Rainbows don't work the way you think they work
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- Опубліковано 29 сер 2024
- A visual explanation of the mathematics at the end of the rainbow.
I learned this explanation from V.I. Arnold's book "Mathematical Understanding of Nature", where you can also find the calculations hidden behind these animations and graphs. bookstore.ams....
The rainbow photo is by Kodyak Tisch on flickr. flickr.com/pho...
#SoME1
FAQs
* How does a double rainbow form?
The secondary (outer) rainbow is formed by rays that reflect twice inside a raindrop. In that situation the graph has a minimum instead of a maximum, but the same reasoning still works. And it's not just the graph that is flipped: the colours of the secondary rainbow are in reverse order too! Starting form the horizon you see white reflected light, then the primary rainbow, then a band without any reflected light, then the secondary rainbow, outside of which there is again white reflected light. I've posted some unedited animations of this here: ua-cam.com/video/3edTSqbqwHg/v-deo.html
* Doesn't some of the light exit at the back instead of reflecting?
Yes it does. In the animation I've only drawn those rays that contribute to the primary rainbow. Which fraction of the light is reflected depends on the angle of incidence. The animation does not take into account this dependence. This is one of several reasons why my animated rainbow is only an approximation of the real thing.
* Does total internal reflection take place?
No. If a ray inside water hits the surface at a very shallow angle, all of it will be reflected. This phenomenon is called total internal reflection. However, none of the rays shown in the animation hit the back at a sufficiently shallow angle for this to take place. A good explanation why total internal reflection can't take place here (with a helpful sketch) is given in this Quora answer: www.quora.com/Does-total-internal-reflection-take-place-in-a-rainbow-or-not/answer/Jeff-Jo-4
Thanks for the great video and your crystal clear follow up answers! This is what brilliant educational UA-cam is all about :)
I- I just wanted to asked those questions, anyway, thank you!
In the diagram you used a circle to represent a raindrop, but raindrops aren’t spherical? Also, great job on the animations they are very well made and I’m sure this video is soon to gain even more traction.
So Would A Double Or Triple Rainbow Be Somehow Significant To The Harmonics Of The Light Or Is It Only Dew To The Properties Of The Prism?
Waw. I hope to see more videos like that. Thank you.
This is brilliant
Hello @SteveMould
Indeed! I knew a lot of the basic explanation, but I didn't know the angles decreased again as you continue to increase the variable *d*
I also never thought about the concentration of rays near the maximum, which is a direct consequence of the derivative going to zero and changing directions at the maximum. Very interesting indeed! It helps explain why the colored portion of the rainbow is bright enough to see against the backdrop of the sky.
It would be wonderful to also get an explanation of the interference supernumerary rings as well.
Thank you, Steve, for sharing!!
You sent me and it was a good watch
Agreed! Thanks Steve 😊👍
I often see double rainbows (most often the outer one is much weaker than the inner one). 50 years ago I saw a triple rainbow, all of which were very bright. It took me 48 years to understand how that could have happened. The answer is that one (probably the innermost) was due to the Sun's reflection from the smooth surface of a lake behind me; so there were in effect two Suns making the three rainbows!
That's crazy, must have been quite a sight!!! As I'm sure is evident in your memory lasting 50 years of it. Nature truly is beautiful ❤️
Were you near the lake, or much higher up?
That's a good guess, but is not the reason double and triple rainbows appear. These are due to the same process, but when the light bounces around twice in the water droplet before exiting. With each bounce, some power is lost, so the double triple (and fourth, fifth, etc) are dimmer and dimmer. That is why you usually see just one or two, and you need exceptionally clear wheather and perfect conditions to see the third rainbow.
I came to the comments to ask about double rainbows. Would love to see a follow-up video on those!
@@frotaurare you sure that it wasn't a contributing factor in this instance? He said that they were all very bright. Maybe the second light source hitting at a significantly different angle caused a similar effect but for a different reason.
this was a ridiculously good explanation! and beautifully animated as well!
