supernumerary bows by katrina brubacher & asha padmanabhan
TRANSCRIPT
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SUPERNUMERARY BOWS
ByKatrina Brubacher
&Asha Padmanabhan
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“Rainbow” commonly refers to a single circular arc of non repeating colors.
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Is the rainbow a spot 42° above your head’s shadow?
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Is the rainbow a spot 42° above your head’s shadow?
A spherical raindrop will not prefer one direction to another.
All locations that lie 42° from the shadow of your head are equally likely to send the concentrated rainbow to you.
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Therefore the primary rainbow is a circle with radius 42° and it’s center at your head’s shadow.
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Alexander’s Dark Band• Most of the rays that come out of a drop are
concentrated at 138° from the sun, but some light is bent through all angles between 180° and 138°
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SUPERNUMERARY BOWS
• Supernumeraries are much more common than you’d think but the number that are seen vary.
• Their colors also vary. The most common colors are pinks and bluish greens but yellow is sometimes also observed as well as violet.
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Newton believed that the behavior of light was best explained as a series of small particles traveling from the light source to the eye but this does not explain the presence of supernumerary bows.
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“Supernumeraries proved to be the midwife that delivered the
wave theory of light to its place of dominance in the 19th
century.” ~Rainbow Bridge
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Young’s Theory
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Young’s Theory
• In the 1800’s most scientists agreed with Newton, but Robert Hooke and Christiaan Huygens believed that light behaved more like waves than particles.
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Young’s Theory
• In the 1800’s most scientists agreed with Newton, but Robert Hooke and Christiaan Huygens believed that light behaved more like waves than particles.
• In 1803 Thomas Young asserted that supernumerary bows could be explained only if light were thought of as a wave phenomenon.
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Interference
• It is the interference of waves that explains supernumerary bows.
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Interference cont.
• It is the interference of waves that explains supernumerary bows.
• If the crests of two waves coincide, they reinforce each other to make a larger wave. If a crest of one wave sits in the trough of another, the two disturbances cancel each other and the medium will be at it’s original level.
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Interference cont.
• It is the interference of waves that explains supernumerary bows.
• If the crests of two waves coincide, they reinforce each other to make a larger wave. If a crest of one wave sits in the trough of another, the two disturbances cancel each other and the medium will be at it’s original level.
• This is called constructive and destructive interference.
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Supernumerary bows are not caused by the interference between two light waves, they are caused by the interference of two different portions of the same light wave.
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Size of raindrops
• Young used the wave theory to account for the color and brightness of the supernumerary bows and to estimated the sizes of raindrops that yielded supernumeraries.
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Size of raindrops cont.
• The size of the raindrops change the appearance of the supernumerary bows.
– A smaller drop gives widely spaced bows, the larger drop gives more tightly spaced bows and each bow is narrower.
– The first supernumerary for the smaller drop occurs at the same deviation angle as the second supernumerary of the larger drop.
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Size of raindrops cont.
• When the drops are small, each bow is broad, including the primary. Hence the bow’s colours overlap and appear pastel
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• Young was able to estimate the raindrop size of a shower based on the spacing between supernumerary bows. The spacing decreases as the drop increases.
• The reason for this is that the spacing of bright and dark bands in the folded wave front depends on the path length the wave has traversed within the drop.
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Size of raindrops cont.
In nature, drops with a radius that is greater than 0.4mm can make the supernumeraries brighter than the primary rainbow.
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• Supernumeraries of the secondary rainbow?
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• Young did not give a quantitative account of the interference theory of the rainbow.
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• Young did not give a quantitative account of the interference theory of the rainbow.
• For a numerical description we must look to Airy’s Integral.
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George Biddel Airy (1801-92)
Airy’s theory of the rainbow extended and mathematically formalized Young’s largely empirical explanations of interference within a raindrop.
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AIRY’S MATHEMATICS
• The explanation for the supernumerary bows come from looking at light exiting a raindrop.
• The light is sharply cut off in the direction of minimum deviation and the effects are similar to those of a shadow along a straight edge. This was first solved by Fresnel.
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Fresnel’s Integral
• Total disturbance given by:
Ao sin pt ∫ cos δ dx + Ao cos pt ∫ sin δ dx
where
• ∫ cosδ dx = B ∫ cos(v²/2) dv
• ∫ sinδ dx = B ∫ sin (v²/2) dv
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• In a rainbow the effects of diffraction are seen just inside the illuminated area. This area is cut off by the cone of minimum deviation.
• This leads to bright and dark bands within the primary bow or outside the secondary bow : supernumerary bows.
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AIRY’S INTEGRAL
A = ( λa²/(4kcosθ))^⅓ ∫ cos ((/2)(u³-zu) du
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bow number z at max intensity z at min intensity
1 1.085 2.4955
2 3.4669 4.3631
3 5.1446 5.8922
4 6.5782 7.2436
5 7.8685 8.4788
6 9.0599 9.6300
7 10.1774 10.7161
8 11.2364 11.7496
9 12.2475 12.7395
10 13.2185 13.6925
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• We’ve computed the values of the table by using a series developed from Airy’s Integral.
Pochhammer
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bow number
z at max intensity
intensityz at min intensity
intensity
1 1 0.2868 2.3 0.0007
2 3.1 0.1392 3.6 0.0016
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Things that are NOT possible
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