Home » Optics

Category Archives: Optics

Soundly Proving the Curvature of the Earth at Lake Pontchartrain

Excerpted from an article by Mick West

A classic experiment to demonstrate the curvature of a body of water is to place markers (like flags) a fixed distance above the water in a straight line, and then view them along that line in a telescope. If the water surface is flat then the markers will appear also in a straight line. If the surface of the water is curved (as it is here on Earth) then the markers in the middle will appear higher than the markers at the ends.

Here’s a highly exaggerated diagram of the effect by Alfred Russel Wallace in 1870, superimposed over an actual photograph.

Lake Pontchartrain power lines demonstrating the curvature Metabunk

This is a difficult experiment to do as you need a few miles for the curvature to be apparent. You also need the markers to be quite high above the surface of the water, as temperature differences between the water and the air tend to create significant refraction effects close to the water.

However Youtuber Soundly has found a spot where there’s a very long line of markers permanently fixed at constant heights above the water line, clearly demonstrating the curve. It’s a line of power transmission towers at Lake Pontchartrain, near New Orleans, Louisiana.

The line of power lines is straight, and they are all the same size, and the same height above the water. They are also very tall, and form a straight line nearly 16 miles long. Far better than any experiment one could set up on a canal or a lake. You just need to get into a position where you can see along the line of towers, and then use a powerful zoom lense to look along the line to make any curve apparent

One can see quite clearly in the video and photos that there’s a curve. Soundly has gone to great lengths to provide multiple videos and photos of the curve from multiple perspectives. They all show the same thing: a curve.

Lake Pontchartrain curve around Earth

One objection you might make is that the towers could be curving to the right. However the same curve is apparent from both sides, so it can only be curving over the horizon.



People have asked why the curve is so apparent in one direction, but not in the other. The answer is compressed perspective. Here’s a physical example:


Compressed perspective on a car

That’s my car, the roof of which is slightly curved both front to back and left to right. I’ve put some equal sized chess pawns on it in two straight lines. If we step back a bit and zoom in we get:

Compressed perspective on a car II

Notice a very distinct curve from the white pieces, but the “horizon” seems to barely curve at all.

Similarly in the front-back direction, where there’s an even greater curve:

Compressed perspective on a car III

There’s a lot more discussion with photos here Soundly Proving the Curvature of the Earth at Lake Pontchartrain




Lord Of The Rings Optics challenge

A great physics problem for senior year students:

In J. R. R. Tolkien’s The Lord of the Rings (volume 2, p. 32), Legolas the Elf claims to be able to accurately count horsemen and discern their hair color (yellow) 5 leagues away on a bright, sunny day.

“Riders!” cried Aragorn, springing to his feet. “Many riders on swift steeds are coming towards us!”
“Yes,” said Legolas,”there are one hundred and five. Yellow is their hair, and bright are their spears. Their leader is very tall.”
Aragorn smiled. “Keen are the eyes of the Elves,” he said.
“Nay! The riders are little more than five leagues distant,” said Legolas.”

Make appropriate estimates and argue that Legolas must have very strange-looking eyes, have some means of non-visual perception, or have made a lucky guess. (1 league ~ 3.0 mi.)

On land, the league is most commonly defined as three miles, though the length of a mile could vary from place to place and depending on the era.
At sea, a league is three nautical miles (3.452 miles; 5.556 kilometres).

Several solutions are possible, depending on the estimating assumptions

Eye focusing rays of light figure_10_24_labeled

When parallel light waves strike a concave lens the waves striking the lens surface at a right angle goes straight through but light waves striking the surface at other angles diverge. In contrast, light waves striking a convex lens converge at a single point called a focal point. The distance from the long axis of the lens to the focal point is the focal length. Both the cornea and the lens of the eye have convex surfaces and help to focus light rays onto the retina. The cornea provides for most of the refraction but the curvature of the lens can be adjusted to adjust for near and far vision.


