PFE023: Waves

After a semester long hiatus, PFE is BACK.

Waves may be a purely mathematical construct and as such confusing, worrisome, and/or boring to most. Yet that doesn't mean that they don't show up everywhere.

Sound waves, light waves, ocean waves, radio waves, and "the wave" are just a few examples that we experience on a regular basis. Some more subtle examples are the vibrating waves on a string or a drum head (see oil slicks and music for more background).

Waves can be classified in a number of ways, but for now we'll just stick to two main categories: standing waves and traveling waves.

For a standing wave, think of a piano string vibrating up and down in any of the following fashions:

Note that the endpoints are fixed as well as certain points in the middle. Standing waves oscillate at a certain frequency. If the wave is on a string in air, it will produce a certain pitch of sound.

The alternative is a traveling wave which moves and does not have fixed points or nodes of the wave. An example of such is shown here

Whoa! PFE goes animated!

An example of such is when you whip the vacuum cleaner cord to get it unstuck from something. You can briefly see a short traveling wave in the cord.

Ocean waves are a form of traveling wave. Keep in mind though, that even as the wave moves across the ocean, the water itself is not moving horizontally, instead it is just moving up and down. In this sense it should start to become clear that when a wave is moving, it is typically not carrying actual stuff, but rather is carrying energy.

In the same way, as sound saves travel through air, the air particles themselves are not traveling any great distance, instead, they merely travel far enough to let the other air particles near them know how the wave is moving. So again, a sound wave is really a transfer of energy.

Finally, we get to light, which is the most confusing wave of all. Light is certainly the transfer of energy [as anyone who has ever tried to cook anything with a 60 Watt light bulb [think easy-bake ovens] knows] that propagates forward not unlike a sound wave.

That's waves.

PFE022: Rainbows

Rainbows are majestic, ethereal visions of color. While possibly (but not likely) not the most beautiful thing in nature, their intangibility has made them an object of interest throughout time. According to the Bible,

I do set my bow in the cloud, and it shall be for a token of a covenant between me and the earth.

Physicists tend to have a more down to earth discussion of the source of rainbows.

But first, I must explain the basis for many popular physicists jokes (yes we do, apparently, have a sense of humor). It is common in physics courses to work problems on a simple shape, say, a sphere, because the mathematics works out more easily. More complicated shapes usually follow in the same direction but with harder math (in practical applications this means computers), but the line "assume a sphere" is very common among physicists.

Anyways, rainbows form all the colors of rainbow by light passing through them. Yet this is different from both the mirage phenomena and the oil slick phenomena (I seem to like self-references, it holds things together?). The physics though is related.

First, consider white light. What we think of as white light is actually a collection of all (or nearly all) colors that we can see. We know this because we can pass it through a prism and it splits white light into all colors.

If you don't recognize this, shame on you.

From playing with prisms, we can see that light travels differently through glass depending on its color (remember oil slicks?).

When light travels through glass (or water) the light bends at the surface from air to glass (water) or vice versa. But how much it bends depends on the color (wavelength) of the light.

The next step is where we get to use our "sphere" approximation. Rainbows require airborne water droplets. These can come from sprinklers, ocean spray, or rain falling. In any case, the droplets may be any number of shapes that don't particularly resemble spheres. That said, for the sake of this exercise, suppose all the water droplets are spherical. Then, as white light enters the drop from the sun,

it bounces around inside the light and comes down towards our eyes. But as the light bounces around through the droplet, the colors eventually split into the rainbow spectrum.

This shot is a beautiful example of a double rainbow which is what happens when some of the light makes two internal bounces instead of just one as usual.

That's rainbows.

PFE021: Oil Slicks

Oil spills under your car,

aren't they lovely?

It's a good thing they're so pretty, how else would we know our cars (holes at the bottom of the ocean) are leaking oil?

But what a unique thing, that oil shimmers, and turns all sorts of colors. Perhaps it is fledgling rainbow? I think not.

Perhaps more interesting is to consider why we never see this in other liquids laying about. The ice outside is melting, but no shimmer. Or milk. I've spilled that before (only once, I swear) and just saw white. Maybe they should show all kinds colors too.

Well, milk is easy to understand since I can't see through it anyways (I drink the hearty stuff, I can't speak for skim milk), but shouldn't we see this in water too?

But of course, we do. Regular rainbows are formed from water droplets and are way prettier anyways. That said, the phenomena leading to rainbows is different from that of oil slicks and I will get to that in a future post.

As we know [hopefully], oil and water do not mix and oil sits on top of water (is less dense than water). This has to do with their chemical properties and isn't of interest at the moment. You can get some oil and water and put them in a glass if you like.

The key property of oil (it turns out you need motor oil for this, cooking oil is a little bit different and won't work) that makes it shimmer is that it forms really thin layers. Unlike water, which likes to bead up, oil is content to run free. Thus, if oil is spilled on a smooth surface (such as a puddle of water) it will create a very thin layer across the surface of the water. It is at this point that the physics kicks in.

The typical diagram for thin films looks like this:

Too many numbers and variables and confusing things.

The main thing here is that some of the light that hits the oil is immediately reflected, and some passes through the oil and is then reflected. [That is, some of the light travels along A->B->C while the rest only travels A->D before lining up again.] So when we look at an oil slick, we see light that has taken two very distinct paths as one smooth image.

Now, we've already learned about mirages and that light doesn't always behave as it should, but that effect is mainly a trick in our heads (combined with the fact that light can apparently bend around things if it so desires). It turns out that light is even wackier than just that. It actually behaves like a wave. Before we get into what that means, suffice it to say, in extremely simplistic terms, it goes up and down in some regular fashion.

Now, we know what the speed of light is exactly

299,792,458. m/s    (duh)

so that is fixed, but that doesn't tell us how fast the wave is going to oscillate. These oscillations are described by wavelength or frequency (if you know one, you can get the other). Moreover, the wavelength (frequency) of light can be just about anything. In fact, the wavelength of light, as you may have guessed, corresponds directly to the color. Aha! We're getting back to our oil slick!

If the wavelength is just right, when the light splits at the top surface of the oil (point A) and then recombines at C and D, the high points of the one will line up with the high points of the other and the low points will line up with the low points. Then this wavelength (color) can be seen nearly as strongly as the original light.

On the other hand, if the wavelength is just wrong, when the two paths of light line up, the high points will line up with the low points and the waves will cancel each other out. Thus this wavelength/color disappears entirely.

Of course, most wavelengths fit somewhere in between, but for slicks or bubbles of just the right thickness, these "right" and "wrong" wavelengths line up perfectly with the wavelengths of light that we can see. Then some colors shine clearly while others disappear entirely and which colors are emphasized change with both the viewing angle to the surface and minute changes in the thickness.

That's oil slicks.