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IJMRR/ Jan 2014/ Volume 4/Issue 1/Article No-2/5-12
ISSN: 2249-7196
INTERNATIONAL JOURNAL OF MANAGEMENT RESEARCH
AND REVIEW
MATHEMATICS RAINBOW
Maryam Shafiee*1, Dr.Zahra Arzajani1, Moheddeseh Tajik1
1
Zakaria Razi Student Research Center, Ray one district, Ministry of Education, Tehran, Iran.
ABSTRACT
In this article the point is why rainbow can be seen in a drop of round water just between
angles 40 to 42 grade when a beam of sunlight enters the drop of water, some part of it is
reflected over it ,and the other part ,after entering the drop, exits from the other side. Some
other part of the light according to Snell's Low is reflected and exits from the drop and form
the rainbow. In the formation of rainbow two main factors play important role: coefficientre
fraction of light, wavelength which are the cause of different colors in rainbow. Each drop of
water can create one color of rainbow color in human vision bound .Then we explain the
reason why a rainbow is curved. After the investigation and proving mathematical
formulation, we came to the conclusion that the intensity of outgoing light, in all the angles
are not identical.And the most part of colorful light which goes out of the drop forms angle
about 42 grade with the direction of sun radiation. Definitely, the angle depends on the color
of the ray and it varies between 40 up to 42 grade for red up to violet .
Keywords: Rainbow, Snell's Low, wavelength, light refraction.
INTRODUCTION
There is no doubt that one of the light representations in the sky is rainbow in which the light
of the sun is diffused by the water drop and can be viewed by viewers.
Rainbow is one of the most beautiful atmospheric phenomenon which can be expected in
rainy days.This phenomenon happens when the white rays of the sun high above in the sky
hits round drops of rain,and after its breakdowntovisual light spectrum ,and reflexing within
the drop, comes out of it.(the same that happens in the spectrum).
It is of course well-known that rainbow has seven colors; however this is a historical mistake.
When IssacNewton,for the first time ,passed white rays of the sun from the spectrum and
found the analyzed spectrum of the light ,having only six colors . But, since number seven
was a holly number, he preferred azure to be located between blue and violet in order to
attract more public attention to this phenomenon. (www.youtube.com)
Rainbow is not a physical phenomenon. It means that we cannot touch it by our hands, or
have show representation, or record it through photograph, or having its track on tape
recorder. No matter how close you get to the rainbow, it moves farther and farther from you,
thus; it remains in a fixed distance from you. Because of this fact, no one could have ever
trapped it.
*Corresponding Author
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IJMRR/ Jan 2014/ Volume 4/Issue 1/Article No-2/5-12
ISSN: 2249-7196
For the first time, Rene Descartes did an experiment, he made an experiment, and he made a
big bow and considered three principles:
1. Rainbow will be formed whenever there is fine water drops, not just after rain.
2. Drops are almost round and circular.
3. The size of water drops (being bigger or smaller) does not affect the rainbow appearance.
When the light hits the round water drop, part of it is reflected while the rest part is refracted
and moves along the length of the drop until it reaches the curved inner surface and mirror
like of the drop. In each of its contact with the inner surface of the drop, part of the light is
reflected and remains in the drop and the rests comes out .Therefore, the rays of the light after
one, two, three or more internal reflection comes out of the drop with another refraction.
However , the light that hit the drop at its back is not subject to generalreflection.And some
part of the light exits at the back of the drop.But the outgoing light from the back side does
not form the rainbow,because the radiation spectrum at the back of the drop ,like the visible
rainbow does not have the required status for the formation of rainbow; therefore, the colors
instead forming a rainbow ,they combine toghter.( H. Nussenzveig 1977)
Rain drop
Fig 1: Supposed shape of the rays of the light radiating at the drop
In fact, rainbow is a continuum spectrum of all colors from violet to red which includes in
the bound of humanvision.The colors of the rainbow are the results of two basic facts:
Firstly,the radiation of the sunlight consists of all visible within human vision bound. The
colors of the sunlight when combined together seem to be white. This property of the sunlight
was first explained by Issac Newton in 1666.Visual spectrum is the name of Electromagnetic
spectrum which is recognizable and visible by human eye.Wavelength of visual spectrum
between 380 to 750 nanometer and having the frequency of 400 to 700 terahertz1.