By Chad Orzel is an Associate Professor in the Department of Physics and Astronomy at Union College in Schenectady, NY

The limiting factor here is the wave nature of light– light passing through any aperture will interfere with itself, and produce a pattern of bright and dark spots.
So even an infinitesimally small point source of light will appear slightly spread out, and two closely spaced point sources will begin to run into one another.
The usual standard for determining whether two nearby sources can be distinguished from one another is the Rayleigh criterion:

Rayleigh Criterion circular aperature

sine of the angular separation between two objects = 1.22 x ratio of the light wavelength to the diameter of the (circular) aperture, through which the light passes.
To get better resolution, you need either a smaller wavelength or a larger aperture.

Legolas says that the riders are “little more than five leagues distant.”
A league is something like three miles, which would be around 5000 meters, so let’s call it 25,000 meters from Legolas to the Riders.
Visible light has an average wavelength of around 500 nm, which is a little more green than the blond hair of the Riders, but close enough for our purposes.

The sine of a small angle can be approximated by the angle itself.

The angle = the size of the separation between objects divided by the distance from the objects to the viewer.

Putting it all together, Legolas’s pupils would need to be 0.015 m in diameter.
That’s a centimeter and a half, which is reasonable, provided he’s an anime character. I don’t think Tolkien’s Elves are described as having eyes the size of teacups, though.

We made some simplifying assumptions to get that answer, but relaxing them only makes things worse. Putting the Riders farther away, and using yellower light would require Legolas’s eyes to be even bigger. And the details he claims to see are almost certainly on scales smaller than one meter, which would bump things up even more.

Any mathematical objections to these assumptions? Sean Barrett writes:

“The sine of a small angle can be approximated by the angle itself, which in turn is given, for this case, by the size of the separation between objects divided by the distance from the objects to the viewer.”

Technically this is not quite right; the separation divided by the distance is not the angle itself, but rather the tangent of the angle. (SOHCAHTOA: sin = opposite/hypoteneuse; tangent = opposite/adjacent.)

Because the cos of a very small angle is very nearly 1, however, the tangent is just as nearly equal the angle as is the sine. But that doesn’t mean you can just skip that step. And there’s really not much need to even mention the angle; with such a very tiny angle, clearly the hypoteneuse and the adjacent side have essentially the same length (the distance to either separated point is also essentially 25K meters), and so you can correctly say that the sine itself is in this case approximated by the separation divided by the distance, and never mention the angle at all.

(You could break out a calculator to be on the safe side, but if you’re going to do that you need to know the actual formulation to compute the angle, not compute it as opposite/adjacent! But, yes, both angle (in radians) and the sine are also 1/25000 to about 10 sig figs.)

II. Another solution

Using the Rayleigh Criterion. In order for two things, x distance apart, to be discernible as separate, at an angular distance θ, to an instrument with a circular aperture with diameter a:

θ > arcsin(1.22 λ/a)

5 leagues is approximately 24000 m.
Sssume that each horse is ~2 m apart from each other
So arctan (1/12000) ≅ θ.
We can use the small-angle approximation (sin(θ) ≅ tan(θ) ≅ θ when θ is small)
So we get 1/12000 ≅ 1.22 λ/a

Yellow light has wavelengths between 570 and 590 nm, so we’ll use 580.

a ≅ 1.22 * (580E-9 m)* 12000 ≅ .0085 m.

8 mm is about as far as a human pupil will dilate, so for Legolas to have pupils this big in broad daylight must be pretty odd-looking.
Edit: The book is Six Ideas that Shaped Physics: Unit Q, by Thomas Moore

III. Great discussion on the Physics StackExchange

Could Legolas actually see that far? Physics StackExchange discussion

Here, Kyle Oman writes:

For a human-like eye, which has a maximum pupil diameter of about mm and choosing the shortest wavelength in the visible spectrum of about 390 nm, the angular resolution works out to about 5.3×105  (radians, of course).