There's other effects also happening here, particularly to do with the internal reflection off the back of the droplet. Until the angle is shallower than a critical angle, most of the energy will be transmitted out the back rather than reflected. Likewise, if that exit refraction is sufficiently shallow, you instead get a second internal reflection, giving a double -- or with even more reflections, a triple -- rainbow.
Critical total reflection for water-air is 48° I think. Below that you dissipate from the back. So indeed the rays hitting the droplet centrally are very attenuated from this effect
Thanks for this comment.
And if the angle of incidence is shallow (near the top in the diadram), the light will reflect off and not enter the drop.
It appears, based on the raw animations posted by Mats in another comment, that 3 reflections would produce the "triple" rainbow on the other side of the raindrop from you, putting it behind the first 2 from your perspective.
(But it's entirely possible that paths with more internal bounces, a third image could be projected in the same area -- between you and the raindrop -- as the first two)
I think the total internal reflection effect is much more dominant than the effect described in this video.
Rays hitting the drop near center (small angles) are likely to hit the back of the inside of the drop at higher than the critical angle and pass through the drop without reflecting. The "critical angle" is where any light completely reflects off the inside surface. This can be demonstrated by filling a glass with water and looking somewhat upward at the surface of the water. At some angle, the surface suddenly looks silvery and reflects what is on the other side of the glass. At this angle and all shallower angles, all the light hitting the bottom of the surface is reflected. At higher angles, we look through the surface of the water and little light is being reflected. If this is considered, the left end of the graph would be missing.
Thanks for your comment! Total internal reflection is a fascinating and underappreciated topic, but I don't think it is relevant here. As you say, it happens when the angle (between the ray and the surface) is shallower than some critical angle, but rays hitting the drop near the centre will hit the back at a large angle, almost perpendicular (see 1:50). Furthermore, if the droplet is perfectly spherical, then a ray inside of it will have the same angle with the surface, whether you look at entry, reflection, or exit. Since it was possible to enter at this angle, it must be larger than the critical angle, so it will be possible to leave the droplet any time it hits the surface.
True and this would remove values from the thin spectra and make the maximum values more pronounced,resulting in a more vivid rainbow. Since all this beautiful work is alredy done i would love to see a more accurate one with a double rainbow if possible,created the same way. Beautiful video anyway.
@@MatsVermeeren Actually if you take critical angles under cosideration double and triple rainbows will result from your graph with not much effort.
Maybe I should have made clear that I only drew those rays that contribute to the primary rainbow. Some (if not most) of the light will indeed exit in the back instead of reflecting. This is true for all values of d!
The reflection coefficient depends somewhat on the angle, and if the animation took this into account the rainbow would possibly be a tiny bit brighter. However, none of the rays will hit the droplet at a shallower-than-critical angle, so for no d will there be total internal reflection.
@@MatsVermeeren i think the light rays exiting the back still looks impressive. To see them you have turn around and look towards the droplets between you and the sun. It looks like the sky is blazing orange because its refracting the colors of the sunset. Whenever i see a rainbow i always make sure to look the other way hoping to see this zero order glow. Its not always possible because there isn’t always water droplets at every direction, also why sometimes half the rainbow is missing :(
Something to note is that in this 2D representation you give all entry locations equal weight, while in 3D you have expanding circles rather than just a moving point. This means the graph should start quite flat and curve upwards, reaching a more pronounced maximum at ~42°, which strengthens the effect, and is why you usually only see a little white glow just inside the main rim of the rainbow. Of course, our vision working on a log scale complicates things even further.
Mobile phone cameras - on the other hand - are very good at detecting those tiny differences; and I finally have the answer to my question why faint rainbows start looking like white disks with a rainbow at the edge when you take a long exposure photo. 🤯
OMG thank you! Man, I wondered multiple times already whether I was just seeing psychological or other phenomenons at the rainbow, but it turns out the default explanation is just off. Thank you for highlighting something that makes much more sense and models the actual observations way more accurately!
Glad you liked it! I wondered if my title was a bit too much like clickbait, but I think your reaction justifies it :-)
@@MatsVermeeren it’s good! It would have been clickbaity if you called it ‘The truth about rainbows’ or ‘What we thought about rainbows is all wrong’.
one of the best explanations of rainbows I ever saw. waiting for the second part, with the 2nd rainbow, the dark part between the two and the additional (quantum?) effects!
