### Author Topic: physics of Rainbow  (Read 607808 times)

#### Fu-Kwun Hwang

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• Posts: 3062
##### physics of Rainbow
« on: January 29, 2004, 11:39:38 pm »
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A most charming example of chromatic dispersion is a rainbow.
When white sunlight is intercepted by a drop of water in the atmosphere, some of the light refracts into the drop, reflects from the drop's inner surface, and then refracts out of the drop.
As with the prism, the first refraction separates the sunlight into its component colors, and the second refraction increases the separation. The result is the rainbow.

You can find similar explanations at textbook. However, the incoming ray can be refracted at different angle, why the rainbow is located at fixed angle of view? Why it is a circle?

This applet shows the physics of the rainbow.

The black circle represents a drop of water in the atmosphere.
Initially, red light is coming from the left; you can click inside the colored blocks to change the color of the incoming light.
The incoming ray is unpolarized, which can be represented as a mixture of two polarized waves whose planes of polarization are perpendicular to each other.
So the notation " 50%| 50%+" means that half is polarized in the up-down direction and half is perpendicular to the screen.
Many things can happen to the light.
• Part of the incoming ray is reflected back to the atmosphere (indicated by ray number 1). The intensity of each polarized component is shown along the ray path.

• Part of the light refracts into the drop, then refracts back to the atmosphere(ray number 2).

• Some reflects from the drop's inner surface, and refracts back to the atmosphere (ray number 3). This gives rise to the ordinary rainbow.

• Some reflects twice inside the drop, then refracts back to the atmosphere.(ray number 4) This gives rise to the secondary rainbow that is sometimes seen.

You can drag the incoming ray, move it up and down, and watch how the relative intensities change. R is the radius of the water drop; b is the vertical distance of the incoming ray from the center of the circle.
The intensity of the light coming from rays 3 and 4 is plotted versus viewing angle. Click ^or vto change the scale.
For ray number 3 there is a maximum scattering angle, and for ray number 4 there is a minimum -- this is why there are strong peaks in the scattered intensity.
The rainbow is actually a disk of scattered light, but it is brightest at the edge; the disk for different wavelengths is a different size, and that is why we see the color effects there.
You can click inside the white box, and see what will happen to white light.
When light refracts, it follows the law of refraction ni sin(ctai)=nr sin( ctar) where n is the index of refraction.
The number in the left-bottom corneri is the angle of incidence, r is the angle of refraction.
Most of the light is refracted out by ray
2. When your eyes intercept the separated colors from raindrops, the red you see comes from drops angled slightly higher in the sky than does the blue.
You see a circular arc of color, with red on the outside and blue on the inside.
Click inside the colored box to show this effect. You can drop one of the rain drops,Try it!

Did you notice that the rainbow consists of partially polarized light?
The following picture are results from experiment (view rainbow with polarizer at different angle).

You are welcomed to check out EJS version of rainbow applet.

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#### Fu-Kwun Hwang

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• Posts: 3062
##### Email discussion
« Reply #1 on: January 29, 2004, 11:41:06 pm »

 name direction of electric field color s-wave perpendicular to the screen ( + ) same color as the light p-wave  in the up-down direction ( | ) yellow

This is an e-mail message from Neal Rasmussen (posted with his permission)

I am a tornado chaser for 20 years. About 4 years ago I was driving through a McDonald's drive thru. As it was just after a pretty good storm, I had mys polarized yellow tinted flip-up sunglasses on. Low and behold I almost dropped my cigarette into my lap. There off in the east was a double rainbow! Not the kind I've seen dozens of times where the second one is reveresed and much outside the primary, but this was immediatly INSIDE the primary!

Gaping with my mouth open and people behind me honking, I flipped my flip-ups up and the second inner one went away! I must have sat there for 2 minutes flipping my sunglasses. The colors were NOT reversed and it looked like one continuous double wide rainbow, with two complete sets of colors. I ruled out that the flip-ups where giving double vision as I looked at lights the next day, streetlights headlights, etc. and no doubles! Can this possibly be? Could a source of polarized light be at a different angle, perhaps the sun shining through a cirrus cloud or noctelucent (sp).

Email message:
Subject: circular rainbows
Date: Fri, 4 Aug 2000 19:03:52 EDT
From: Jrs3290@aol.com
To: hwang@phy03.phy.ntnu.edu.tw

Hi, I am facinated by rainbows but am not smart enough to understand it all.
I do know that the most amazing one I have ever seen was when I was in a
small single engine aircraft above a layer of white clouds. The cloud cover
was smooth and the sun made a shadow of my aircraft . As I looked at the
shadow of the aircraft on the clouds I noticed it was encircled by a
circular rainbow. The rainbow followed until the cloud cover changed and then
disappeared. Sincerly, Jim Spencer

My response:

I was always dreaming to be able to see the same view as you have seen (whenever was on an airplane).
What you have seen is a true rainbow. Circular rainbow can only be viewed from above the ground.
When we were on the ground, at least half of them were missing.
The viewing angle is about 42 degree.