At a distance of 24 km, this corresponds to a linear resolution (θd, where is the distance) of about 1.2m1. So counting mounted riders seems plausible since they are probably separated by one to a few times this resolution.

Comparing their heights which are on the order of the resolution would be more difficult, but might still be possible with dithering.

Does Legolas perhaps wiggle his head around a lot while he’s counting? Dithering only helps when the image sampling (in this case, by elven photoreceptors) is worse than the resolution of the optics. Human eyes apparently have an equivalent pixel spacing of something like a few tenths of an arcminute, while the diffraction limited resolution is about a tenth of an arcminute, so dithering or some other technique would be necessary to take full advantage of the optics.

An interferometer has an angular resolution equal to a telescope with a diameter equal to the separation between the two most widely separated detectors. Legolas has two detectors (eyeballs) separated by about 10 times the diameter of his pupils75 mm or so at most. This would give him a linear resolution of about 15cm at a distance of 24 km, probably sufficient to compare the heights of mounted riders.

However, interferometry is a bit more complicated than that. With only two detectors and a single fixed separation, only features with angular separations equal to the resolution are resolved, and direction is important as well.

If Legolas’ eyes are oriented horizontally, he won’t be able to resolve structure in the vertical direction using interferometric techniques. So he’d at the very least need to tilt his head sideways, and probably also jiggle it around a lot (including some rotation) again to get decent sampling of different baseline orientations. Still, it seems like with a sufficiently sophisticated processor (elf brain?) he could achieve the reported observation.

Luboš Motl points out some other possible difficulties with interferometry in his answer, primarily that the combination of a polychromatic source and a detector spacing many times larger than the observed wavelength lead to no correlation in the phase of the light entering the two detectors. While true, Legolas may be able to get around this if his eyes (specifically the photoreceptors) are sufficiently sophisticated so as to act as a simultaneous high-resolution imaging spectrometer or integral field spectrograph and interferometer. This way he could pick out signals of a given wavelength and use them in his interferometric processing.

A couple of the other answers and comments mention the potential difficulty drawing a sight line to a point 24 km away due to the curvature of the Earth. As has been pointed out, Legolas just needs to have an advantage in elevation of about 90 meters (the radial distance from a circle 6400 km in radius to a tangent 24 km along the circumference; Middle-Earth is apparently about Earth-sized, or may be Earth in the past, though I can’t really nail this down with a canonical source after a quick search). He doesn’t need to be on a mountaintop or anything, so it seems reasonable to just assume that the geography allows a line of sight.

Finally a bit about “clean air”. In astronomy (if you haven’t guessed my field yet, now you know…) we refer to distortions caused by the atmosphere as “seeing”.

Seeing is often measured in arcseconds (3600 arcse60 arcmi13600 arcsec = 60arcmin = 1∘), referring to the limit imposed on angular resolution by atmospheric distortions.

The best seeing, achieved from mountaintops in perfect conditions, is about arcsec,
or in radians 4.8×106 . This is about the same angular resolution as Legolas’ amazing interferometric eyes.

I’m not sure what seeing would be like horizontally across a distance of 24 km. On the one hand there is a lot more air than looking up vertically; the atmosphere is thicker than 24 km but its density drops rapidly with altitude. On the other hand the relatively uniform density and temperature at fixed altitude would cause less variation in refractive index than in the vertical direction, which might improve seeing.

If I had to guess, I’d say that for very still air at uniform temperature he might get seeing as good as 1 arcsec, but with more realistic conditions with the Sun shining, mirage-like effects probably take over limiting the resolution that Legolas can achieve.