This is the best and clearest explanation of rainbows I’ve ever seen. Thank you.
This is exactly how the media and documentaries present rainbows to the general public, so the title is a bit of a stretch, but thanks for the video.
I appreciate that the title is a bit click-baity and doesn't apply to everyone, but in my experience most explanations for a general audience incorrectly suggest that *all* light of a given wavelength is reflected at the angle where we see it in the rainbow.
A title like that makes me think that the video will first present a simple and naive explanation that I know to be inaccurate. But the initial explanation was the most intuitive and comprehensive explanation I've ever heard, bar the rest of the video.
Never had such a ”Eureka!” moment watching a UA-cam video before. Brilliant video.
I've never before a good explanation of why the inside of a rainbow is brighter than the outside. Thanks!
Yes, the brightness of the area inside the rainbow is often dramatic compared to the darkness around it. This was a very good explanation for why that is.
How about a double rainbow?
Almost the same thing. The secondary (outer) rainbow is formed by rays that reflect *twice* inside a raindrop. In that situation the graph has a minimum instead of a maximum, but the same reasoning still works. And it's not just the graph that is flipped: the colours of the secondary rainbow are in reverse order too!
@@MatsVermeeren Cool thanks. Double rainbow visualization for next video?
The darkest part of a double rainbow is in-between the two bands.
Nice! Double 🌈 next pls!
@@jameshart4867 Yes, exactly. I've noticed that every time I've seen one.
The best explanation on rainbows I have seen. You sir really know how to satisfy curiosity.
Wow. One of the best explanation videos I've ever seen! The best part is that I actually understood it! Well kind-off...
But I'm hooked, to say the least. Thank you so much.
The brightness difference between inside and outside of the rainbow is something I'd been wondering about for a couple years. I am really glad I just randomly stumbled upon your video Mats. Thank you for the clear explanation :)
Absolutely, it's nice to finally have an answer for this
This explanation leaves out the important pot o' gold, where it can be found and how long it lasts.
I knew the inner part of the rainbow was slightly lighter. But the best explanation until now is that it had something to do with the same effect creating the colored bands. Kort en bonding; Your description was well done!
I have always wondered why the insides of rainbows are brighter. Thank you for finally letting me learn why
That is the best explanation of rainbows I have ever seen. Thank you.
I've acrually always noticed that the inside is brighter, and always wondered if that was purely perceptional or if there was a physical explanation. Now I know! Thank you!
Fascinating! I hope you get a lot more views! (I found you on 3Blue1Brown's playlist.)
Man just popped up 10 years ago, made this banger of a video 10 years later, and then never uploaded again. God.
I've actually noticed this is bright disk for a while, so when I saw the graph, I immediately knew where you were going with it.
Fantastic - Always fun to take something you think you know and go to the next level, especially when there are effects you haven’t noticed that are only explained by that deeper knowledge.
I've often noticed that the inside of a rainbow looks a little different than the outside, but never gave it much thought. So when the video got to that part at the end, I audibly gasped, recognizing what I've noticed so many times!
I was always wondering why it's brighter, and now I now!
Thank you, really awesome🌟
I have to admit, I fell into poor form and started responding down here before listening to the whole thing because I assumed, based on low sub count and my algorithm history, that you were some version of a flerfer; when I first heard you say "reflection". I knew all of this information already and was gonna go in on it on you.
I am very pleased I was incorrect and out of turn, instead.
Great explanation.
Well done, friendo.
Keep up proper discussions about your passions. And have a sub to make up for my snootyness.
That is awesome! I think this video has finally been blessed by the algorithm, and deservedly so
I have ALWAYS wondered why the inside of a rainbow is brighter. Thank you so much, this makes me really happy to learn!!
I knew there was something inside the rainbow arc! I just didn't know it was really there or at least wasn't really sure about it. Always thought I made it up or so. So thanks for the explenation! Great video :D
Beautifully explained! Thank you!
