Abstract

For an accurate modeling of natural rainbows, it is necessary to take into account the flattened shape of falling raindrops. Larger drops do also oscillate, and their axes exhibit tilt angles with respect to the vertical. In this paper, I will discuss two rare rainbow phenomena that are influenced by these effects: bright spots belonging to various rainbow orders, but appearing at remarkable angular distances from their traditional locations, as well as triple-split primary rainbows. While the former have not been observed in nature so far, the latter have been documented in a few photographs. This paper presents simulations based on natural drop size distributions using both a geometric optical model, as well as numerically calculated Möbius shifts applied to Debye series data.

© 2020 Optical Society of America

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  1. J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
    [Crossref]
  2. C. B. Boyer, The Rainbow: from Myth to Mathematics (Yoseloff, 1959), pp. 159–160 and 223–224.
  3. J. A. Lock and G. P. Können, “Rainbows by elliptically deformed drops. I. Möbius shift for high-order rainbows,” Appl. Opt. 56, G88–G97 (2017).
    [Crossref]
  4. W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentellen Prüfung,” Abh. Kgl. Sächs. Ges. Wiss. Math.-Phys. Kl. 30, 108–254 (1907).
  5. A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?” J. Opt. Soc. Am. 73, 1626–1628 (1983).
    [Crossref]
  6. G. P. Können and J. A. Lock, “Rainbows by elliptically deformed drops. II. The appearance of supernumeraries of high-order rainbows in rain showers,” Appl. Opt. 56, G98–G103 (2017).
    [Crossref]
  7. G. P. Können, “Appearance of supernumeraries of the secondary rainbow in rain showers,” J. Opt. Soc. Am. A 4, 810–816 (1987).
    [Crossref]
  8. F. E. Volz, “Some aspects of the optics of the rainbow and the physics of rain,” in Physics of Precipitation, H. Weickmann, ed., Geophysical Monograph 5 (American Geophysical Union, 1960), pp. 280–286.
  9. A. Haußmann, “Observation, analysis, and reconstruction of a twinned rainbow,” Appl. Opt. 54, B117–B127 (2015).
    [Crossref]
  10. A. Haußmann, “Polarization-resolved simulations of multiple-order rainbows using realistic raindrop shapes,” J. Quant. Spectrosc. Radiat. Transfer 175, 76–89 (2016).
    [Crossref]
  11. J. A. Lock, “Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. I. Theory,” Appl. Opt. 39, 5040–5051 (2000).
    [Crossref]
  12. C. L. Adler, D. Phipps, K. W. Saunders, J. K. Nash, and J. A. Lock, “Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. II. Experiment,” Appl. Opt. 40, 2535–2545 (2001).
    [Crossref]
  13. K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [Crossref]
  14. P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
    [Crossref]
  15. H. J. Simpson and P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468–3473 (1991).
    [Crossref]
  16. D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt. 37, 1520–1526 (1998).
    [Crossref]
  17. A. Haußmann, “Light scattering from sessile water drops and raindrop-shaped glass beads as a validation tool for rainbow simulations,” Appl. Opt. 56, G136–G144 (2017).
    [Crossref]
  18. T. Nousiainen and K. Muinonen, “Light scattering by Gaussian, randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
    [Crossref]
  19. T. Nousiainen, “Scattering of light by raindrops with single-mode oscillations,” J. Atmos. Sci. 57, 789–802 (2000).
    [Crossref]
  20. M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of falling raindrops—a review,” Atmos. Res. 97, 416–425 (2010).
    [Crossref]
  21. K. V. Beard, R. J. Kubesh, and H. T. Ochs, “Laboratory measurements of small raindrop distortion. Part I: Axis ratios and fall behavior,” J. Atmos. Sci. 48, 698–710 (1991).
    [Crossref]
  22. K. V. Beard and R. J. Kubesh, “Laboratory measurements of small raindrop distortion. Part 2: Oscillation frequencies and modes,” J. Atmos. Sci. 48, 2245–2264 (1991).
    [Crossref]
  23. T. Timusk, “Flashes of light below the dripping faucet: an optical signal from capillary oscillations of water drops,” Appl. Opt. 48, 1212–1217 (2009).
    [Crossref]
  24. http://dl.meteoros.de/haussmann/AHaussmann_DisTripRainbows_supp_inf.pdf .
  25. L. Cowley, “Anomalous bows,” http://atoptics.co.uk/fz845.htm .
  26. A. Haußmann, “Triple-split rainbow observed and photographed in Japan, August 2012,” https://atoptics.wordpress.com/2015/03/11/triple-split-rainbow-observed-and-photographed-in-japan-august-2012/ .
  27. A. Boonsinsuk, “Triple-split rainbow,” (in Thai language) http://www.cloudloverclub.com/triple-spit_rainbow-chiang_dao/ .
  28. A. Haußmann, “A multi-split rainbow from south-east China, August 12th, 2014,” https://atoptics.wordpress.com/2019/06/27/a-multi-split-rainbow-from-south-east-china-august-12th-2014/ .
  29. S. D. Gedzelman, “Rainbows in strong vertical electric fields,” J. Opt. Soc. Am. A 5, 1717–1721 (1988).
    [Crossref]
  30. F. Pattloch and E. Tränkle, “Monte Carlo simulation and analysis of halo phenomena,” J. Opt. Soc. Am. A 1, 520–526 (1984).
    [Crossref]
  31. V. N. Bringi, M. Thurai, and D. A. Brunkow, “Measurements and inferences of raindrop canting angles,” Electron. Lett. 44, 1425–1426 (2008).
    [Crossref]
  32. G.-J. Huang, V. N. Bringi, and M. Thurai, “Orientation angle distributions of drops after an 80-m fall using a 2D video disdrometer,” J. Atmos. Ocean. Technol. 25, 1717–1723 (2008).
    [Crossref]
  33. R. L. Lee, “Mie theory, Airy theory, and the natural rainbow,” Appl. Opt. 37, 1506–1519 (1998).
    [Crossref]
  34. S. D. Gedzelman, “Simulating rainbows in their atmospheric environment,” Appl. Opt. 47, H176–H181 (2008).
    [Crossref]
  35. S. Kanamori, K. Fujiwara, T. Yoshinobu, B. Raytchev, T. Tamaki, and K. Kaneda, “Physically based rendering of rainbows under various atmospheric conditions,” in 18th Pacific Conference on Computer Graphics and Applications (2010).
    [Crossref]
  36. I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
    [Crossref]
  37. P. Laven, “Supernumerary arcs of rainbows: Young’s theory of interference,” Appl. Opt. 56, G104–G112 (2017).
    [Crossref]
  38. P.-E. Ouelette, “Supernumerary bows: caustics of a refractive sphere and analysis of the relative overall Gouy phase shift of supernumerary rays,” Appl. Opt. 58, 712–722 (2019).
    [Crossref]
  39. A corrected version of the supporting information discussing the mathematical details of the simulation is linked in [17].
  40. S. D. Gedzelman, “Rainbow brightness,” Appl. Opt. 21, 3032–3037 (1982).
    [Crossref]
  41. H. E. Edens and G. P. Können, “Probable photographic detection of the natural seventh-order rainbow,” Appl. Opt. 54, B93–B96 (2015).
    [Crossref]