IV. Also on StackExchange, the famous Luboš Motl writes:

Let’s first substitute the numbers to see what is the required diameter of the pupil according to the simple formula:

I’ve substituted the minimal (violet…) wavelength because that color allowed me a better resolution i.e. smaller θθ. The height of the knights is two meters.
Unless I made a mistake, the diameter DD is required to be 0.58 centimeters. That’s completely sensible because the maximally opened human pupil is 4-9 millimeter in diameter.
Just like the video says, the diffraction formula therefore marginally allows to observe not only the presence of the knights – to count them – but marginally their first “internal detailed” properties, perhaps that the pants are darker than the shirt. However, to see whether the leader is 160 cm or 180 cm is clearly impossible because it would require the resolution to be better by another order of magnitude. Just like the video says, it isn’t possible with the visible light and human eyes. One would either need a 10 times greater eye and pupil; or some ultraviolet light with 10 times higher frequency.
It doesn’t help one to make the pupils narrower because the resolution allowed by the diffraction formula would get worse. The significantly more blurrier images are no helpful as additions to the sharpest image. We know that in the real world of humans, too. If someone’s vision is much sharper than the vision of someone else, the second person is pretty much useless in refining the information about some hard-to-see objects.

The atmospheric effects are likely to worsen the resolution relatively to the simple expectation above. Even if we have the cleanest air – it’s not just about the clean air; we need the uniform air with a constant temperature, and so on, and it is never so uniform and static – it still distorts the propagation of light and implies some additional deterioration. All these considerations are of course completely academic for me who could reasonably ponder whether I see people sharply enough from 24 meters to count them. 😉

Even if the atmosphere worsens the resolution by a factor of 5 or so, the knights may still induce the minimal “blurry dots” at the retina, and as long as the distance between knights is greater than the distance from the (worsened) resolution, like 10 meters, one will be able to count them.

In general, the photoreceptor cells are indeed dense enough so that they don’t really worsen the estimated resolution. They’re dense enough so that the eye fully exploits the limits imposed by the diffraction formula, I think. Evolution has probably worked up to the limit because it’s not so hard for Nature to make the retinas dense and Nature would be wasting an opportunity not to give the mammals the sharpest vision they can get.

Concerning the tricks to improve the resolution or to circumvent the diffraction limit, there aren’t almost any. The long-term observations don’t help unless one could observe the location of the dots with the precision better than the distance of the photoreceptor cells. Mammals’ organs just can’t be this static. Image processing using many unavoidably blurry images at fluctuating locations just cannot produce a sharp image.

The trick from the Very Large Array doesn’t work, either. It’s because the Very Large Array only helps for radio (i.e. long) waves so that the individual elements in the array measure the phase of the wave and the information about the relative phase is used to sharpen the information about the source. The phase of the visible light – unless it’s coming from lasers, and even in that case, it is questionable – is completely uncorrelated in the two eyes because the light is not monochromatic and the distance between the two eyes is vastly greater than the average wavelength. So the two eyes only have the virtue of doubling the overall intensity; and to give us the 3D stereo vision. The latter is clearly irrelevant at the distance of 24 kilometers, too. The angle at which the two eyes are looking to see the 24 km distant object are measurably different from the parallel directions. But once the muscles adapt into this slightly non-parallel angles, what the two eyes see from the 24 km distance is indistinguishable.


V. Analyzed in “How Far Can Legolas See?” by minutephysics (Henry Reich)



Ray Tracing

This lesson is from Rick Matthews, Professor of Physics, Wake Forest University.

Lesson 1, convex lens
The object is far from the lens.

Convex Lens ray tracing GIF


Lesson 2, convex lens
The object is near the lens.

Convex Lens ray tracing GIF Object near lens

The rules for concave lenses, are similar:

A horizontal ray is refracted outward, as if emanating from the near focal point.

A ray that strikes the middle of the lens continues in a straight line.

A ray coming from the object, far from the far focal point, will leave the lens horizontal.

Lesson 3, concave lens.
Note that object placement has little effect on the nature of the image.
The rays diverge.

Concave Lens ray tracing GIF


In every case:

if the rays leaving the lens actually intersect then the image is real.

