Sunday, 31 January 2016

Things that waves do, part 1: Reflection and refraction

Right now we're looking at waves, and how studying waves ties into, well, everything. Before we move onto things like resonance, which will let us collapse bridges* (and sometimes resonance does collapse them, just by accident).....


 
.....lets have  quick recap on the main behaviours waves have. In this post we'll look at the two best known:

Reflection:
Waves bounce off things, and we call it reflection... but there's a little bit more to it than that: In two dimensions, or three, waves follow a law of "angle of incidence equals angle of reflection" (the law of reflection). To make it easier to see the law of reflection in diagrams we often use ray diagrams (explained in this post). Drawn with a ray diagram the law of reflection looks like this:

 
When doing calculations we will almost always use ray diagrams, but it's important to remember that light actually moves in waves. A good way of visualising reflection with waves is using a ripple tank with a light behind - so the shadows of the waves are projected onto a screen. Sainnt Mary's University show this nicely: 



When we combine reflection with phase and interference (see this post) we can get some unexpected phenomena, such as standing waves.

Refraction:

Let's go back to our equation that relates wave speed, frequency and wavelength (from this post):

...where V is wavespeed, f is frequency, lambada is wavelength.

One of the things that happens when a relationship binds quantities like these together is that when one of them changes, at least one of the others needs to change to maintain balance in the relationship.
And guess what? In refraction they change: When a wave passes from one medium to a different medium its speed changes - like a cats speed would change if swam from water into treacle.

You're planning me to do what?!

The speed changes but the frequency does not (otherwise the light would change colour).  Because it's speed is different but not its frequency the wavelength must change.That doesn't cause any complications when the wave hits the edge of the new medium  straight on....

 
...but when it comes in at an angle this means that the wave fronts would be broken.....


...unless the angle of the waves changes. And, because the wave fronts don't like to be broken, the angle of them does change: 


To make things a bit simpler we'll draw this out as a ray diagram (see here for a quick run down on ray diagrams)



There is a law that predicts how the angle of the rays change, called Snells' law. It uses a property that every medium has, called 'refractive index'. Refractive index determines how light moves through a medium. For example, empty space has a refractive index of 1, so light moves through it easiest, air as an index of 1.000277 and most glass has an index of around 1.5.
To use Snell's law we need to draw a line (called a 'normal' line) that is at right angles to the surface the rays are passing through, and measure the angles the incoming and refracted rays make to it. When we've done that, Snell's law is this:


 
Where:

Angle 1 = angle between the incoming rays and the line that is at right angle to the surface
Angle 2 is the angle between the refracted rays and the line that is at right angle to the surface
n1 = index of refraction for the material the light starts in
n2 = index of refraction of the medium the light ends up in

So, as well as direction, refraction changes wavelength and speed. But by how much? There're related equations that let us work out the change in wavelength and wave speed, using the angles the waves make to the normal line:









...Where:
V = wavespeed
Lambada = wavelength
Angle 1 = angle between the incoming rays and the line that is at right angle to the surface
Angle 2 is the angle between the refracted rays and the line that is at right angle to the surface.

Practice Questions:
 
1: A ray strikes a reflecting surface at a 40 degree angle to that surface. What angle does the reflected ray make with the surface?
 
2: A ray passes through vacuum, then into a block of glass with a refractive index of 1.4. It strikes the glass at an angle of 45 degrees to the normal. What angle does it make with the normal beneath the surface of the glass?
 
3: The light from question 2 has a wavelength of 560 nanometers in the vacuum.
     a) What is its wavelength inside the glass?
     b)What is its speed inside the glass?

* I've mentioned before that a Physicist should have a touch of supervillian.

Sunday, 24 January 2016

Introducing ray diagrams....

Before we go any further we need to introduce the concept of ray diagrams. Don't worry, this is fairly painless!
In other posts we've just run with the idea that light travels in waves. And that's true.... but if we're trying to calculate the path light takes, especially when we get to things like diffraction and refraction, accurately drawing waves themselves quickly gets difficult. To simplify this we use ray diagrams: These aren't meant to indicate that light is made up of rays, it's just a convenient way of showing the direction of the waves. 

To turn a wave diagram into a ray diagram is simple: Draw a line at right angles from the line of the wave, pointing in the direction the waves is going. Like so:
This is the 'ray' associated with that wave. If the the wave front is very broad, add some more. If the wave front is curved draw your ray at right angles to the tangent of the wave, and do it at several points, like so:

...which gives you a ray diagram like this:


...and that's all a ray diagram is, a way of showing the direction of motion of the waves!

Tuesday, 19 January 2016

Phase and Coherence...and a seasick cat.