  42. G. P. Können, “Polarization and visibility of higher-order rainbows,” Appl. Opt. 54, B35–B40 (2015).
    [Crossref]
  43. Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
    [Crossref]
  44. F. Xu, J. A. Lock, and C. Tropea, “Debye series for light scattering by a spheroid,” J. Opt. Soc. Am. A 27, 671–686 (2010).
    [Crossref]
  45. J. A. Lock, C. L. Adler, B. R. Stone, and P. D. Zajak, “Amplification of high-order rainbows of a cylinder with an elliptical cross section,” Appl. Opt. 37, 1527–1533 (1998).
    [Crossref]
  46. C. L. Adler, J. A. Lock, and B. R. Stone, “Rainbow scattering by a cylinder with a nearly elliptical cross section,” Appl. Opt. 37, 1540–1550 (1998).
    [Crossref]
  47. H. R. Pruppacher and R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
    [Crossref]
  48. K. V. Beard and C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
    [Crossref]
  49. C. C. Chuang and K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
    [Crossref]
  50. In [9], the value for the parameter $f_3$f3 in Eq. (5) was given mistakenly without its minus sign, i.e., it should read correctly $f_{3}=-0.2911$f3=−0.2911.
  51. E. Becker, W. J. Hiller, and T. A. Kowalewski, “Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets,” J. Fluid Mech. 231, 189–210 (1991).
    [Crossref]
  52. A. Prosperettia, “Linear oscillations of constrained drops, bubbles, and plane liquid surfaces,” Phys. Fluids 24, 032109 (2012).
    [Crossref]
  53. For real-valued spherical harmonics, $m \gt 0$m>0 modes have a $\cos (m\,\varphi)$cos⁡(mφ) dependence, while the ones with $m \lt 0$m<0 have a $\sin (m\,\varphi)$sin⁡(mφ) dependence. These are linearly independent, and both are necessary to build the complete basis function set. However, they can be arithmetically combined to a single $m \gt 0$m>0 mode in a rotated (and depending on the phase offsets, possibly rotating) coordinate system. Moreover, in video disdrometer measurements, only a projection of the drop surface is seen. This explains why most papers on drop oscillations neglect modes with negative $m$m.
  54. J. Q. Feng and K. V. Beard, “A perturbation model of raindrop oscillation characteristics with aerodynamic effects,” J. Atmos. Sci. 48, 1856–1868 (1991).
    [Crossref]
  55. M. Szakáll, S. Kessler, K. Diehl, S. K. Mitra, and S. Borrmann, “A wind tunnel study of the effects of collision processes on the shape and oscillation for moderate-size raindrops,” Atmos. Res. 142, 67–78 (2014).
    [Crossref]
  56. S. Müller, M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of raindrops with reduced surface tensions: measurements at the Mainz vertical wind tunnel,” Atmos. Res. 119, 38–45 (2013).
    [Crossref]
  57. H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation, 2nd ed. (Kluwer Academic, 1997), pp. 400–408.
  58. For the (2,0) mode, the temporally averaged axis ratio corresponds to the axis ratio of the nonoscillating equilibrium state. This is not necessarily the case for $m \neq 0 $m≠0 modes.
  59. K. V. Beard and A. R. Jameson, “Raindrop canting,” J. Atmos. Sci. 40, 448–454 (1983).
    [Crossref]
  60. M. Agrawal, A. R. Premlata, M. K. Tripathi, B. Karri, and K. C. Sahu, “Nonspherical liquid droplet falling in air,” Phys. Rev. E 95, 033111 (2017).
    [Crossref]
  61. M. Balla, M. K. Tripathi, and K. C. Sahu, “Shape oscillations of a nonspherical water droplet,” Phys. Rev. E 99, 023107 (2019).
    [Crossref]
  62. C. W. Ulbrich, “Natural variations in the analytical form of the raindrop size distribution,” J. Clim. Appl. Meteorol. 22, 1764–1775 (1983).
    [Crossref]
  63. The raw RGB simulation images were subjected to a gamma increase (to $\gamma = 2 $γ=2) (Figs. 3,4, and 6) and a subsequent contrast enhancement (Figs. 3 and 4) before display. No artificial background was added to the simulations.
  64. The actual eDSD for these calculations is given by the formula $n( a_{0} )={{n}_{0}}\cdot \exp ( -2\cdot {{a}_{0}}\cdot \Lambda )+{{n}_{1}}\cdot \exp ( -{{{( {{a}_{0}}-{{a}_{1}} )}^{2}}}/{2\sigma _{1}^{2}}\; )+{{n}_{2}}\cdot \exp ( -{{{( {{a}_{0}}-{{a}_{2}} )}^{2}}}/{2\sigma _{2}^{2}}\; )$n(a0)=n0⋅exp⁡(−2⋅a0⋅Λ)+n1⋅exp⁡(−(a0−a1)2/2σ12)+n2⋅exp⁡(−(a0−a2)2/2σ22) and this set of parameters: $n_{0}=1$n0=1 (normalization factors and actual physical units are not relevant), $\Lambda= 4\,\,{\rm mm}^{-1}$Λ=4mm−1, $n_1=0.066$n1=0.066, $a_1=0.55\,\,{\rm mm}$a1=0.55mm, $\sigma_1=0.025\,\,{\rm mm}$σ1=0.025mm, $n_2=0.06$n2=0.06, $a_2=0.68\,\,{\rm mm}$a2=0.68mm, $\sigma_2=0.01\,\,{\rm mm}$σ2=0.01mm.
  65. They can, however, be relevant for artificial-light rainbows, and maybe for reflection bows.
  66. M. Theusner, “Photographic observation of a natural fourth-order rainbow,” Appl. Opt. 50, F129–F133 (2011).
    [Crossref]
  67. M. Großmann, E. Schmidt, and A. Haußmann, “Photographic evidence for the third-order rainbow,” Appl. Opt. 50, F134–F141 (2011).
    [Crossref]
  68. There is one older observation of a red spot appearing inside the primary rainbow above the antisolar point, recorded 20 September 1896, in Vlissingen, Netherlands, at a Sun elevation of about 13.3°. The elevation of the spot above the horizon was roughly estimated to be 14.5°, and it was interpreted as a reflected-light glory by the observer [69]. However, at this Sun elevation, no such spot does appear in the current simulations based on the BC model.
  69. Koninklijk Nederlandsch Meteorologisch Instituut. Onweders, Optische Verschijnselen, Enz. in Nederland. Naar vrijwillige waarnemingen in 1896, Deel 17 (Amsterdam, 1897), pp. 48–49.
  70. I am aware of only one single photograph showing a candidate twinned secondary rainbow (with the splitting point shifted off the center to the right), but more details concerning the observation and photographic equipment are needed for a definite assessment. It should be noted that modern cellphone and other digital cameras can produces artifacts when several frames are combined by an optimization algorithm, and the camera orientation was not kept fixed during the multiexposure.
  71. L. Cowley and T. Wooten, “Puzzling triple rainbow,” https://www.atoptics.co.uk/fza149.htm .
  72. C. Hinz and M. Worme, “Twinned rainbow,” https://atoptics.wordpress.com/2011/05/27/twinned-rainbow-2/ .
  73. L. Cowley and M. Worme, “Peculiar rainbows,” https://www.atoptics.co.uk/fz517.htm .
  74. J. Q. Feng and K. V. Beard, “Raindrop shape determined by computing steady axisymmetric solutions for Navier-Stokes equations,” Atmos. Res. 101, 480–491 (2011).
    [Crossref]
  75. J. Appleyard, “Computational simulation of freely falling water droplets on graphics processing units,” Ph.D. dissertation (School of Engineering, Cranfield University, 2013).
  76. C. Hinz, “Moving ripples,” https://www.meteoros.de/themen/halos/arbeit-des-akm/moving-ripples/ .

