Physics, explained by cats: 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)