Up to now we've established that waves are oscillations moving through a medium (this post), that they have properties we can measure, like wave speed and frequency (this post), and that when two waves meet at a point they can interact constructively (reinforcing each othe), or destructively, (cancelling each other out, this post). We've also established that cats are integral to physics, although not always in ways cats would like.

That's all true, but it's a bit of an oversimplification: Two waves don't have to be in exactly the right position to reinforce each other (all the troughs lined up), or in exactly the right position to cancel each other out completely (troughs lined up with peaks). In fact it's unlikely that they will exactly match up either way, in a real situation.
To get to grips with waves interacting in more realistic ways like this (by realistic I mean awkward and complicated) we need to introduce a new wave property, phase, and a new property of two interacting waves coherence.

Lets look at coherence first: Imagine we have  two long trains of waves that are superimposed over each other:


Above: Two superimposed waves -  we'll come back to them in a bit.

Both trains of waves have similar amplitudes, the same frequencies, and the same wavelengths. Yet they are clearly "out-of-step" with each other - the peaks and troughs don't line up. When waves with the same frequency and wavelength (and similar amplitudes) don't 'line up' we say they are coherent - they are the same except that their peaks and troughs do not coincide.

The difference between the waves in the above situation is termed a phase difference between the two waves. Each wave, A and B, has a 'phase' at every point on the graph. Because they're out of step, their phases are different. 

But what is phase? Phase is basically a measure of how far through it's cycle the wave is at any given point.... well, it's easier to show you. Here's a wave:




Now we pick a point on the wave, and to make it stand out we put a cat on it. Because, well, why not?



As the wave moves, that cat moves up and down, in a cycle:


The point the cat is riding moves up and down in a cycle. The phase of the wave at that point can be thought of as: 'How far through it's cycle the cat is'. 

OK, fine (aside from a sea sick moggy). But to use this new quantity we need to be able to put a number to it. And that turns out to be a bit tricky: How far up or down the cat is won't work, because two cats on different waves could be at the exact same height, but one moving up and the other down - so at very different points on their cycles. 

Above: These two cats are at the same 'height' (displacemnt from the X-axis technically), but at different points of their up/down cycle, hence one is moving up and one is moving down - so we can't use X-axis displacemnet to measure phase.

Velocity doesn't work well either, or acceleration: It's hard to put a simple number to how far through it's up-down cycle the cat is, because the speed that a point is propelled up then down by a wave isn't constant, nor is it's acceleration either: Any point with a wave passing through it moves fast, slows down near the peak of the wave, then reverses direction and moves the other way while speeding up

But there is a way we can get a number that changes at a steady rate out of this, and so have useful way of measuring phase: For a simple sine wave the variation in the cats vertical velocity actually matches the variation in vertical component of the velocity of a point on a spinning wheel - like this:



So, using this imaginary wheel, we can attach an angle  to each position of the point the cat is riding on it's cycle. So, as the whole wave is made of points going up and down in this way, every point on a wave has a phase that can be expressed as an angle:


We can go a step further with this idea, and plot the angles onto the x - axis, so now the wave is just defined by it's amplitude (maximum height off the X axis) and it's phase:

With this way of measuring the phase of a wave (it's not very intuitive I know, but it makes the maths simpler in the long run and trust me that makes this worth the effort) we can also measure phase difference.

Phase difference:
What if a point has two waves passing through it? If you have two waves passing through a point then the 'phase difference ' is the difference in phases of the two waves at that point. So if there were just wave 'A' present the cat would be here.....
Above: Wave A.

....and if there were just wave B present the cat would be here.
Above: Wave B
Now we can add our imaginary phase wheel in. At any point where they meet two out of phase waves will be at different angles around the rim of the imaginary wheel. The difference between the two angles is the phase difference.


Let's go back to the two waves we saw earlier:
Above: The two wavetrains form the start, because I'm too lazy to draw them again.

Now lets add in a point where we'll measure both their phases, and the wheel for measuring phase:


What's the phase difference between the two waves? It's the angle between the two angles on the wheel. Or, if we were drawing the angles onto the X-axis directly, it's the separation in degrees between a point on wave 1, and its corresponding point on wave 2 (a peak on wave one and the nearest peak in wave 2, for example). If the waves are coherent (so they are identical apart from their phase difference) then their imaginary wheels spin at the same speeds, and the phase difference between them is constant along their entire lengths.

Two more terms you'll need: If two waves have no phase difference (so the peaks align) then they are said to be 'in phase'. If they're exactly 180 degrees out of phase (so each peak perfectly aligns with a trough) they're said to be in 'anti-phase'.

So, to sum up:
  • If two waves are coherent they are of identical frequency, similar amplitude, but have a phase difference between them.

  • Phase is a measure of how far through it's cycle of oscillation a point on the wave is.

