SUPERNUMERARY BOWS

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SUPERNUMERARY BOWS

• By
• Katrina 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 heads
  shadow?

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• Is the rainbow a spot 42 above your heads
  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 its center at your heads
shadow.
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Alexanders 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 youd
  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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Youngs Theory
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Youngs Theory

• In the 1800s most scientists agreed with Newton,
  but Robert Hooke and Christiaan Huygens believed
  that light behaved more like waves than
  particles.

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Youngs Theory

• In the 1800s 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 its 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 its 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 bows 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
  Airys Integral.

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George Biddel Airy (1801-92)

• Airys theory of the rainbow extended and
  mathematically formalized Youngs largely
  empirical explanations of interference within a
  raindrop.

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AIRYS 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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Fresnels Integral

• Total disturbance given by
• Ao sin pt ? cos d dx Ao cos pt ? sin d dx
• where
• ? cosd dx B ? cos(?v²/2) dv
• ? sind 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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AIRYS INTEGRAL

• A ( ?a²/(4kcos?))? ? 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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• Weve computed the values of the table by using a
  series developed from Airys Integral.

Pochhammer


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bow number z at max intensity intensity z 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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T H E E N D
T H E E N D
T H E E N D