Fig 2: Visual wavelength between 400 to 700 terahertz
1
Terahertz equals 10 Hertz (Hertz is the unit of measurement for wavelength in SIsystem)
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IJMRR/ Jan 2014/ Volume 4/Issue 1/Article No-2/5-12
ISSN: 2249-7196
Another point is that when the light of different colors pass from an obstacle e.g; air and
reaches another environment e.g; water or glass, it diffuses with different rate.
We have different rainbows including:
Binary rainbow:It forms as a result of two reflections and two refractions within water drop
which leads to the formation two rainbows ; therefore, the first rainbow color is the reverse of
the next rainbow.
Rainbow with additional bows: This happens when the outgoing light interfere each other
and based on whether this interference is builder or desolater a series of additional bows may
be built the colors of which may not belong to the rainbow.
Lunar rainbow: In which complete moon has enough light for the formation of rainbow to
make a lunar rainbow. But, as the moonlight is weaker than the sunlight, the light of this
rainbow is also weaker.(Jearl Walker, 1980)
DISCUSSION AND RESULTS
A rainbow is not actually in a specific place in the sky .The place of its appearance depends
on the place of the viewer and the situation of the sun .All the rain drops refract and reflect
the sunlight identically.But only the light of some rain drops reach the viewer’s eye.In our
investigation we noticed that the refraction of the light in the rain drop is the most important
factor in the formation of the rainbow. The amount of refraction generally depends on two
factors: wavelength and the environment in which the light enters.
When the light moves from an environment into another one in which the rate of light differs
from another one,the angle of light beam in the contour of the two environment changes and
it is so called –refracte.( www.tebyan.net)
Vylbrt Snell in his refraction Iow calculated that how the light curves or diffuses while
passing from one environment to another environment with different density(for example,
passing through air and entering water).In fact this low shows the angle of the light before
hitting and after the refraction at the contour of the two environment. (fa.wikipedia)
Fig 3: Snell Refraction Rule
When we study the direction of the light ray inside the water drop, we realize that the
deviation(drift) angle for red and violet colors differs from each other . Because violet which
has shorter wavelength,it refracts with a larger angle in comparison with the red light and
while the water drop cracks because of the light radiation from its back ,violet shines with a
smaller angle in comparison with the original radiated white color angle and reaches the
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IJMRR/ Jan 2014/ Volume 4/Issue 1/Article No-2/5-12
ISSN: 2249-7196
viewers’ eyes. As a result , red color is at the top of the rainbow while violet color is at the
bottom. (C. Boyer, 1987)
Fig 4: Light Refraction from red to violet
When we observe the rainbow and its pack of colors in fact we are looking at the light
which is refracted from scattered water drops . Each drop of water can create one color of
the rainbow which is at the human vision bound. For example, if the outgoing red light from
one drop can be seen by viewer, no blue light can be viewed from the same drop. Therefore,
according to figure 5 , we can see red light from the higher drop , yellow from the middle
one, blue and violet from the lower drop .Thus, the top of the rainbow is red and the bottom
part of it, is seen as violet. (R. A. R. Tricker, 1970&H. Nussenzveig 1977)
Fig 5: Presentation of rainbow colors by round water drops
Knowing the fact that the coefficient refraction of light in water is 1.33 to 1.34 grade ,the
reflected spectrum is an angle about 40 to 42 grade (this fact is provable by mathematical
formulation).This angle does not have any relation with the size of water drops ,no matter
whether it is big or small. But the purity of the rainbow colors depends on the size of rain
drops. Bigger drops (with diameter about a few millimeter) give brighter rainbows with
separated and beautiful colors, small drops of rain (with diameter about 0.1) give rainbows
with overlapping colors which can always be seen close to white and transparent colors.
(Jearl Walker, 1980)
Rainbows are concentric circles which we cannot see its whole circle because of the earth
movement ,and it is only possible for us to see its arc. As much as the sun is at the lowest
level of the horizon , the rainbow will be more complete. This phenomenon does not happen
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IJMRR/ Jan 2014/ Volume 4/Issue 1/Article No-2/5-12
ISSN: 2249-7196
unless the sun , the viewers’ eyes, and the middle part of the rainbow all three, be in one
direction. Therefore,if the sun is very high in the sky ,this straight line will never be made.