2019 (2)

2017 (5)

2016 (1)

A. Haußmann, “Polarization-resolved simulations of multiple-order rainbows using realistic raindrop shapes,” J. Quant. Spectrosc. Radiat. Transfer 175, 76–89 (2016).
[Crossref]

2015 (3)

2014 (1)

M. Szakáll, S. Kessler, K. Diehl, S. K. Mitra, and S. Borrmann, “A wind tunnel study of the effects of collision processes on the shape and oscillation for moderate-size raindrops,” Atmos. Res. 142, 67–78 (2014).
[Crossref]

2013 (1)

S. Müller, M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of raindrops with reduced surface tensions: measurements at the Mainz vertical wind tunnel,” Atmos. Res. 119, 38–45 (2013).
[Crossref]

2012 (2)

A. Prosperettia, “Linear oscillations of constrained drops, bubbles, and plane liquid surfaces,” Phys. Fluids 24, 032109 (2012).
[Crossref]

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

2011 (3)

2010 (2)

F. Xu, J. A. Lock, and C. Tropea, “Debye series for light scattering by a spheroid,” J. Opt. Soc. Am. A 27, 671–686 (2010).
[Crossref]

M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of falling raindrops—a review,” Atmos. Res. 97, 416–425 (2010).
[Crossref]

2009 (1)

2008 (3)

V. N. Bringi, M. Thurai, and D. A. Brunkow, “Measurements and inferences of raindrop canting angles,” Electron. Lett. 44, 1425–1426 (2008).
[Crossref]

G.-J. Huang, V. N. Bringi, and M. Thurai, “Orientation angle distributions of drops after an 80-m fall using a 2D video disdrometer,” J. Atmos. Ocean. Technol. 25, 1717–1723 (2008).
[Crossref]

S. D. Gedzelman, “Simulating rainbows in their atmospheric environment,” Appl. Opt. 47, H176–H181 (2008).
[Crossref]

2002 (2)

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
[Crossref]

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[Crossref]

2001 (1)

2000 (2)

J. A. Lock, “Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. I. Theory,” Appl. Opt. 39, 5040–5051 (2000).
[Crossref]

T. Nousiainen, “Scattering of light by raindrops with single-mode oscillations,” J. Atmos. Sci. 57, 789–802 (2000).
[Crossref]

1999 (1)

T. Nousiainen and K. Muinonen, “Light scattering by Gaussian, randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
[Crossref]

1998 (4)

1991 (5)

K. V. Beard, R. J. Kubesh, and H. T. Ochs, “Laboratory measurements of small raindrop distortion. Part I: Axis ratios and fall behavior,” J. Atmos. Sci. 48, 698–710 (1991).
[Crossref]

K. V. Beard and R. J. Kubesh, “Laboratory measurements of small raindrop distortion. Part 2: Oscillation frequencies and modes,” J. Atmos. Sci. 48, 2245–2264 (1991).
[Crossref]

H. J. Simpson and P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468–3473 (1991).
[Crossref]

J. Q. Feng and K. V. Beard, “A perturbation model of raindrop oscillation characteristics with aerodynamic effects,” J. Atmos. Sci. 48, 1856–1868 (1991).
[Crossref]

E. Becker, W. J. Hiller, and T. A. Kowalewski, “Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets,” J. Fluid Mech. 231, 189–210 (1991).
[Crossref]

1990 (1)

C. C. Chuang and K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
[Crossref]

1988 (1)

1987 (2)

G. P. Können, “Appearance of supernumeraries of the secondary rainbow in rain showers,” J. Opt. Soc. Am. A 4, 810–816 (1987).
[Crossref]

K. V. Beard and C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
[Crossref]

1984 (2)

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
[Crossref]

F. Pattloch and E. Tränkle, “Monte Carlo simulation and analysis of halo phenomena,” J. Opt. Soc. Am. A 1, 520–526 (1984).
[Crossref]

1983 (3)

A. B. Fraser, “Why can the supernumerary bows be seen in a rain shower?” J. Opt. Soc. Am. 73, 1626–1628 (1983).
[Crossref]

K. V. Beard and A. R. Jameson, “Raindrop canting,” J. Atmos. Sci. 40, 448–454 (1983).
[Crossref]

C. W. Ulbrich, “Natural variations in the analytical form of the raindrop size distribution,” J. Clim. Appl. Meteorol. 22, 1764–1775 (1983).
[Crossref]

1982 (1)

1979 (1)

1971 (1)

H. R. Pruppacher and R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
[Crossref]

1907 (1)

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentellen Prüfung,” Abh. Kgl. Sächs. Ges. Wiss. Math.-Phys. Kl. 30, 108–254 (1907).

Adam, J. A.

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
[Crossref]

Adler, C. L.

Agrawal, M.

M. Agrawal, A. R. Premlata, M. K. Tripathi, B. Karri, and K. C. Sahu, “Nonspherical liquid droplet falling in air,” Phys. Rev. E 95, 033111 (2017).
[Crossref]

Appleyard, J.

J. Appleyard, “Computational simulation of freely falling water droplets on graphics processing units,” Ph.D. dissertation (School of Engineering, Cranfield University, 2013).

Balla, M.

M. Balla, M. K. Tripathi, and K. C. Sahu, “Shape oscillations of a nonspherical water droplet,” Phys. Rev. E 99, 023107 (2019).
[Crossref]

Beard, K. V.