If the rays leaving the lens diverge then someone looking back through the lens
would see a virtual image:
Your mind would extrapolate where you think the image should be,
even though one isn’t really there, as shown below with the dotted lines.


image from Giancoli Physics, 6th edition



Light pollution


Source: http://www.pbs.org/seeinginthedark/astronomy-topics/light-pollution.html

This is what we see on a night without clouds, if there was no light pollution:

Some camera filters can filter out some of the glare


Enter a caption

from http://photography-on-the.net/forum/showthread.php?t=1063821

Here are the various levels of polluted vs dark skies:


This video from Sunchaser Pictures shows what LA night skies could look like without light pollution.

“An experimental timelapse created for SKYGLOWPROJECT.COM, a crowdfunded quest to explore the effects and dangers of urban light pollution in contrast with some of the most incredible Dark Sky Preserves in North America. Visit the site for more!
Inspired by the “Darkened Cities” stills project by Thierry Cohen, this short film imagines the galaxy over the glowing metropolis of Los Angeles through composited timelapse and star trail astrophotography. Shot by Gavin Heffernan (SunchaserPictures.com) and Harun Mehmedinovic (Bloodhoney.com). SKYGLOW is endorsed by the International Dark Sky Association”

also at


This lesson is from http://darksky.org/light-pollution/

Less than 100 years ago, everyone could look up and see a spectacular starry night sky. Now, millions of children across the globe will never experience the Milky Way where they live. The increased and widespread use of artificial light at night is not only impairing our view of the universe, it is adversely affecting our environment, our safety, our energy consumption and our health.

What is Light Pollution?

Most of us are familiar with air, water, and land pollution, but did you know that light can also be a pollutant?

The inappropriate or excessive use of artificial light – known as light pollution – can have serious environmental consequences for humans, wildlife, and our climate. Components of light pollution include:

  • Glare – excessive brightness that causes visual discomfort
  • Skyglow – brightening of the night sky over inhabited areas
  • Light trespass – light falling where it is not intended or needed
  • Clutter – bright, confusing and excessive groupings of light sources

Light pollution is a side effect of industrial civilization. Its sources include building exterior and interior lighting, advertising, commercial properties, offices, factories, streetlights, and illuminated sporting venues.

The fact is that much outdoor lighting used at night is inefficient, overly bright, poorly targeted, improperly shielded, and, in many cases, completely unnecessary. This light, and the electricity used to create it, is being wasted by spilling it into the sky, rather than focusing it on to the actual objects and areas that people want illuminated.

Glossary of Lighting Terms

How Bad is Light Pollution?

With much of the Earth’s population living under light-polluted skies, over lighting is an international concern. If you live in an urban or suburban area all you have to do to see this type of pollution is go outside at night and look up at the sky.

According to the 2016 groundbreaking “World Atlas of Artificial Night Sky Brightness,” 80 percent of the world’s population lives under skyglow.

In the United States and Europe 99 percent of the public can’t experience a natural night!


If you want to find out how bad light pollution is where you live, use this interactive map created from the”World Atlas” data or the NASA Blue Marble Navigator for a bird’s eye view of the lights in your town. Google Earth users can download an overlay also created from the “World Atlas” data. And don’t forget to check out the Globe at Night interactive light pollution map data created with eight years of data collected by citizen scientists.

Effects of Light Pollution

For three billion years, life on Earth existed in a rhythm of light and dark that was created solely by the illumination of the Sun, Moon and stars. Now, artificial lights overpower the darkness and our cities glow at night, disrupting the natural day-night pattern and shifting the delicate balance of our environment. The negative effects of the loss of this inspirational natural resource might seem intangible. But a growing body of evidence links the brightening night sky directly to measurable negative impacts including

Light pollution affects every citizen. Fortunately, concern about light pollution is rising dramatically. A growing number of scientists, homeowners, environmental groups and civic leaders are taking action to restore the natural night. Each of us can implement practical solutions to combat light pollution locally, nationally and internationally.