And because of that we can only see rainbows only early in the morning or in the evening. In
such situation the colorful light which reach our eyes ,is the collection of exited colors from
all the drops which its connective line and that of our eyes along with the straight direction of
the sun , makes angle about 40 to 42 grade.
Therefore, if a person want to see rainbow,he has to stand back to the sun and opposite to
rain in order to see the analyzed and radiated rays of the rainbow. (H. Nussenzveig 1977)
Fig 6: The approximate angle and location of the viewer to see rainbow
In this state ,the person sees the rays of six colors :red, orange, yellow, green, blue and violet
Which reach his eyes with round symmetry.Our eyes extend these rays ,and associate bows
of six colors in the sky which is called rainbow.
PROVING RAINBOW THROUGH MATHEMATICAL FORMULA
To calculate the angle of rainbow we cannot just use Snell Rule and refraction, rather we
should usesin. And geometry and derivation.In this part ,we only do the calculations which
are related to the original rainbow.
In calculating the angle of the rainbow we should have some of the additional lines in the
assumed circle of water drop e.g; altitude, diameter of the circle, radiation angle, refraction
and reflexion angles.
Fig 7: Angles and assumed lines in the water drop
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IJMRR/ Jan 2014/ Volume 4/Issue 1/Article No-2/5-12
ISSN: 2249-7196
"a" is angle of incidence
"b" is angle of transmitted
"r" is raindrop radious
The angle between the horizontal dash line radius "r" and is also "a"
"e" is the angle for seeing the colors of the rainbow by the viewer. (The aim is to calculate
this angle)
"h" is the altitude of the light where it hits the drop.
Notice: There is a isosceles between the angle of radiation and the two radius of the circle,
and the angles of b are of equal size.
The central angles of the circle totally is 360 grade. Therefore, we have:
And also we have:
majour fourmula
From the calculated formula, with regard to “a “we take differential coefficient. And we
equal it by zero. (www.phys.uwosh.edu)
*Snell Refraction Rule in point “a “(where the beam of the light hits the drop):
Because the first environment is air ,we take the refraction coefficient of the light in the air,
one.
1
Take derivative of with Snell's Low refraction the respect to a:
We know that:
We replace it in the obtained formula:
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IJMRR/ Jan 2014/ Volume 4/Issue 1/Article No-2/5-12
ISSN: 2249-7196
Snell's Low refraction in the point b (where the beam of light comes out from the drop):
Here the first environment is water and the second environment is air:
1
Therefore From these two formulas we conclude that:
we take ‘Arc’ from the formula in order to replace it in the final formula:
both side will be given second power and will be replaced for ‘
by *
’ in the formula marked
We replace it
If we use the obtained amount instead of ‘sin a’, the final formula will be the following:
If the coefficient refraction of light in water be 1.33,we will have:
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IJMRR/ Jan 2014/ Volume 4/Issue 1/Article No-2/5-12
ISSN: 2249-7196
For red light
If the coefficient refraction of light in water be 1.34, we will have:
For violet light
In other words ,we conclude from the above calculation that the intensity of the outgoing
light is not identical in all the angles,and most of the colorful light which exits from the drop
makes
An angle about 42 grade.This angle certainly depends on the color of the ray, and for red and
violet ,the angle is between 40 to 42.Therefore , we suppose that colorful rays can effectively
exit the drop only in angles about 42 grade.
REFERENCES
Boyer C. The rainbow: from myth to mathematics, Princeton University Press. Atmospheric
Sciences 1987; 29: 211.
Fraiser, Alistair B. In homogenieties in the Color and Intensity of the Rainbow.
Greenler, Robert, Rainbows, Halos, and Glories, Cambridge University Press, 1980.
Nussenzveig H. The theory of the rainbow, Scientific American, April 1977.
Humphreys WJ. Physics of the Air, McGraw-Hill Book Co. 1929.
Humphreys WJ. Weather Proverbs and Paradoxes, Williams and Wilkins Company. 1929.
Tricker RA. Introduction to meteorological optics, American Elsevier, 1970 .
-fa.wikipedia.org/wiki/
-www.youtube.com
-www.tebyan.net
-www.phys.uwosh.edu/rioux/genphysii/pdf/rainbows.pd
-www.phys.uwosh.edu
-www.konjkav.com
www.ams.org/samplings/feature-column/fcarc-rainbows
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