SUPERNUMERARY
BOWS
By
Katrina Brubacher
&
Asha Padmanabhan
“Rainbow” commonly refers to a single circular
arc of non repeating colors.
Is the rainbow a spot 42° above your head’s
shadow?
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.
Therefore the primary rainbow is a circle
with radius 42° and it’s center at your head’s
shadow.
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°
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.
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.
“Supernumeraries proved to be
the midwife that delivered the
wave theory of light to its place
of dominance in the 19th
century.”
~Rainbow Bridge
Young’s Theory
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.
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.
Interference
• It is the interference of waves that explains
supernumerary bows.
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.
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.
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.
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.
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.
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
• 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.
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.
• Supernumeraries of the secondary rainbow?
• Young did not give a quantitative account of
the interference theory of the rainbow.
• 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.
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.
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.
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
• 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.
AIRY’S INTEGRAL
A = ( λa²/(4kcosθ))^⅓ ∫ cos ((/2)(u³-zu) du
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
• We’ve computed the values of the table by
using a series developed from Airy’s
Integral.
Pochhammer
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
Things that are NOT possible
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