J. Q. Feng and K. V. Beard, “Raindrop shape determined by computing steady axisymmetric solutions for Navier-Stokes equations,” Atmos. Res. 101, 480–491 (2011).
[Crossref]

K. V. Beard and R. J. Kubesh, “Laboratory measurements of small raindrop distortion. Part 2: Oscillation frequencies and modes,” J. Atmos. Sci. 48, 2245–2264 (1991).
[Crossref]

K. V. Beard, R. J. Kubesh, and H. T. Ochs, “Laboratory measurements of small raindrop distortion. Part I: Axis ratios and fall behavior,” J. Atmos. Sci. 48, 698–710 (1991).
[Crossref]

J. Q. Feng and K. V. Beard, “A perturbation model of raindrop oscillation characteristics with aerodynamic effects,” J. Atmos. Sci. 48, 1856–1868 (1991).
[Crossref]

C. C. Chuang and K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
[Crossref]

K. V. Beard and C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
[Crossref]

K. V. Beard and A. R. Jameson, “Raindrop canting,” J. Atmos. Sci. 40, 448–454 (1983).
[Crossref]

Becker, E.

E. Becker, W. J. Hiller, and T. A. Kowalewski, “Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets,” J. Fluid Mech. 231, 189–210 (1991).
[Crossref]

Borrmann, S.

M. Szakáll, S. Kessler, K. Diehl, S. K. Mitra, and S. Borrmann, “A wind tunnel study of the effects of collision processes on the shape and oscillation for moderate-size raindrops,” Atmos. Res. 142, 67–78 (2014).
[Crossref]

S. Müller, M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of raindrops with reduced surface tensions: measurements at the Mainz vertical wind tunnel,” Atmos. Res. 119, 38–45 (2013).
[Crossref]

M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of falling raindrops—a review,” Atmos. Res. 97, 416–425 (2010).
[Crossref]

Boyer, C. B.

C. B. Boyer, The Rainbow: from Myth to Mathematics (Yoseloff, 1959), pp. 159–160 and 223–224.

Bringi, V. N.

G.-J. Huang, V. N. Bringi, and M. Thurai, “Orientation angle distributions of drops after an 80-m fall using a 2D video disdrometer,” J. Atmos. Ocean. Technol. 25, 1717–1723 (2008).
[Crossref]

V. N. Bringi, M. Thurai, and D. A. Brunkow, “Measurements and inferences of raindrop canting angles,” Electron. Lett. 44, 1425–1426 (2008).
[Crossref]

Brunkow, D. A.

V. N. Bringi, M. Thurai, and D. A. Brunkow, “Measurements and inferences of raindrop canting angles,” Electron. Lett. 44, 1425–1426 (2008).
[Crossref]

Chuang, C.

K. V. Beard and C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
[Crossref]

Chuang, C. C.

C. C. Chuang and K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
[Crossref]

Diehl, K.

M. Szakáll, S. Kessler, K. Diehl, S. K. Mitra, and S. Borrmann, “A wind tunnel study of the effects of collision processes on the shape and oscillation for moderate-size raindrops,” Atmos. Res. 142, 67–78 (2014).
[Crossref]

S. Müller, M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of raindrops with reduced surface tensions: measurements at the Mainz vertical wind tunnel,” Atmos. Res. 119, 38–45 (2013).
[Crossref]

M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of falling raindrops—a review,” Atmos. Res. 97, 416–425 (2010).
[Crossref]

Edens, H. E.

Feng, J. Q.

J. Q. Feng and K. V. Beard, “Raindrop shape determined by computing steady axisymmetric solutions for Navier-Stokes equations,” Atmos. Res. 101, 480–491 (2011).
[Crossref]

J. Q. Feng and K. V. Beard, “A perturbation model of raindrop oscillation characteristics with aerodynamic effects,” J. Atmos. Sci. 48, 1856–1868 (1991).
[Crossref]

Fraser, A. B.

Fujiwara, K.

S. Kanamori, K. Fujiwara, T. Yoshinobu, B. Raytchev, T. Tamaki, and K. Kaneda, “Physically based rendering of rainbows under various atmospheric conditions,” in 18th Pacific Conference on Computer Graphics and Applications (2010).
[Crossref]

Gedzelman, S. D.

Gouesbet, G.

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[Crossref]

Gréhan, G.

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[Crossref]

Großmann, M.

Gutierrez, D.

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Han, Y. P.

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[Crossref]

Haußmann, A.

Hiller, W. J.

E. Becker, W. J. Hiller, and T. A. Kowalewski, “Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets,” J. Fluid Mech. 231, 189–210 (1991).
[Crossref]

Huang, G.-J.

G.-J. Huang, V. N. Bringi, and M. Thurai, “Orientation angle distributions of drops after an 80-m fall using a 2D video disdrometer,” J. Atmos. Ocean. Technol. 25, 1717–1723 (2008).
[Crossref]

Jameson, A. R.

K. V. Beard and A. R. Jameson, “Raindrop canting,” J. Atmos. Sci. 40, 448–454 (1983).
[Crossref]

Jarosz, W.

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Jensen, H. W.

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Kanamori, S.

S. Kanamori, K. Fujiwara, T. Yoshinobu, B. Raytchev, T. Tamaki, and K. Kaneda, “Physically based rendering of rainbows under various atmospheric conditions,” in 18th Pacific Conference on Computer Graphics and Applications (2010).
[Crossref]

Kaneda, K.

S. Kanamori, K. Fujiwara, T. Yoshinobu, B. Raytchev, T. Tamaki, and K. Kaneda, “Physically based rendering of rainbows under various atmospheric conditions,” in 18th Pacific Conference on Computer Graphics and Applications (2010).
[Crossref]

Karri, B.

M. Agrawal, A. R. Premlata, M. K. Tripathi, B. Karri, and K. C. Sahu, “Nonspherical liquid droplet falling in air,” Phys. Rev. E 95, 033111 (2017).
[Crossref]

Kessler, S.

M. Szakáll, S. Kessler, K. Diehl, S. K. Mitra, and S. Borrmann, “A wind tunnel study of the effects of collision processes on the shape and oscillation for moderate-size raindrops,” Atmos. Res. 142, 67–78 (2014).
[Crossref]

Klett, J. D.

H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation, 2nd ed. (Kluwer Academic, 1997), pp. 400–408.

Können, G. P.

Kowalewski, T. A.

E. Becker, W. J. Hiller, and T. A. Kowalewski, “Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets,” J. Fluid Mech. 231, 189–210 (1991).
[Crossref]

Kubesh, R. J.

K. V. Beard, R. J. Kubesh, and H. T. Ochs, “Laboratory measurements of small raindrop distortion. Part I: Axis ratios and fall behavior,” J. Atmos. Sci. 48, 698–710 (1991).
[Crossref]

K. V. Beard and R. J. Kubesh, “Laboratory measurements of small raindrop distortion. Part 2: Oscillation frequencies and modes,” J. Atmos. Sci. 48, 2245–2264 (1991).
[Crossref]

Langley, D. S.