Subject: Re: circular rainbows
Date: Sat, 5 Aug 2000 10:47:47 EDT
From: Jrs3290@aol.com
To: hwang@phy03.phy.ntnu.edu.tw

Hi, yes you may use the letter any way you like. If you do I wish you would
let me know so I can look at it. The angle from where I was sitting in the
aircraft and the shadow was about 45 degrees as I remember it and I wonder if
the angle had been greater or less do you think I would still have seen a
rainbow or was I just fortunate that day to be at the right place at the
right time ?? Jim Spencer

• Guest
##### topic44
« Reply #2 on: January 30, 2004, 09:09:50 am »
Subject: I saw a Polarized Rainbow!
Date: Sat, 14 Mar 1998 01:18:24 -0800
From: Neal Rasmussen <nealras1@airmail.net>
To: hwang@phy03.phy.ntnu.edu.tw
I am, as far as I know, the first to see a "Polarized Rainbow"!
I think we should name this phenomona the "Rasmussen-Wang Rainbow". :-)
I am a tornado chaser for 20 years. About 4 years ago I was driving through a McDonald's drive thru.
As it was just after a pretty good storm, I had mys polarized yellow tinted flip-up sunglasses on.
Low and behold I almost dropped my cigarette into my lap.
There off in the east was a double rainbow!
Not the kind I've seen dozens of times where the second one is reveresed and much outside the primary,
but this was immediatly INSIDE the primary!
Gaping with my mouth open and people behind me honking,
I flipped my flip-ups up and the second inner one went away!
I must have sat there for 2 minutes flipping my sunglasses.
The colors were NOT reversed and it looked like one continuous double wide rainbow,
with two complete sets of colors.
I ruled out that the flip-ups where giving double vision as I looked at lights the next day,
streetlights headlights, etc. and no doubles!
Can this possibly be? Could a source of polarized light be at a different angle,
perhaps the sun shining through a cirrus cloud or noctelucent (sp).
Neal Rasmussen nealras1@airmail.net

• Guest
##### topic44
« Reply #3 on: January 30, 2004, 05:38:12 pm »
From: NAPLES96@aol.com
Date: Thu, 28 Oct 1999 08:58:38 EDT
Subject: Interesting find!
To: hwang@phy03.phy.ntnu.edu.tw

Just stumbled upon your observation while looking for reasons for a rainbow.
Actually I had seen a "round" rainbow and was unable to find any comments on
the subject. Have you any knowledge of one?

I'm just an admirer of the beauty of the sky, here in Naples, FL.

Thank you,

• Guest
##### topic44
« Reply #4 on: January 30, 2004, 05:49:45 pm »
From: Ayuyroq@aol.com
Date: Thu, 2 Dec 1999 16:48:07 EST
Subject: (no subject)
To: HWANG@phy03.phy.ntnu.edu.tw

My youngest daughter did an elementary school science fair project on the
effects of pollution on rainbows and she found the applet on how rainbows are
formed, which is copyrighted by you, on line.

Could you tell us how to credit this work done on line in her Bibliography?

Thank you,

A.

• Guest
##### rainbow physics
« Reply #5 on: February 22, 2004, 12:34:00 pm »
Great site with clear explanation.

I thought the double rainbow that one sees emanates from light that enters very low on the raindrop and then reflected twice within the raindrop. Is that true?

Your diagram seems to suggest that the primary and secondary rainbows are created by different paths of the same light and not seperate rays of light.

Is there any way to add the perspective of the observer to your diagram to show the paths of the light producing what the observor sees (primary and secondary rainbows) and the angles at which the rainbows would be seen in nature.

Is there anyway to also add the perspective of the observer to show why the color spectrum is seen as a "bow" or arc and why a rainbow would be seen as a circle from just the right perspective (e.g. airplane).

Great site and thanks.

#### Fu-Kwun Hwang

• Hero Member
• Posts: 3062
##### topic44
« Reply #6 on: February 22, 2004, 12:45:13 pm »
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from the following picture, you will find that the intensity of the first rainbow is about 2*4.5% of the incoming red light. So the intensity is indeed small.
The rainbow you saw was coming from many different raindrop.
The primary and the second rainbow also create from differnet raindrop and different light path. Remember light need to enter your eye for you to see the image.
The angle shown on the applets is assume the sun light is horizontal.
If the sun is higher you need to substract the angle of the sun ray relative to horizontal.
If the angle is less than zero , you would not be able to see rainbow on the ground.
That why you never saw rainbow at noon (after 9am or before 3pm) on the ground.

The angle for the second also shown on the applet, as indicated in the following picture.

The following applet show you why why the color spectrum is seen as a "bow" or arc.
(The other half will be underground if there are raidrops and can refelect sun light, which is impossible.) However, it is possible when you were on an airplane.

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• Computer Engineer
• Newbie
• Posts: 1
• Music, Computers, Rainbows.
##### topic44
« Reply #7 on: February 22, 2004, 09:43:38 pm »
Nice explanations of the mathematics of rainbows.

