Nnnyyyooowwwn!

Did you get it? YES! The Doppler Effect, as described by Sheldon Cooper, is "the apparent change in the frequency of a wave caused by relative motion between the source of the wave and the observer."

To help us better understand the Doppler Effect, we are going to continuously use a firetruck siren as a wave source. We're also going to use a lot of animations! (I borrowed all of them from http://www.acs.psu.edu/drussell/Demos/doppler/doppler.html)

A stationary source emits waves that are completely symmetrical. This means that if a firetruck were stopped, every observer would hear the same frequency of its siren.

                                                  





When the firetruck is moving to the right, the center of each wave is displaced as the source catches up to them. Even though the waves from the siren are still being emitted with the same frequency, they are closer together on the right side than they are on the left. So if you were to go stand in front of the firetruck, you would hear a higher frequency than I would standing behind the firetruck.

Let's pretend that this firetruck has super powers and can travel at the speed of sound. You wouldn't hear the siren until the firetruck actually got to you because the waves are no longer ahead of the truck! Also, right as the truck is passing you, you wouldn't really hear the siren. Instead, you would hear this huge "thump" because of all the wavefronts adding together. When it finally passed you, the siren would appear to have a very low frequency because the waves behind it are so far apart. Some food for thought: this actually happens with bullets!

What if the firetruck is traveling faster than the speed of sound? Holy cow. We're going to have to do more pretending. The firetruck with super powers has been working out a bit. Now it can travel faster than the speed of sound! In other words, the firetruck would pass you before you even heard the siren! Also, immediately after it passed, you still wouldn't hear the siren. You would hear a BOOM! A sonic boom to be exact. Obviously you're never going to see a firetruck moving faster than the speed of sound; however, some aircrafts can actually move this fast!



Sadly, our journey through the World of Waves is over. I hope you leave this world feeling confident in the knowledge you now possess. You are a master of the waves around you! You did a good job, Young Physicist. Your eyes are open and your mind is, too. I hope to hear about your many physics adventures to come. Don't forget "the important thing is to not stop questioning. Curiosity has its own reason for existing." - Albert Einstein  

Expanding Interference

Interference- there's a word you already know! Let's expand on it a bit, shall we?

Interference is the variation of wave amplitude that occurs when waves of the same or different frequency come together.

Regarding waves, destructive interference is very similar to the interference that we think about. Destructive interference means wave pulses are on two opposite sides (one up, one down) while they pass through the same place at the same time. If they have the same amplitude what do you think will happen? Oh, you're very close! They will completely counteract each other. You subtract them. For example, let's say Pulse 1 has an amplitude of 5mm and Pulse 2 also has an amplitude of 5mm. The Resultant Pulse would have an amplitude of 0mm: 5mm-5mm=0mm. When the resultant pulse is equal to 0 it is called complete destructive interference. Not to hard, right?

If they pass each other on two opposite sides, but the amplitudes of the pulses are different, the resultant pulse is still found in the same way- by subtraction! If Pulse 1 has an amplitude of 7mm and Pulse 2 has an amplitude of 3mm, the Resultant Pulse would have an amplitude of 4mm: 7mm-3mm=4mm. The Resultant Pulse will be facing the same direction as the pulse with the bigger amplitude, in this case, Pulse 1.

Fantastic question! That's next: constructive interference. Constructive interference is also when pulses travel through the same point at the same time. However, unlike destructive interference, they are on the same side. In this case, you add the pulses together to get the resultant pulse. Let's take two pulses with amplitudes of 7mm and 3mm again. The Resultant Pulse would be 10mm: 7mm+3mm=10mm.

What does all of this mean? Well, basically when two pulses interfere with one another, a new pulse is created and that is the pulse observed! It can be bigger or smaller. However, immidiately after the pulses pass through each other, they snap back to themselves! The resultant pulse is gone and once again the original pulses are back.

Pretty simple, I know! If you remember to subtract for destructive interference and add for constructive interference, you'll be golden!

Surprisingly enough, this happens in real life, too! Think about the roadtrip you took two summers ago. You were listening to your favorite radio station, 104.3. All of a sudden, it starts to get a bit scratchy. You hear some weird Latin song, and then it goes back to your Pop, then to Latin, then to Pop, then to Latin and then... nothing. Where did the music go? In that time frame when no music was coming from your radio, the Pop station and the Latin American station cancelled each other out. Since they're on the same frequency, the waves coming from each station interfered with the other, leaving with you no music to listen to. Aw man.