Laven, P.

P. Laven, “Supernumerary arcs of rainbows: Young’s theory of interference,” Appl. Opt. 56, G104–G112 (2017).
[Crossref]

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Lee, R. L.

Lock, J. A.

Marston, P. L.

Méès, L.

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[Crossref]

Mitra, S. K.

M. Szakáll, S. Kessler, K. Diehl, S. K. Mitra, and S. Borrmann, “A wind tunnel study of the effects of collision processes on the shape and oscillation for moderate-size raindrops,” Atmos. Res. 142, 67–78 (2014).
[Crossref]

S. Müller, M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of raindrops with reduced surface tensions: measurements at the Mainz vertical wind tunnel,” Atmos. Res. 119, 38–45 (2013).
[Crossref]

M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of falling raindrops—a review,” Atmos. Res. 97, 416–425 (2010).
[Crossref]

Möbius, W.

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentellen Prüfung,” Abh. Kgl. Sächs. Ges. Wiss. Math.-Phys. Kl. 30, 108–254 (1907).

Muinonen, K.

T. Nousiainen and K. Muinonen, “Light scattering by Gaussian, randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
[Crossref]

Müller, S.

S. Müller, M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of raindrops with reduced surface tensions: measurements at the Mainz vertical wind tunnel,” Atmos. Res. 119, 38–45 (2013).
[Crossref]

Munoz, A.

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Nash, J. K.

Nousiainen, T.

T. Nousiainen, “Scattering of light by raindrops with single-mode oscillations,” J. Atmos. Sci. 57, 789–802 (2000).
[Crossref]

T. Nousiainen and K. Muinonen, “Light scattering by Gaussian, randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
[Crossref]

Ochs, H. T.

K. V. Beard, R. J. Kubesh, and H. T. Ochs, “Laboratory measurements of small raindrop distortion. Part I: Axis ratios and fall behavior,” J. Atmos. Sci. 48, 698–710 (1991).
[Crossref]

Ouelette, P.-E.

Pattloch, F.

Phipps, D.

Pitter, R. L.

H. R. Pruppacher and R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
[Crossref]

Premlata, A. R.

M. Agrawal, A. R. Premlata, M. K. Tripathi, B. Karri, and K. C. Sahu, “Nonspherical liquid droplet falling in air,” Phys. Rev. E 95, 033111 (2017).
[Crossref]

Prosperettia, A.

A. Prosperettia, “Linear oscillations of constrained drops, bubbles, and plane liquid surfaces,” Phys. Fluids 24, 032109 (2012).
[Crossref]

Pruppacher, H. R.

H. R. Pruppacher and R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
[Crossref]

H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation, 2nd ed. (Kluwer Academic, 1997), pp. 400–408.

Raytchev, B.

S. Kanamori, K. Fujiwara, T. Yoshinobu, B. Raytchev, T. Tamaki, and K. Kaneda, “Physically based rendering of rainbows under various atmospheric conditions,” in 18th Pacific Conference on Computer Graphics and Applications (2010).
[Crossref]

Ren, K. F.

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[Crossref]

Sadeghi, I.

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Sahu, K. C.

M. Balla, M. K. Tripathi, and K. C. Sahu, “Shape oscillations of a nonspherical water droplet,” Phys. Rev. E 99, 023107 (2019).
[Crossref]

M. Agrawal, A. R. Premlata, M. K. Tripathi, B. Karri, and K. C. Sahu, “Nonspherical liquid droplet falling in air,” Phys. Rev. E 95, 033111 (2017).
[Crossref]

Sassen, K.

Saunders, K. W.

Schmidt, E.

Seron, F.

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Simpson, H. J.

Stone, B. R.

Szakáll, M.

M. Szakáll, S. Kessler, K. Diehl, S. K. Mitra, and S. Borrmann, “A wind tunnel study of the effects of collision processes on the shape and oscillation for moderate-size raindrops,” Atmos. Res. 142, 67–78 (2014).
[Crossref]

S. Müller, M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of raindrops with reduced surface tensions: measurements at the Mainz vertical wind tunnel,” Atmos. Res. 119, 38–45 (2013).
[Crossref]

M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of falling raindrops—a review,” Atmos. Res. 97, 416–425 (2010).
[Crossref]

Tamaki, T.

S. Kanamori, K. Fujiwara, T. Yoshinobu, B. Raytchev, T. Tamaki, and K. Kaneda, “Physically based rendering of rainbows under various atmospheric conditions,” in 18th Pacific Conference on Computer Graphics and Applications (2010).
[Crossref]

Theusner, M.

Thurai, M.

V. N. Bringi, M. Thurai, and D. A. Brunkow, “Measurements and inferences of raindrop canting angles,” Electron. Lett. 44, 1425–1426 (2008).
[Crossref]

G.-J. Huang, V. N. Bringi, and M. Thurai, “Orientation angle distributions of drops after an 80-m fall using a 2D video disdrometer,” J. Atmos. Ocean. Technol. 25, 1717–1723 (2008).
[Crossref]

Timusk, T.

Tränkle, E.

Trinh, E. H.

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
[Crossref]

Tripathi, M. K.

M. Balla, M. K. Tripathi, and K. C. Sahu, “Shape oscillations of a nonspherical water droplet,” Phys. Rev. E 99, 023107 (2019).
[Crossref]

M. Agrawal, A. R. Premlata, M. K. Tripathi, B. Karri, and K. C. Sahu, “Nonspherical liquid droplet falling in air,” Phys. Rev. E 95, 033111 (2017).
[Crossref]

Tropea, C.

Ulbrich, C. W.

C. W. Ulbrich, “Natural variations in the analytical form of the raindrop size distribution,” J. Clim. Appl. Meteorol. 22, 1764–1775 (1983).
[Crossref]

Volz, F. E.

F. E. Volz, “Some aspects of the optics of the rainbow and the physics of rain,” in Physics of Precipitation, H. Weickmann, ed., Geophysical Monograph 5 (American Geophysical Union, 1960), pp. 280–286.

Wu, S. Z.

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[Crossref]

Xu, F.

Yoshinobu, T.

S. Kanamori, K. Fujiwara, T. Yoshinobu, B. Raytchev, T. Tamaki, and K. Kaneda, “Physically based rendering of rainbows under various atmospheric conditions,” in 18th Pacific Conference on Computer Graphics and Applications (2010).
[Crossref]

Zajak, P. D.

Abh. Kgl. Sächs. Ges. Wiss. Math.-Phys. Kl. (1)

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentellen Prüfung,” Abh. Kgl. Sächs. Ges. Wiss. Math.-Phys. Kl. 30, 108–254 (1907).

