., J. atmos. sci., 29, 211, (1972)
[146] Minnaert, M.G.J., Light and colour in the open air, (1954), Dover New York
[147] Können, G.P., J. opt. soc. am. A, 4, 810, (1987)
[148] Minnaert, M.G.J., Light and colour in the outdoors, (1993), Springer New York
[149] Green, A.W., J. appl. meteorol., 14, 1578, (1975)
[150] Voltz, F.E., Handbuch der geophysik, vol. 8, (1961), Borntrager Berlin
[151] Marston, P.L.; Trinh, E.H., Nature, 312, 529, (1984)
[152] Pruppacher, H.R.; Klett, J.D., Microphysics of clouds and precipitation, (1978), Reidel Dordrecht
[153] Lock, J.A., Appl. opt., 26, 5291, (1987)
[154] Wood, R.W., Physical optics, (1934), Macmillan New York
[155] Gedzelman, S.D., J. opt. soc. am. A, 5, 1717, (1988)
[156] Zrnić, D.S.; Doviak, R.J.; Mahapatra, P.R., Radio sci., 19, 75, (1984)
[157] Rasmussen, R.; Walcek, C.; Pruppacher, H.R.; Mitra, S.K.; Lew, J.; Levizzani, V.; Wang, P.K.; Barth, U., J. atmos. sci., 42, 1647, (1985)
[158] Shipley, S.T.; Weinman, J.A., J. opt. soc. am., 68, 130, (1978)
[159] Liou, K-N.; Hansen, J.E., J. atmos. sci., 28, 995, (1971)
[160] Bosanac, S., Molec. phys., 35, 1057, (1978)
[161] Connor, J.N.L., (), 45
[162] Thylwe, K.E., J. phys. A, 16, 1141, (1983)
[163] Thylwe, K.E.; Connor, J.N.L., J. phys. A, 18, 2957, (1985)
[164] Connor, J.N.L.; Delos, J.B.; Carlson, C.E., Mol. phys., 31, 1181, (1976)
[165] Bosanac, S., J. math. phys., 19, 789, (1978)
[166] Connor, J.N.L.; Marcus, R.A., J. chem. phys., 55, 5636, (1971)
[167] Mullen, J.M.; Thomas, B.S., J. chem. phys., 58, 5216, (1973)
[168] Connor, J.N.L.; Farrelly, D.; Mackay, D.C., J. chem. phys., 74, 3278, (1981)
[169] Beck, D., J. chem. phys., 37, 2884, (1962)
[170] Buck, U.; Pauly, H., J. chem. phys., 54, 1929, (1971)
[171] Buck, U., Rev. mod. phys., 46, 369, (1974)
[172] Buck, U., Adv. chem. phys., 30, 313, (1975)
[173] Buck, U., (), 499
[174] Kleyn, A.W., Comm. at. mol. phys., 19, 133, (1987)
[175] Neskovic, N., Rainbows and catastrophes, (1990), Boris Kidric Institute of Nuclear Science Belgrade
[176] van den Biesen, J.J.H.; Hermans, R.M.; van den Meijdenberg, C.J.N., Physica A, 115, 396, (1982)
[177] Bernstein, R.B., (), 313
[178] see also [92].
[179] Kong, P.; Mason, E.A.; Munn, R.J., Am. J. phys., 38, 294, (1970)
[180] Levinson, N., Kgl. danske videnskab. selskab., mat.-fys. medd., 25, 9, (1949)
[181] Mason, E.A.; Nyeland, C.; van den Biesen, J.J.H.; van den Meijdenberg, C.J.N., Physica A, 116, 133, (1982)
[182] Schutte, A.; Bassi, D.; Tommassini, F.; Scoles, G., Phys. rev. lett., 29, 979, (1972)
[183] Bosanac, S., Mol. phys., 36, 453, (1978)
[184] Bosanac, S., Phys. rev. A, 19, 125, (1979)
[185] Korsch, H.J.; Thylwe, K.E., J. phys. B, 16, 793, (1983)
[186] Brink, D.M., Semiclassical methods for nucleus-nucleus scattering, (1985), Cambridge University Press Cambridge · Zbl 0586.47004
[187] Frahn, W.E., Diffractive processes in nuclear physics, (1985), Oxford University Press Oxford
[188] da Silveira, R., (), 103
[189] Pato, M.P.; Hussein, M.S., Phys. rep., 189, 127, (1990)
[190] M.S. Hussein, K.W. McVoy, in: D. Wilkinson (Ed.), Progress in Nuclear and Particle Physics, Vol. 12, Pergamon Press, Oxford, 1984, p. 103; see also R.C. Fuller, Phys. Rev. C 12 (1975) 1561.
[191] Put, L.W.; Paans, A.M.J., Nucl. phys., A 291, 93, (1977)
[192] Fuller, R.C.; Moffa, P.J., Phys. rev., C 15, 266, (1977)
[193] Brink, D.M.; Takigawa, N., Nucl. phys., A 279, 159, (1977)
[194] Takigawa, N.; Lee, Y.S., Nucl. phys., A 292, 173, (1977)
[195] Hussein, M.S.; Nussenzveig, H.M.; Villari, A.C.C.; Cardaso, J.L., Phys. lett., B 114, 1, (1982)
[196] Lipperheide, R., Nucl. phys., A 469, 190, (1987)
[197] Marty, C., Z. phys., A 309, 261, (1983)
[198] Barrette, J.; Alamanos, N., Nucl. phys., A 441, 733, (1985)
[199] Ueda, M.; Takigawa, N., Nucl. phys., A 598, 273, (1996)
[200] Ueda, M.; Pato, M.P.; Hussein, M.S.; Takigawa, N., Phys. rev. lett., 81, 1809, (1998)
[201] Ueda, M.; Pato, M.P.; Hussein, M.S.; Takigawa, N., Nucl. phys., A 648, 229, (1999)
[202] B. Schrempp, F. Schrempp, Phys. Lett. B 70 (1977) 88; see also B. Schrempp, F. Schrempp, Nuovo Cimento Lett. 20 (1977) 95.