Describing Diffraction

We're half way through our adventure, Young Physicist. You've come a long way so far! Let's keep going.

I can see you getting ahead of yourself. Diffraction, unfortunately, is not a lot like reffraction. In fact, it's not like it at all! Sorry. Oh, no. Don't get upset. Diffraction is easy, I promise!
What is diffraction? Oh, well, diffraction is the spreading of waves into a region behind a barrier. Also very important to diffraction is Huygens' Principle which states that any point on a wave front can be treated as a point source of waves. You see, a wavefront can be broken up into little "wavelets" which form a new wavefront. This is a lot easier to understand through a diagram.

Because wavefronts are made up of even smaller wavelets, when a wave passes through a small opening, it can change its shape and bend. Notice that in the diagram, the one on the left has a big opening between the barriers. The waves on the other side of the barrier bend only a little bit around the edges. The diagram on the right shows a smaller opening between the two barriers. The waves on the other side of this one are very circular.

http://upload.wikimedia.org/wikipedia/commons/b/bb/Water_ripples_Diffraction.png

Why? Good question! If the opening is very, very small, only one wavelet can squeeze through, right? Right! Therefore, when the wavelet breaks off from the rest of the wavefront, it forms its own wavefront. This wavefront is now circluar and made up of even more wavelets that spread out in a circular fashion. If the opening is large, nine or ten wavelets can fit through. When the wavelets are together, they keep the wavefront straight. The ends of the new wavefront curve slightly because they are the ends of the wavelets.

Although you may not realize it right away, this actually does happen in real life! Who would have thought? Perhaps one summer evening you were outside playing catch with your beloved neighbors. The windows of your house were open to let in the cool, most needed breeze. All of a sudden, you hear your mom talking! It sounds as if she's at the window speaking directly to you! How could this be if she's not actually right there? Diffraction, young one, diffraction. Her voice travels through the kichen and squeezes itself through the open window, only to make it into your ears.

Interesting, huh?

Predicting Refraction

Have you ever tried running in a pool? It's a lot harder than running outside on the grass, so in result, you move a lot slower. Waves do the same thing!

When waves pass from one medium into another, they change speed which causes them to "bend." This is refraction. You may think that you've never seen this before, but I'm sure you have! Take for example the pool skimmer. When you put it into the water, it appears to bend, when in reality it remains perfectly straight!

https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0HEs7a57C10BMad0bubH4tMNEDPnJt_dBhhOF0VvcEy4KnPzFg9q7m6OK62S4vZPu0e1075uZJD0Bg-0Ub85aGHZghdeKu8B32fd_zWHGFdDzEapUx_65hrEYOmvpqF2hyphenhyphenx15HfVYmALo/s1600/IMG_0090.JPG


Why is this, you ask? The air and the water are two separate mediums with two distinct refractive indices (a number that indicates how fast waves can travel through the medium). Depending on the index of refraction, waves will appear to bend either towards or away from the normal line when they enter a new medium. Pretty neat, don't you think?

If a wave travels from a fast medium (air) to a slow medium (water), it would bend toward the normal line, like the picture below. If a wave travels from a slow medium to a fast medium, it would bend away from the normal line.

Good question! Yes, you can find the angle of refraction by using a formula, thanks to Willebrord Snell - hey, don't laugh at his name! Stop that! He realized that the index of refraction of the first medium times the sine of the angle of incidence is equal to the index of refraction of the second medium times the sine of the angle of refraction.

http://www.math.ubc.ca/~cass/courses/m309-01a/chu/Fundamentals/snell01.gif

We can also find the speed of light will travel through a certain medium by using the formula n=c/v where n is the index of refraction, c is the speed of light, and v is the speed. Refraction is very interesting, and quite predictable, you're right!