A "Round" Rainbow will always be observed when the sun is directly behind you and the water droplets are directly in front of you and you are at an altitude that enables you to see the circle of light. I'm not entirely sure if the distance you are from the water droplets determines the size of the circle, I'll have to investigate more. The "circle" is a "Halo" effect similar to the effect you get on a foggy night when the moon is at its fullest.

#### Le

• Newbie
• Posts: 1
##### topic44
« Reply #8 on: October 10, 2005, 10:59:27 pm »
Hi!

Came across this site when researching light and the rainbow. Have found some really interesting information here. I dont actually work in or study science, but I am fascinated by it and always trying to inform myself in many scientific theories which I try to incorporate into my artwork(am in my final yr in art college, specializing in interdisciplinary studies). Anyway...
:oops: This will possibly be a stupid question but its something someone has recently told me about. I have been told that on one side of the rainbow there is a kind of blank space or void which is said to be unexplanable, romantically described as holding the mysteries of the universe, this fascinated me so much and am wondering if there is infact such a void?...

Thanks

#### Fu-Kwun Hwang

• Hero Member
• Posts: 3062
##### email from users
« Reply #9 on: February 25, 2006, 05:03:41 pm »
I receive the folllowing message from a user :

Hi there, I took these shots, and more besides flying into Bristol on a commercial flight.

The following are pictures attached with the email message
[img:02ee8304f5]http://www.phy.ntnu.edu.tw/ntnujava/pics/pic20060225a.png[/img:02ee8304f5]
and
[img:02ee8304f5]http://www.phy.ntnu.edu.tw/ntnujava/pics/pic20060225b.png[/img:02ee8304f5]

But I might say that the above two picture are not the same as the rainbow in the simulation.
I also saw it once when I am in an airplane,too. I can not recall it's name, now.

#### gon

• Newbie
• Posts: 5
##### Re: physics of Rainbow
« Reply #10 on: October 11, 2007, 04:39:38 pm »
Mr. Fu-Kwun Hwang,

You are such a genius. My question is, how does double rainbow forms? I was once see a double rainbow when I am lying on the floor while fixing my jeep engine parts saturday morning. And I was amazed because usual rainbows are single. How do we know about it? My friends are also amazed with it. Thanks in advance.

I have here an image of double rainbow:  http://www.photocentric.net/gallery/rainbows/RPrainbow2.jpg

-Ms. Gon.

#### Fu-Kwun Hwang

• Hero Member
• Posts: 3062
##### Re: physics of Rainbow
« Reply #11 on: October 11, 2007, 10:00:19 pm »
I am not a genius. I learned it from standard textbook for physics. If you have enough resolution device, you even can see the third ranibow. But it will be very difficult, because the first two are pointing back to the direction of the sun, so you can see it while your back is face to the sun. (for the light coming from the sun, the intensity of the rainbow light  will be too weak compared to the light from the sun, so you would not be able to see it).

Intensity for the second rainbow is even weaker(2%) than the first one (about 5% of the intensity of sun light).
So normally, you might not notice it.

If you drag the horizontal red line with your mouse from the top and move it downward slowly.
Check out the angle for #3 ray (at the bottom you will find 3):
you will find the angle increase first, then at about 42 degree it become smaller again as you continue to drag the line down.
And the maximum angle is the angle you will find rainbow for red light (when the ray of incoming sun light is horizontal).

You will find #4 ray at the top. But when you drag the red line lower than the center of the water drop(big white circle)
the # 4 ray starts to point download. when it reach near 50 degree(also the maximum angle), that is the angle you will find the second rainbow.
Normally, when you watch the rainbow, the ray of the sun light is not horizontal.
If it is 15 degree higher, you need to substrate the 15 from 42 (or 50). So when the sun is too high, there is no way you will find rainbow in the ground (you might see rainbow from high enough mountain).
So you might be able to see the rainbow in the early morning or late in the afternoon.

I hope the above information will help.  I upload the picture for the rainbow, so more user can see the beautiful rainbow.
And I hope many of you can enjoy the fun of physics as I do.

#### gon

• Newbie
• Posts: 5
##### Re: physics of Rainbow
« Reply #12 on: October 12, 2007, 08:10:15 am »
Wow... That was the double rainbow photo I have given you. It was really great to see. I can't really imagine that there is still a third rainbow.    Wow ...... Thanks a lot. I gonna tell this to my friends and also ofcourse to meet you Mr. Fu-Kwun Hwang.  -*-
« Last Edit: October 23, 2007, 09:51:09 am by gon »

#### nonymou

• Newbie
• Posts: 1
##### Re: physics of Rainbow
« Reply #13 on: May 14, 2008, 12:33:16 am »
Dear Fu-Kwun,
Thank you for sharing this excellent work you have done.  This is a tremendous resource for understanding physics!  I am not a physicist, but I am working on a solar energy project and need to understand something about optics for concentrating sunlight with a lens, including the prismatic effect, as demonstrated by your rainbow applet.  I hope to be able to create an applet of the project I'm working on, building on the 'thick lens' and the 'rainbow'.  If I am successful, I'll post it for others to see.

Thanks again.