You Can Help!

The good news is that light pollution, unlike many other forms of pollution, is reversible and each one of us can make a difference! Just being aware that light pollution is a problem is not enough; the need is for action. You can start by minimizing the light from your own home at night. You can do this by following these simple steps.

  • Learn more. Check out our Light Pollution blog posts
  • Only use lighting when and where it’s needed
  • If safety is concern, install motion detector lights and timers
  • Properly shield all outdoor lights
  • Keep your blinds drawn to keep light inside
  • Become a citizen scientist and helping to measure light pollution

Learn more about Outdoor Lighting Basics

Then spread the word to your family and friends and tell them to pass it on. Many people either don’t know or don’t understand a lot about light pollution and the negative impacts of artificial light at night. By being an ambassador and explaining the issues to others you will help bring awareness to this growing problem and inspire more people to take the necessary steps to protect our natural night sky. IDA has many valuable resources to help you including Public Outreach Materials, How to Talk to Your Neighbor, Lighting Ordinances and Residential and Business Lighting.

Want to do more? Get Involved Now


Blue sky

This is the outline for a future lesson on Rayleigh Scattering: Why the sky is blue

– Rayleigh scattering occurs when light is scattered off many very small particles.
– Mie scattering occurs when light is scattered off of many larger particles.


Addressing misconceptions

Question: Particles in the air cause shorter wavelengths (blue-ish0 to scatter more than the longer wavelengths (reddish.) This causes us to see the sky as being blue. So why does the sunrise (or sunset) and sun look red/orange?

Answer: “When you look at the sky and see blue you’re seeing blue light being scattered towards your eye.”

“When you look at the sun and it looks red or orange that’s because the blue light is being scattered away from your eye – leaving the remaining light to enter your eye.”

“The blue light is being scattered in all directions by Raleigh scattering. The colors you see depend on what direction you’re looking.”

Reference Physicsforums.com How-does-rayleigh-scattering-work


External resources





Brownian motion app  galileoandeinstein Brownian motion app

Lesson EarthRef.org Digital Archive ematm.lesson3.scattering.pptx

EM in the Atmosphere: Reflection, Absorption, and Scattering Lesson Plan

Powerpoint for the lesson plan

Learning standards

SAT subject test in Physics: Waves and optics

• General wave properties, such as wave speed, frequency, wavelength, superposition, standing wave diffraction, and Doppler effect
• Reflection and refraction, such as Snell’s law and changes in wavelength and speed
• Ray optics, such as image formation using pinholes, mirrors, and lenses
• Physical optics, such as single-slit diffraction, double-slit interference, polarization, and color

AP Learning Objectives

IV.A.2.b: Students should understand the inverse-square law, so they can calculate the intensity of waves at a given distance from a source of specified power and compare the intensities at different distances from the source.

IV.B.2.b: Know the names associated with electromagnetic radiation and be able to arrange in order of increasing wavelength the following: visible light of various colors, ultraviolet light, infrared light, radio waves, x-rays, and gamma rays.

L.2: Observe and measure real phenomena: Students should be able to make relevant observations, and be able to take measurements with a variety of instruments (cannot be assessed via paper-and-pencil examinations).

L.3: Analyze data: Students should understand how to analyze data, so they can:
– a) Display data in graphical or tabular form.
– b) Fit lines and curves to data points in graphs.

L.5: Communicate results: Students should understand how to summarize and communicate results, so they can:
– a) Draw inferences and conclusions from experimental data.
– b) Suggest ways to improve experiment.
– c) Propose questions for further study


http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/rbowpri.htmlThe Sumerian hero Gilgamesh traveled the world in search of a way to cheat death. On one of his journeys, he came across an old man, Utnapishtim, who told Gilgamesh a story from centuries past. The gods brought a flood that swallowed the earth.