Thank you for this "enlightening" explanation 😊. Now I can visualise a funnel behind the rainbow! 🌈 👌🏼
I was fully expecting this to be another UA-cam video that says things are what you think, only to be exactly what you think. But no it was really clear explanation and I had never actually considered the detail of how they form. Nice!
This works exactly how i thought they worked. We did the refraction calculations in calculus and it was really interesting to learn
Thank you, rainbows are neat! The arc of the rainbow describes the base of a light cone whos apex is your eye. A person standing next to you sees a different rainbow.
Great video! I already noticed that things doesn't add up in the way I used to think about rainbows, but I never bothered to sit down and actually do the math to see what's really happening. Now I don't have to anymore, thanks!
That's amazing!!! As a geologist, I work daily with refraction in thin section of translucent minerals, and refraction never stop amazing me. Great work!!!
this explanation is worth a pot of Gold. Brief, clear, and colorful. Humanity benefits most from bright people making vaccins or explaining things clearly, and the song " I can see clearly now the rain has gone" OS perhaps an appropriate tribute to this contribution about the rainbow. So, I bow for this excellent presentation. I have one question, and forgive me if I did not get the point. Only a small fraction of light hits a rain droplet spot on. But this fraction will hardly be reflected at all ? Whatever, thank you as many times as there are raindrops making a rainbow.
Great explanation. Would love to to see a similar explanation for "pilot's glory", the circular but smaller rainbow you see from airplane around its shadow on clouds.
The first explanation was better than anything I have heard before but the second one was even better. Great video!
This is the best video on UA-cam.
I was recently flying into Scotland and the sun was at the perfect angle behind the plane from my point of view. There was a full rainbow encircling the shadow the plane cast on a nearby cloud. Most magical thing I've ever seen.
You made my day, some years ago I had this same question running my mind but I didn't have the math knowledge to solve it, wish I had this video at that time cause I always felt the original explanation was uncompleted, you deserve a like.
I've always wondered why rainbows are brighter on the inside. Such a great intuitive explanation!
Thank you! I've wondered about this for decades. It all makes much for sense now.
Nice explanation. I hope you continue to make this types of videos. I also liked the way you suggested an math exercise for the viewer
This is amzing! Too bad there is only 2 videos on your channel, I can't wait untill the next one comes out
Wonderful video, I actually always wondered why inside of the rainbow is brighter, but never investigated that. Now I know! And from the hindsight, I don't know how I could just accept that there is only one way light reflects in a droplet.
Loved that! Fascinating and easily broken down to make sense with the animations, nicely done!
I have, once, seen 3 independent rainbows in the sky at the same time...
I was 12 years old and it was about 3 days after the local area had recorded about 1½ times its average annual rainfall in 36 hours, filling 2 normally dry lakes nearby...
The morning was extremely still, the smoke from the neighbour's farmhouse (4 miles to the north) was going straight up. I was heading to school and was at a point about ½ way between the two lakes, the sun rising over the hills behind me ...
The largest and clearest of the rainbows was directly west of me...
The other two were _about_ 22° to the north & south of that line and both looked "dirty" as if there was some mud mixed in with the rainbow...
It took me a while to realise that the surfaces of the two lakes were probably the light sources for these and *they* were very muddy bodies of water which explained the colour mix.
you've got an amazing channel that deserves far more subscribers and recognition for your hard work. I hope you'll make more videos in the future at a pace that works for you :)
I knew about the brighter inside and also correctly understood that is because of how the rays reflect. But this is a much more compete explanation.
The algorithm has blessed me with selection of videos, and now i return the favour. For the algorithm!
Excellent visualization! Shame that you didn’t include how double rainbows happen. But this was such a great way of showing it without having to introduce big words like dispersion
Life is brighter inside a rainbow, poetic
I liked the ending but i think you cut the video too soon. After drawing the primary rainbow, i think there was a great opportunity to draw the secondary rainbow and show that the gap in between looks dark while the space in between looks bright. This is very noticeable and shows that the inside of a rainbow isn’t empty, its white because all the colors are mixed. Alexander’s band is dark because no visible colors can reach those angles as shown in your graphs. 👍
Great video; thanks for taking the time.