ACM Trans. Graph. (1)

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Appl. Opt. (20)

J. A. Lock, “Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. I. Theory,” Appl. Opt. 39, 5040–5051 (2000).
[Crossref]

S. D. Gedzelman, “Simulating rainbows in their atmospheric environment,” Appl. Opt. 47, H176–H181 (2008).
[Crossref]

J. A. Lock and G. P. Können, “Rainbows by elliptically deformed drops. I. Möbius shift for high-order rainbows,” Appl. Opt. 56, G88–G97 (2017).
[Crossref]

T. Timusk, “Flashes of light below the dripping faucet: an optical signal from capillary oscillations of water drops,” Appl. Opt. 48, 1212–1217 (2009).
[Crossref]

M. Großmann, E. Schmidt, and A. Haußmann, “Photographic evidence for the third-order rainbow,” Appl. Opt. 50, F134–F141 (2011).
[Crossref]

H. E. Edens and G. P. Können, “Probable photographic detection of the natural seventh-order rainbow,” Appl. Opt. 54, B93–B96 (2015).
[Crossref]

A. Haußmann, “Observation, analysis, and reconstruction of a twinned rainbow,” Appl. Opt. 54, B117–B127 (2015).
[Crossref]

D. S. Langley and P. L. Marston, “Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination,” Appl. Opt. 37, 1520–1526 (1998).
[Crossref]

G. P. Können and J. A. Lock, “Rainbows by elliptically deformed drops. II. The appearance of supernumeraries of high-order rainbows in rain showers,” Appl. Opt. 56, G98–G103 (2017).
[Crossref]

R. L. Lee, “Mie theory, Airy theory, and the natural rainbow,” Appl. Opt. 37, 1506–1519 (1998).
[Crossref]

J. A. Lock, C. L. Adler, B. R. Stone, and P. D. Zajak, “Amplification of high-order rainbows of a cylinder with an elliptical cross section,” Appl. Opt. 37, 1527–1533 (1998).
[Crossref]

C. L. Adler, D. Phipps, K. W. Saunders, J. K. Nash, and J. A. Lock, “Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. II. Experiment,” Appl. Opt. 40, 2535–2545 (2001).
[Crossref]

S. D. Gedzelman, “Rainbow brightness,” Appl. Opt. 21, 3032–3037 (1982).
[Crossref]

H. J. Simpson and P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468–3473 (1991).
[Crossref]

G. P. Können, “Polarization and visibility of higher-order rainbows,” Appl. Opt. 54, B35–B40 (2015).
[Crossref]

P. Laven, “Supernumerary arcs of rainbows: Young’s theory of interference,” Appl. Opt. 56, G104–G112 (2017).
[Crossref]

C. L. Adler, J. A. Lock, and B. R. Stone, “Rainbow scattering by a cylinder with a nearly elliptical cross section,” Appl. Opt. 37, 1540–1550 (1998).
[Crossref]

M. Theusner, “Photographic observation of a natural fourth-order rainbow,” Appl. Opt. 50, F129–F133 (2011).
[Crossref]

A. Haußmann, “Light scattering from sessile water drops and raindrop-shaped glass beads as a validation tool for rainbow simulations,” Appl. Opt. 56, G136–G144 (2017).
[Crossref]

P.-E. Ouelette, “Supernumerary bows: caustics of a refractive sphere and analysis of the relative overall Gouy phase shift of supernumerary rays,” Appl. Opt. 58, 712–722 (2019).
[Crossref]

Atmos. Res. (4)

J. Q. Feng and K. V. Beard, “Raindrop shape determined by computing steady axisymmetric solutions for Navier-Stokes equations,” Atmos. Res. 101, 480–491 (2011).
[Crossref]

M. Szakáll, S. Kessler, K. Diehl, S. K. Mitra, and S. Borrmann, “A wind tunnel study of the effects of collision processes on the shape and oscillation for moderate-size raindrops,” Atmos. Res. 142, 67–78 (2014).
[Crossref]

S. Müller, M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of raindrops with reduced surface tensions: measurements at the Mainz vertical wind tunnel,” Atmos. Res. 119, 38–45 (2013).
[Crossref]

M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, “Shapes and oscillations of falling raindrops—a review,” Atmos. Res. 97, 416–425 (2010).
[Crossref]

Electron. Lett. (1)

V. N. Bringi, M. Thurai, and D. A. Brunkow, “Measurements and inferences of raindrop canting angles,” Electron. Lett. 44, 1425–1426 (2008).
[Crossref]

J. Atmos. Ocean. Technol. (1)

G.-J. Huang, V. N. Bringi, and M. Thurai, “Orientation angle distributions of drops after an 80-m fall using a 2D video disdrometer,” J. Atmos. Ocean. Technol. 25, 1717–1723 (2008).
[Crossref]

J. Atmos. Sci. (8)

J. Q. Feng and K. V. Beard, “A perturbation model of raindrop oscillation characteristics with aerodynamic effects,” J. Atmos. Sci. 48, 1856–1868 (1991).
[Crossref]

T. Nousiainen, “Scattering of light by raindrops with single-mode oscillations,” J. Atmos. Sci. 57, 789–802 (2000).
[Crossref]

K. V. Beard, R. J. Kubesh, and H. T. Ochs, “Laboratory measurements of small raindrop distortion. Part I: Axis ratios and fall behavior,” J. Atmos. Sci. 48, 698–710 (1991).
[Crossref]

K. V. Beard and R. J. Kubesh, “Laboratory measurements of small raindrop distortion. Part 2: Oscillation frequencies and modes,” J. Atmos. Sci. 48, 2245–2264 (1991).
[Crossref]

K. V. Beard and A. R. Jameson, “Raindrop canting,” J. Atmos. Sci. 40, 448–454 (1983).
[Crossref]

H. R. Pruppacher and R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
[Crossref]

K. V. Beard and C. Chuang, “A new model for the equilibrium shape of raindrops,” J. Atmos. Sci. 44, 1509–1524 (1987).
[Crossref]

C. C. Chuang and K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
[Crossref]

J. Clim. Appl. Meteorol. (1)

C. W. Ulbrich, “Natural variations in the analytical form of the raindrop size distribution,” J. Clim. Appl. Meteorol. 22, 1764–1775 (1983).
[Crossref]

J. Fluid Mech. (1)

E. Becker, W. J. Hiller, and T. A. Kowalewski, “Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets,” J. Fluid Mech. 231, 189–210 (1991).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

J. Quant. Spectrosc. Radiat. Transfer (2)

A. Haußmann, “Polarization-resolved simulations of multiple-order rainbows using realistic raindrop shapes,” J. Quant. Spectrosc. Radiat. Transfer 175, 76–89 (2016).
[Crossref]

T. Nousiainen and K. Muinonen, “Light scattering by Gaussian, randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
[Crossref]