Gilgamesh Rainbow Magic Dawkins

The Magic of Reality: How We Know What’s Really True [Richard Dawkins]

The gods were angry at mankind so they sent a flood to destroy mankind. The god Ea, warned Utnapishtim and instructed him to build an enormous boat to save himself, his family, and “the seed of all living things.” He does so, and the gods brought rain which caused the water to rise for many days.

When the rains subsided, the boat landed on a mountain, and Utnapishtim set loose first a dove, then a swallow, and finally a raven, which found land. The god Ishtar, created the rainbow and placed it in the sky, as a reminder to the gods and a pledge to mankind that there would be no more floods.  A similar story, with theological modifications, is in the Hebrew Bible as the story of Noah and the Ark.

This story comes from the Epic of Gilgamesh

In physics we learn that rainbows are produced by electromagnetic radiation – visible light – reflecting in marvelous ways from the dispersion of light.

Let’s start with the basics:

A prism separates white light into many colors

How? Each wavelength of light refracts by a different amount

The result is dispersion – each wavelength is bent by a different amount



The physics of rainbow formation

Rainbows: At Atmospheric optics



Rebecca McDowell  How rainbows form


Do rainbows have reflections?

It certainly seems like rainbows can have reflections. Consider this great photo by Terje O. Nordvik, September ’04 near Sandessjøen, Norway.

But rainbows aren’t real objects – and so they literally can’t have reflections!
So what are we seeing here? See Rainbow reflections: Rainbows are not Vampires


Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

HS-PS4-3. Evaluate the claims, evidence, and reasoning behind the idea that electromagnetic radiation can be described by either a wave model or a particle model, and that for some situations involving resonance, interference, diffraction, refraction, or the photoelectric effect, one model is more useful than the other.

SAT subject test in Physics: Waves and optics

• General wave properties, such as wave speed, frequency, wavelength, superposition, standing wave diffraction, and Doppler effect
• Reflection and refraction, such as Snell’s law and changes in wavelength and speed
• Ray optics, such as image formation using pinholes, mirrors, and lenses
• Physical optics, such as single-slit diffraction, double-slit interference, polarization, and color.


Waves in 2 dimensions

Waves in 2-dimensions

Glencoe Physics, Chapter 14.

Throw a stone into water: observe the circular crests and troughs

Waves propagate in 2 dimensions:

along the X-axis, and Y-axis, simultaneously.

a wavefront represents the crest of a wave in 2 dimensions

2-dimensional waves always travel perpendicular to their wavefronts

wave’s direction is represented by a ray

Here, water waves, or light waves, hit a curved surface

The wavefronts reflect to a point, called the focus.

Refraction of 2-D waves

Refraction is the change in direction of wave propagation due to a change in its transmission medium.

Often seen with light.

Seen with water waves, when they move from deep water into shallow water.

Here light waves refrac as they move from one medium into another (from air into diamond)

In this simplified case, the light waves (or water waves) are all parallel to each other.

Here we see the same thing, but now the rays of light are more realistic.

They emanate from a source, so they are circular, not parallel.

Yet when the rays hit the water, they are approximately parallel, so the result is the same.


Snell’s law at PhysicsClassroom.com

Water waves can be refracted

Animation: Wiley Refraction

Animation GCSE Light and water refraction

Here we see water waves changing direction, as they enter shallower waters.


From a presentation by Luo Yanjie.

From a presentation by Luo Yanjie.


Details on the cause of refraction (PhysicsClassroom.Com)


Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

HS-PS4-1. Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling within various media. Recognize that electromagnetic waves can travel through empty space (without a medium) as compared to mechanical waves that require a medium

SAT subject test in Physics: Waves and optics

• General wave properties, such as wave speed, frequency, wavelength, superposition, standing wave diffraction, and Doppler effect
• Reflection and refraction, such as Snell’s law and changes in wavelength and speed
• Ray optics, such as image formation using pinholes, mirrors, and lenses
• Physical optics, such as single-slit diffraction, double-slit interference, polarization, and color