There is a point at which no light will exit the medium. Here the refracted light is perpendicular to the normal line. This occurs at the critical angle. If the angle of incidence is bigger than the critical angle what do you think happens? Yes! That's correct! The light is reflected back into the medium. This is known as total internal reflection. It acts just like reflection, which we know all about! Therefore, the angle of incidence must be equal to the angle of reflection. However, in order for this to happen, the light must be traveling from a medium with a high refractive index to one with a lower refractive index. Confused yet? Look at this diagram. I think it will help!

http://micro.magnet.fsu.edu/primer/java/refraction/criticalangle/criticalanglejavafigure1.jpg

Can you calculate the critical angle? What do you think? We already said that refraction is quite predictable! So yes, you can determine the critical angle. All we need is Snell's Law.

You see waves being refracted all the time! Even in movies. For example, have you ever watched Brother Bear? It's good, isn't it? A bit sad though... anyway. You know the seen where Koda tries to teach Kenai to catch fish? Well, Kenai isn't very good at it. He's looking into the water at the fish and trying to reach right down to get them. He must not have realized that the light from the fish is refracted; therefore, the fish isn't exactly where it appears to be. Of course, this isn't only true for animated bears. It happens to people, too!

http://www.daviddarling.info/images/spear_fishing_refraction.jpg

Reflecting on Reflection

Ah, on to the next topic now. Go ahead and take a seat, but take this mirror with you. You're going to want it as we study... dun dun dun... Reflection!

Before we begin, I'd like you to look in that mirror I gave you. Looking back is, well, you! Now smile... wave with your left hand... wave with your right hand. Good, just like that. What did you notice? Yes, you're correct! When you smile, nothing really seems to change. But when you wave with your left hand, the image of you waves with its right hand and when you wave with your right hand, it waves with its left! Funny little thing, reflection is. It seems confusing now, but don't worry. Soon you'll understand perfectly!

The first you thing you need to know is that when a wave is reflected there is no change in its speed, frequency, or wavelength. The only difference between the incoming wave and reflected wave is that they are inverted! I'll show you what I mean.


http://www.acs.psu.edu/drussell/Demos/reflect/reflect.html

You see, on the way to the barrier, the wave is going up, but after it hits the barrier and turns around, the wave is going down. The entire time the wave is perpendicular to the barrier.

But what if the wave and the barrier aren't perpendicular to each other? Great question! First, look at this diagram.


http://aplusphysics.com/courses/regents/waves/images/law-of-reflection.gif

This picture shows us the Law of Reflection. This law states that the angle in which the incoming ray, or incident ray, hits the reflecting surface, is equal the the angle in which the reflected ray leaves. The angles are measured from the "normal line" which is perpendicular to the reflecting surface. This is true for any wave that is being reflected off of any surface at any angle. Don't forget it!

A bit more tricky to understand are concave and convex barriers. But don't you fret, by remembering the Law of Reflection, you'll get it in no time!

When straight waves crash into a concave barrier, the waves reflect off of the barrier at the same angle that they came in at each individual point (Ahem, the Law of Reflection).  Here's where it gets interesting: instead of being straight waves, the waves are now circular because concave barriers curve inwards! Also, every wave is reflected to a common point called the "focus point." It must be magic you say? Naw, it's just physics!

Another interesting fact about the focus point: if circular waves start at the focus point and hit a concave barrier, they reflect off as straight waves!

Why yes, you're right! You're really thinking on your feet now. Spoons are concave! Why are you upside down in the reflection on a spoon? I think this picture can briefly help you to understand, now that you know a bit more about reflection.

http://blissfullydomestic.com/wp-content/uploads/mirrors-concave-1-300x140.jpg

Also concave are satellite dishes. The straight waves come down and hit the dish in all different locations. The waves are then reflected off of the dish and are, well, focused to the focus point which then sends the waves to your tv!

Convex barriers reflect waves in a very similarly as concave barriers. The Law of Reflection still applies, of course. But since the barrier is curved outwards, the reflected waves are all directed out. However, there is still a focus point! This time, it's on the other side of the barrier. If you were to extend the waves through the barrier, they would all meet at this focus point.

"OBJECTS IN MIRROR ARE CLOSER THAN THEY APPEAR" Have you seen that before? Well duh! If you've ever driven in a car you definitely have! Mirrors on the passenger side of a car are convex. Since the focus point is "inside" the mirror, the image is also created inside. The image also appears smaller because the rays are being pushed closer together.

http://blissfullydomestic.com/wp-content/uploads/mirrors-convex-1-300x137.jpg

Look at all this wave stuff affecting your every day life!