Nature (1)

P. L. Marston and E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
[Crossref]

Opt. Commun. (1)

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, “Scattering of light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[Crossref]

Phys. Fluids (1)

A. Prosperettia, “Linear oscillations of constrained drops, bubbles, and plane liquid surfaces,” Phys. Fluids 24, 032109 (2012).
[Crossref]

Phys. Rep. (1)

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
[Crossref]

Phys. Rev. E (2)

M. Agrawal, A. R. Premlata, M. K. Tripathi, B. Karri, and K. C. Sahu, “Nonspherical liquid droplet falling in air,” Phys. Rev. E 95, 033111 (2017).
[Crossref]

M. Balla, M. K. Tripathi, and K. C. Sahu, “Shape oscillations of a nonspherical water droplet,” Phys. Rev. E 99, 023107 (2019).
[Crossref]

Other (24)

For real-valued spherical harmonics, $m \gt 0$m>0 modes have a $\cos (m\,\varphi)$cos⁡(mφ) dependence, while the ones with $m \lt 0$m<0 have a $\sin (m\,\varphi)$sin⁡(mφ) dependence. These are linearly independent, and both are necessary to build the complete basis function set. However, they can be arithmetically combined to a single $m \gt 0$m>0 mode in a rotated (and depending on the phase offsets, possibly rotating) coordinate system. Moreover, in video disdrometer measurements, only a projection of the drop surface is seen. This explains why most papers on drop oscillations neglect modes with negative $m$m.

H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation, 2nd ed. (Kluwer Academic, 1997), pp. 400–408.

For the (2,0) mode, the temporally averaged axis ratio corresponds to the axis ratio of the nonoscillating equilibrium state. This is not necessarily the case for $m \neq 0 $m≠0 modes.

S. Kanamori, K. Fujiwara, T. Yoshinobu, B. Raytchev, T. Tamaki, and K. Kaneda, “Physically based rendering of rainbows under various atmospheric conditions,” in 18th Pacific Conference on Computer Graphics and Applications (2010).
[Crossref]

A corrected version of the supporting information discussing the mathematical details of the simulation is linked in [17].

In [9], the value for the parameter $f_3$f3 in Eq. (5) was given mistakenly without its minus sign, i.e., it should read correctly $f_{3}=-0.2911$f3=−0.2911.

The raw RGB simulation images were subjected to a gamma increase (to $\gamma = 2 $γ=2) (Figs. 3,4, and 6) and a subsequent contrast enhancement (Figs. 3 and 4) before display. No artificial background was added to the simulations.

The actual eDSD for these calculations is given by the formula $n( a_{0} )={{n}_{0}}\cdot \exp ( -2\cdot {{a}_{0}}\cdot \Lambda )+{{n}_{1}}\cdot \exp ( -{{{( {{a}_{0}}-{{a}_{1}} )}^{2}}}/{2\sigma _{1}^{2}}\; )+{{n}_{2}}\cdot \exp ( -{{{( {{a}_{0}}-{{a}_{2}} )}^{2}}}/{2\sigma _{2}^{2}}\; )$n(a0)=n0⋅exp⁡(−2⋅a0⋅Λ)+n1⋅exp⁡(−(a0−a1)2/2σ12)+n2⋅exp⁡(−(a0−a2)2/2σ22) and this set of parameters: $n_{0}=1$n0=1 (normalization factors and actual physical units are not relevant), $\Lambda= 4\,\,{\rm mm}^{-1}$Λ=4mm−1, $n_1=0.066$n1=0.066, $a_1=0.55\,\,{\rm mm}$a1=0.55mm, $\sigma_1=0.025\,\,{\rm mm}$σ1=0.025mm, $n_2=0.06$n2=0.06, $a_2=0.68\,\,{\rm mm}$a2=0.68mm, $\sigma_2=0.01\,\,{\rm mm}$σ2=0.01mm.

They can, however, be relevant for artificial-light rainbows, and maybe for reflection bows.

There is one older observation of a red spot appearing inside the primary rainbow above the antisolar point, recorded 20 September 1896, in Vlissingen, Netherlands, at a Sun elevation of about 13.3°. The elevation of the spot above the horizon was roughly estimated to be 14.5°, and it was interpreted as a reflected-light glory by the observer [69]. However, at this Sun elevation, no such spot does appear in the current simulations based on the BC model.

Koninklijk Nederlandsch Meteorologisch Instituut. Onweders, Optische Verschijnselen, Enz. in Nederland. Naar vrijwillige waarnemingen in 1896, Deel 17 (Amsterdam, 1897), pp. 48–49.

I am aware of only one single photograph showing a candidate twinned secondary rainbow (with the splitting point shifted off the center to the right), but more details concerning the observation and photographic equipment are needed for a definite assessment. It should be noted that modern cellphone and other digital cameras can produces artifacts when several frames are combined by an optimization algorithm, and the camera orientation was not kept fixed during the multiexposure.

L. Cowley and T. Wooten, “Puzzling triple rainbow,” https://www.atoptics.co.uk/fza149.htm .

C. Hinz and M. Worme, “Twinned rainbow,” https://atoptics.wordpress.com/2011/05/27/twinned-rainbow-2/ .

L. Cowley and M. Worme, “Peculiar rainbows,” https://www.atoptics.co.uk/fz517.htm .

http://dl.meteoros.de/haussmann/AHaussmann_DisTripRainbows_supp_inf.pdf .

L. Cowley, “Anomalous bows,” http://atoptics.co.uk/fz845.htm .

A. Haußmann, “Triple-split rainbow observed and photographed in Japan, August 2012,” https://atoptics.wordpress.com/2015/03/11/triple-split-rainbow-observed-and-photographed-in-japan-august-2012/ .

A. Boonsinsuk, “Triple-split rainbow,” (in Thai language) http://www.cloudloverclub.com/triple-spit_rainbow-chiang_dao/ .

A. Haußmann, “A multi-split rainbow from south-east China, August 12th, 2014,” https://atoptics.wordpress.com/2019/06/27/a-multi-split-rainbow-from-south-east-china-august-12th-2014/ .

C. B. Boyer, The Rainbow: from Myth to Mathematics (Yoseloff, 1959), pp. 159–160 and 223–224.

J. Appleyard, “Computational simulation of freely falling water droplets on graphics processing units,” Ph.D. dissertation (School of Engineering, Cranfield University, 2013).

C. Hinz, “Moving ripples,” https://www.meteoros.de/themen/halos/arbeit-des-akm/moving-ripples/ .

F. E. Volz, “Some aspects of the optics of the rainbow and the physics of rain,” in Physics of Precipitation, H. Weickmann, ed., Geophysical Monograph 5 (American Geophysical Union, 1960), pp. 280–286.