  • Phase is measured as an angle, by comparing the oscillation of the wave to a point on a wheel that spins once for every complete wave oscillation. 

  • Where two waves meet the difference between their phases is called the phase difference between them.

  • Coherent waves will have the same phase difference along their whole length. 


Revision questions:

Answers here



1) Just by looking:



a) Which of these waves are likely to be coherent with each other?

b) Which of these waves are likely to be in phase (no phase difference) with each other?


2)


a) These two coherent waves are out of phase, but by how much?

b) How much will they be out of phase by 100 wavelengths further down the graph?


3)


How many degrees of phase are between the points A and B on this wave?


Next time, we'll look at what happens when we combine the ideas of phase and interference....






Thursday, 14 January 2016

Constructive and destructive interference:

Above: The anatomy of a wave - remember it, it's really a useful thing to have in your brain for a Physics exam.

If waves are moving oscillations (I promise they are, see this post), what happens when they run into each other? 

That might seem a fairly obscure question to ask, but it turns out that:
 
A) Asking obscure questions that no-one else thinks to ask is a big part of science, and occasionally leads to amazing discoveries (or very large explosions).

B) This question is fairly important for a lot of problems*. Especially if you're a sailor caught between two tidal waves.

Physicists don't just have an explanation for two waves colliding (here's a hint: The sailor's in trouble), they have a principle (which is much classier). It's called the principle of superposition, and it goes:

The total disturbance at a point where waves meet is the sum of the disturbances due to each of the waves. 

'The disturbance' is the oscillation of the wave, so when two waves meet at a point  you can find how much the medium they are travelling in is disturbed by taking the  size of each wave at that point and adding them together (be very careful with your plus and minus signs!)

Let's say two waves on a string  run straight into each other - that's a very simple example but we'll look are more complex situations in a later post. At any point where the waves are meeting they will add together, and what you actually get depends on whether the waves are both of the same sign or not, and how big they each are at that point. 
To keep things simple lets look at just two situations: two waves where the peaks and troughs line up, and two waves where the peaks align with the troughs. In the first situation the two waves have the same sign everywhere, so are pulling the wave medium in the same direction.
If the waves are pulling the same direction and are aligned then you just add the amplitudes of the waves at every point, and the wave oscillation will be bigger at every point (which is called constructive interference), like in the picture below...

Above: When two waves meet and the peaks and troughs align we get a bigger wave - it's really just that simple.
 
.... if the waves are pulling in opposite directions at that point the total disturbance will be smaller (destructive interference).....

Above: Where a peak lines up with a trough the waves will cancel out, either partially (for different size waves) or completely (for waves the same size).
 
And, thanks to the magic of Youtube, I can show you slow-mo examples of both. As I like destroying things, lets look at destructive interference first:



Above: When the two pulses meet in  the middle they momentarily cancel out, in a process called destructive interference
 
See how, right in the middle, the two waves both seem to disappear? That's because, at that moment, the trough of one wave is right over the peak of the other, and visa versa - the result is that the two waves momentarily cancel out, and almost vanish.  But when we get two peaks (or two troughs - it's the same thing just upside down) that line up we don't get a cancelling effect reducing the height of the wave, we get the height of both waves combined:


Above: A very similar situation, where two waves run into each other - except that, now, the two pulses are pulling the medium the same way, so they reinforce each other causing a big peak in the middle.
 
The difference between the first and second videos is the direction that medium (the string in this case) is being disturbed - if the two waves are disturbing the medium in the same direction at that point the overall disturbance is bigger, and if they're disturbing it in the opposite direction the overall disturbance is smaller than either of them. This is why it's very important to keep track of your plus and minus signs, because these are what's telling you which way the wave is pulling: If you sum two waves that have an amplitude of 3 cm you can have a resultant combined disturbance of 3cm + 3cm = 6cm, when two peaks align. But you can also have a resultant disturbance of 3cm + (-3cm) = 0 cm when a peak and a trough align.

That's basic constructive and destructive interference.... for very simple waves that just go backwards and forwards on a string. As we go on we'll meet more complex problems involving waves in two, or even three dimensions, but the underlying principle of superposition won't change: 


The total disturbance at a point where waves meet is the sum of the disturbances due to each of the waves


Or, in other words, where two waves meet their amplitudes will add, and the resulting disturbance will be bigger if they both have the same sign at that spot, and smaller if they have different signs at that spot. It's important to remember that idea, and be careful with the plus and minus signs.
Why learn all this? Firstly, in the world, you can get paid for actually knowing something useful (you didn't think teachers were torturing.. er... training.. your brain in that classroom for nothing did you? ). Secondly, if this doesn't come up on your exams at some point I'll eat my hat.
And it's a nice hat.



* For example, the one's you'll get asked in your exam.