Speed, Speed, Speed

Since you're new to this whole "physics" thing, I figured we should start off with a somewhat simpler topic: speed! Basically the entire population of Earth understands the concept of speed. It's how fast (or maybe how slow) you can run from one end of the kitchen to the other. It's how long it takes the Blue Jay to fly from the small tree next to the window to the big one across the yard. The speed of a wave is the same thing!

Have you heard of a Ripple Tank? No? I didn't think so. Don't worry, it sounds scary and intimidating, but it's really not. It's just a shallow tank filled with water that has a wave generator attached. A light shines down on top of the system so the shadows of the waves are recreated on the whiteboard underneath. Not so threatening any longer, right?

By using this contraption, we can measure the speed of a wave not one way... not two ways... but three ways! Can you believe it? Pretty cool, huh? Guess what else! All three ways should give you just about the same speed! Woah! Physics is getting crazy up in here! Why? Because the one factor that changes wave speed is the depth of the water in the tank and we're keeping that constant.

Let's begin! After you get the waves going so that you can actually watch them on the whiteboard without getting a migraine, draw a 30cm line perpendicular to the waves on the whiteboard. There's a picture below for your reference. Choose a wave and follow it across the line with your finger. It's pretty tricky at first, but after you do it a few times, you'll be a pro! Once you have become one with the waves, that is, once you have a steady pace going, we'll time how long it takes you to get from one end of the line to the other. We'll do this a few times just to make sure we're accurate. After you calculate an average time, which should be about .57 seconds, you can determine the speed by dividing the distance (.3 meters) by the time (.57 seconds). The speed equals about .53 m/s. Did you get the same? Fantastic! Easy, right? I told you so.

Method 1

The next way is a bit more complicated and takes some practice. But that's okay. We've got time. First, we need to find the wavelength- the distance a wave travels during one cycle. The easiest way to find this is to measure from crest to crest (the highest point above the equilibrium). You really just need to eyeball this one. It looks to be about 2cm or .02m. Okay, next! See the spinny thing in the picture below? It's a strobe disk. What do you do with it, you ask? You turnandturnandturnandturnandturnandturnandturn. Good! Just like that! Now look through the slits at the waves. You know you're spinning the strobe disk right when the waves look stationary and are the same size through the slits as they are on the whiteboard. No, no! Don't stop. It'll take a while to get right, but when you do, it's so cool! I'm going to go make some more tea. You keep working.

Method 2

Okay, I'm back. Look at you go! Now that you've got this down, we'll find the frequency. The textbook definition of frequency is the number of cycles or vibrations per unit of time. For waves, this means the number of crests that hit a certain point in a specific unit of time. It's measured in 1/second because it's the number of cylces/time. This unit is also known as Hertz (Hz). To find frequency using the strobe disk, time how long it takes to rotate it 10 times. I calculated about 4.2 seconds. Then, divide this by 10 to determine how long one rotation would take (.42 seconds). Divide the number of slits in the strobe disk (12) by the time (.42s). The frequency should equal 28.6Hz.

Since we have both frequency and wavelength, we can now calculate the speed by multiplying the two together: .02 m * 28.6 1/s = .57 m/s. That's pretty close to the speed the calculated the first way! Nice! We are on fire.

The third way to calculate speed isn't as much fun- there's no spinny thing, excuse me, there's no strobe disk to play with and you don't need to chase waves with your finger. However, you do need a computer and webcam. Technology is always interesting within itself... For this video analysis method, you place a ruler down the same way you did in the first method. The computer then videos the waves moving and doing their thing. To determine the speed, the video is analyzed by Vernier which allows us to see each individual point a wave passes! This is then made into a Distance (m) vs Time (s) graph and VOILA! The slope of the graph is the speed! I made the graph below as big as possible, but just in case you still can't see, it says the slope is about .59m/s.

Method 3

The last thing we have to do is compare the three values we calculated for slope. The video analysis is the most accurate method, so we'll compare the first two to the last one. The percent error formula is ((calculated value - accepted value)/accepted value)*100%. By plugging and chugging a few numbers, we determined the percent error of the first method to be about 10%. The percent error of the second one is only about 3%. You're getting good at this physics stuff!