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Figures (6)

Fig. 1.
Fig. 1. (a) Extremal negative, (b) zero, and (c) extremal positive elongation of the (2,0) mode for an ${a_0} = 1.5\;{\rm mm}$ BC model drop at relative amplitude of $A_{2,0}^* = 0.15$; (d) monodisperse amplitude distribution from a numerical experiment involving ${10^5}$ random (2,0) oscillation states; (e) corresponding elongation distribution, following Eq. (10); (f) Gaussian amplitude distribution with a relative standard deviation of 0.3; (g) corresponding elongation distribution; (h) Gaussian distribution of drop axis directions around the vertical (standard deviation 7°, 250 samples).
Fig. 2.
Fig. 2. Gamma-type DSD according to Eq. (12) for $\mu = 1$ and $\Lambda = 3\;{{\rm mm}^{- {1}}}$, including a significant portion of larger drops. Cross section weighting was achieved by the multiplication with $a_0^2$, as valid in the geometric optics regime. This eDSD was used for the GO simulations (Figs. 3 and 4).
Fig. 3.
Fig. 3. GO simulations of dislocated bright spots from a $\mu = 1$, $\Lambda = 3\;{\rm mm}^{- {1}}$ eDSD without oscillations and at zero tilts. Equal-area (Lambert) projections encompassing one hemisphere are shown, with either the Sun or antisolar point in the center (white squares). The white line indicates the horizon. The right column depicts the responsible dominant ray paths for $n = 1.335$ (green light). “T” denotes a total internal reflection. Bright spot from the second order, antisolar hemisphere, $h_S = 36^\circ$: (a) all orders, (b) only second order, (c) ray path for $a_0 = 1.6 \,{\rm mm}$. Bright spot from the fifth order, sunward hemisphere, $h_S = 28^\circ$: (d) all orders, (e) only third–seventh order, (f) ray path for $a_0 = 1.6 \,{\rm mm}$. Bright spot from the sixth order, antisolar hemisphere, $h_S = 54^\circ$: (g) all orders, (h) only third–seventh order, (i) ray path for $a_0 = 0.8 \,{\rm mm}$. Bright spot from the seventh order (marked by the arrow), antisolar hemisphere, $h_S = 32^\circ$: (j) all orders, (k) only third–seventh order, (l) ray path for $a_0 = 1.6 \,{\rm mm}$ [63].
Fig. 4.
Fig. 4. Influence of (2,0) oscillations and Gaussian tilts on the second-order bright spot, ${h_S} = 36^\circ$. (a) Zero oscillations, zero tilts; (b) zero oscillations, tilts $\sigma = 7^\circ$; (c) corresponding CS-eDSOD; (d) monodisperse amplitudes according to Eq. (9), zero tilts; (e) monodisperse amplitudes, tilts $\sigma = 7^\circ$; (f) corresponding CS-eDSOD; (g) polydisperse amplitudes with ${\sigma _{\rm rel}} = 0.3$, zero tilts; (h) polydisperse amplitudes with ${\sigma _{\rm rel}} = 0.3$, tilts $\sigma = 7^\circ$; (i) corresponding CS-eDSOD. Color scaling for the CS-eDSODs runs from white to black, i.e., the darker a point, the higher the corresponding density of drops.
Fig. 5.
Fig. 5. (a) Triple-split rainbow photographed 5 August 2012, 18:24 JST at Yobuko, Kyushu island, Japan, by Kunihiro Tashima (${h_S} = 9.7^\circ$). For orientation, the arrow indicates the intersection point of the 138° small circle (measured from the Sun, corresponding to the familiar “rainbow size” value of 42° from the antisolar point) with the right edge of the image. (b) Inhomogeneous split rainbow photographed 18 November 2009, 15:17 AST at Bridgetown, Barbados, by Mark Worme (${h_S} = 28.3^\circ$); (c) unsharp masked and contrast enhanced version of (b), with labels indicating the upper branch (A), recently found lower branch (B), celestial region of conventional rainbow appearance (1) and region of split appearance of the primary (2).
Fig. 6.
Fig. 6. DMK simulations corresponding to Fig. 5(a) for various parameter combinations, based on a $\mu = 0$, $\Lambda = 4\;{{\rm mm}^{- {1}}}$ eDSD. (a) Mixture of 83 % drops with monodisperse (2,0) amplitudes following Eq. (13) and 17% nonoscillating drops, zero tilts; inset, corresponding CS-eDSOD; (b) mixture of 83% drops with Gaussian polydisperse (2,0) amplitudes following Eq. (13), ${\sigma _{\rm rel}} = 0.15$, and 17% drops with exponentially distributed amplitudes, scaling parameter (expected value) 0.002, Gaussian tilts $\sigma = 7^\circ$; inset, corresponding CS-eDSOD; (c) nonoscillating drops, two narrow peaks centered at ${a_0} = 0.55\;{\rm mm}$ and ${a_0} = 0.68\;{\rm mm}$ added to the eDSD [64], zero tilts; inset, corresponding CS-eDSOD; (d) same as (c), but with Gaussian tilts $\sigma = 7^\circ$; inset, corresponding eDSD (solid line) and CS-eDSD (dashed line, shows identical information as inset in (c). As in Fig. 5(a), the arrow indicates the position along the right image edge at which the scattering angle amounts to 138°.

Equations (13)

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r 0 ( ϑ ) = a 0 ( 1 + n = 0 n max c n cos ( n ( π ϑ ) ) ) .
r ( ϑ , φ , t ) = r 0 ( ϑ ) + l = 2 m = l l A l , m sin ( ω l , m t + ψ l , m ) Y l , m ( ϑ , φ ) .
ω l = σ l ( l 1 ) ( l + 2 ) ρ a 0 3 .
τ l = ρ a 0 2 η ( l 1 ) ( 2 l + 1 ) .
r ( ϑ , t ) = r 0 ( ϑ ) + a 0 A 2 , 0 sin ( ω 2 , 0 t + ψ 2 , 0 ) [ 1 2 ( 3 cos 2 ϑ 1 ) ] .
c 0 = c 0 + 1 4 A 2 , 0 q c 2 = c 2 + 3 4 A 2 , 0 q .
ξ ξ ¯ 3 2 A 2 , 0 q .
ξ max ξ min = 0.0036 ( 2 a 0 1 m m ) 2 + 0.0213 ( 2 a 0 1 m m ) .
A 2 , 0 = 0 .001133 ( a 0 1 m m ) 3 + 0 .004776 ( a 0 1 m m ) 2 + 0 .01407 ( a 0 1 m m ) .
d P d q = 1 π 1 q 2 .
d P d ϑ D = N sin ( ϑ D ) exp ( ϑ D 2 2 σ 2 ) ,
n ( a 0 ) = n 0 ( 2 a 0 ) μ e x p ( Λ 2 a 0 ) .
A 2 , 0 = 0 .025 ( a 0 1 m m ) 0.556 .

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