[203] B. Schrempp, F. Schrempp, Nucl. Phys. B 163 (1980) 397; see also P.D.B. Collins, An Introduction to Regge Theory and High Energy Physics, Cambridge University Press, Cambridge, 1977. · Zbl 1384.60044
[204] Levy, B.R.; Keller, J.B., Canad. J. phys., 38, 128, (1960)
[205] Fricke, S.H.; Brandan, M.E.; McVoy, K.W., Phys. rev., C 38, 682, (1988)
[206] Brandan, M.E.; Satchler, G.R., Phys. rep., 285, 143, (1997)
[207] Feynman, R.P.; Leighton, R.B.; Sands, M., The Feynman lectures in physics, (1964), Addison-Wesley Reading, MA · Zbl 0131.38703
[208] Berry, M.V.; Upstill, C., (), 257
[209] Thom, R., Structural stability and morphogenesis, (1989), Addison-Wesley New York
[210] Arnold, V.I., Russian math. surveys, 26, 1, (1968)
[211] Gilmore, R., Catastrophe theory for scientists and engineers, (1981), Wiley New York · Zbl 0497.58001
[212] Poston, T.; Stewart, I., Catastrophe theory and its applications, (1978), Pitman Boston · Zbl 0382.58006
[213] Trinkaus, H.; Drepper, F., J. phys. A, 10, L11, (1977)
[214] Connor, J.N.L., Mol. phys., 31, 33, (1976)
[215] Connor, J.N.L.; Child, M.S., Mol. phys., 18, 653, (1970)
[216] Connor, J.N.L., Mol. phys., 26, 1217, (1973)
[217] Berry, M.V., J. phys., A 8, 566, (1975)
[218] Garibaldi, U.; Levi, A.C.; Spadacini, R.; Tommei, G.E., Surf. sci., 48, 649, (1975)
[219] Connor, J.N.L., Mol. phys., 27, 853, (1974)
[220] Berry, M.V., Adv. phys., 25, 1, (1976)
[221] Larmor, J., Proc. camb. phil. soc., 7, 131, (1891)
[222] Pearcey, T., Phil. mag., 37, 311, (1946)
[223] Berry, M.V., J. phys., A 10, 2061, (1977)
[224] Marston, P.L.; Trinh, E.H., Nature, 312, 529, (1984)
[225] Marston, P.L., Opt. lett., 10, 588, (1985)
[226] Marston, P.L., J. acoust. soc. am., 81, 226, (1987)
[227] Dean, C.L.; Marston, P.L., Appl. opt., 30, 3443, (1991)
[228] Simpson, H.J.; Marston, P.L., Appl. opt., 30, 3468, (1991)
[229] Marston, P.L., Appl. opt., 19, 680, (1980)
[230] Nye, J.F., Nature, 312, 531, (1984)
[231] Lock, J.A., Appl. opt., 26, 5291, (1987)
[232] Walker, J.D., Am. J. phys., 44, 421, (1976)
[233] Walker, J.D., Sci. am., 237, 138, (1977)
[234] Walker, J.D., Sci. am., 242, 174, (1980)
[235] Berry, M.V.; Mount, K.E., Rep. prog. phys., 35, 315, (1972)
[236] Miller, W.H., Adv. chem. phys., 25, 63, (1974)
[237] George, T.F., Ann. rev. phys. chem., 24, 263, (1973)
[238] Landau, L.D.; Lifshitz, E.M., Mechanics, (1960), Pergamon Press Oxford
[239] Newton, R.G., Scattering theory of waves and particles, (1982), Springer New York · Zbl 0496.47011
[240] Myers, W.D., (), 1
[241] Sir Isaac Newton, Opticks, Q.31, Book 3, Part 1, Dover, New York, 1979, p. 1704.
[242] Haberman, R., Elementary applied differential equations, (1987), Prentice-Hall Englewood Cliffs, NJ
[243] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1972), Dover New York · Zbl 0515.33001
[244] Bowman, J.J.; Senior, T.B.A.; Uslenghi, P.L.E., Electromagnetic and acoustic scattering by simple shapes, (1969), North-Holland Amsterdam
[245] Fock, V.A., Diffraction of radio waves around the Earth’s surface, (1946), Publishers of the USSR Academy of Sciences Moscow · Zbl 0063.01395
[246] Frautschi, S.C., Regge poles and S-matrix theory, (1963), Benjamin New York · Zbl 0129.42501
[247] J.R. Taylor, Scattering Theory, Wiley, New York, 1972 (Chapter 15).
[248] Adam, J.A., Astrophys. space sci., 220, 179, (1994)
[249] Regge, T., Nuovo cimento, 18, 947, (1960)
[250] Barut, A.O.; Calogero, F., Phys. rev., 128, 1383, (1962)
[251] Calogero, F., Nuovo cimento, 28, 701, (1963)
[252] Cheng, H., Nuovo cimento, 45, 487, (1966)
[253] Titchmarsh, E.C., Introduction to the theory of Fourier integrals, (1937), Oxford University Press Oxford · Zbl 0017.40404
[254] Morse, P.M.; Feshbach, H., Methods of theoretical physics, vol. 1, (1953), McGraw-Hill New York · Zbl 0051.40603
[255] Pinsky, M.A., Introduction to partial differential equations with applications, (1984), McGraw-Hill New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.