Always wondered why the inside was slightly brighter, always noticed it growing up but asking people never seemed to yield any answer lol, not even teachers, thanks for the explanation
considering my idea of how rainbows work was "something something sunlight something something raindrops", You could definitely say the title was accurate
World class explanation! Thank you!
The animation made this incredibly easy to understand
Good job!
Just became your 300th subscriber; congrats!
As others have already said, this explanation video is excellent and I hope to see and hear more insights you might have.
Total internal reflection left that chat, and the 3d nature of the droplets, too.
Such an awesome video.
Briljante video 👌
Slick animation, great visualization.
Great explanation, my professor tried explaining it in college, I remember it left me more confused. Waiting for the next one from you.
Beautiful explanation! Super interesting!
I'd always wondered why the area inside a rainbow was bright. Great explanation!
Impressive video for such a short and clear content!
I knew the basics of how rainbows formed but so much I didn't know.
Subscribed and looking foward for more great content like this
Another effect is that more light hits near the edge of droplets than at the center. This is because there's more area at a larger radius (the area integral formulation of a circle being r*dr*dθ). Your visual integration of the alpha-vs-d graph seems to be missing this effect. Taking it into account would give more weight to light reflected at a higher d, making the rainbow itself appear even brighter than the interior region. You could show this by plotting alpha vs d^2 then doing the visual integral on that.
Looks like 3b1B’s animations. Very clean. More please!
Amazing first video
"...it has infinitely many. We're using six here, which is a reasonable approximation."
Thank you for you content, we really needed something new
Thank you Steve Mould for the discover
I always noticed the inside of the rainbow was brighter than the outside but I never knew why until now!
Excellent video and explanation!
Fascinating. Thank you for taking the time to make such an interesting vid.
This is a very well illustrated video, it deservers way more views. Actually, I thought this had way more views.
I have noticed it and always wondered thank you.
I've never seen a rainbow where the outside is brighter than the inside. I always was taught prisms refract light, and water droplets are like a crystal prism, in that they separate white light into its base components.
Nice explanation... but that was what how I "thought" they worked
Best animation ever!
Such a great video!
Superb work man. The animations are verrryy precise. If you don't mind, could you tell me which software/s did you use for that?
I used manim, which is very good for animating geometry and graphs, but it actually wasn't great for handling the colours. To get the mixing of colours approximately right I had to use a large number of dots/arcs with very low opacity.
@@MatsVermeeren ohhh, great.. you've taken manim to another level. ❤️
Intriguing. I'll definitely have to go over the graph to understand it better, but the graphics explain the concepts pretty clearly.
It's also interesting to see photographic images of rainbows made with cameras capable of reception in ultraviolet & infrared wavelengths. I suspect that the rules for some wavelengths of light would be somewhat different as they may move transmit/reflect/refract though water significantly differently - even passing through the droplet entirely.
Thanks for this nice explanation. Years ago I was interested in computer rendering skies and one the things I came across was conditions for rainbows of course, and I looked at simulations and saw, that makes no sense, it's much brighter on the inside than on the outside.. Until the months to come I saw, this was indeed the case. Never understood why tough (I wasn't even so far to to worry about actually rendering rainbow conditions). I failed already on Rayleigh scattering realizing this was a much, much harder task than I imagined, as refracted light can again be reflected etc.
I always noticed that the sky was brighter inside a rainbow than out, and now I know why.
Actually the a parallel beam hitting a circular section will have the sin(theta) uniformly distributed (not the theta). So using the "d" was accurate and for other angles there will be overlap from all wavelengths.
Great explanation. Thanks.
how is this video so good... hats off to you
Best explanation
Explaining all doubts
Like why rain bow is curved
Why Red is at the top
I guess Mats Verneeren found the rainbow connection, I hope Kermit isn’t disappointed
This playlisy is so cool
Would be good to have a follow up and discuss other kinds of rainbows; double rainbow (with a dark band), upside down rainbow, full circle rainbow, reflection rainbow, monochrome rainbow, fog/cloud bow, rainbow, twin rainbow (non-concentric), supernumerary rainbow (multiple concentric rainbows, fading out towards the center),
Amazing… good explanation, sir