© Matt Strassler [August 29, 2012]
This is article 3 in the sequence entitled Fields and Particles: with Math. Here is the previous article.
Once we’ve understood the equations for an oscillator — which describe pretty much everything that bounces or wiggles or rolls back and forth, such as a ball on a spring — we can move on to learn about something equally common in nature: waves. Waves are everywhere: sound and light, the rolling motion of earthquakes, ripples on the surface of a pond, etc.
Before we start, a caution: the term “a wave” can be confusing, because what it means in physics isn’t what it means in English. In physics, it doesn’t mean what we often, in English, would call a wave at the ocean’s edge — a single crest and trough. Instead, what “wave” often means in physics is what we might colloquially in English (or in physics too) call a “wave train”: a series of crests and troughs all moving together in the same direction with the same speed. The simplest form of wave would be one where all the crests are the same height and the same distance apart, and we’ll focus our attention on this case.
Waves are truly remarkable things, once you start to think about them. Imagine you and a friend take a long rope and stretch it rather tightly across a room (Figure 2). Then imagine your friend moves one end of the rope (the green end) up and down a few times. A wave will form at his end of the rope, and it will travel across the room (click the figure to animate it) to your end (the red end.)
This is amazing. I mean, really: amazing, profound, and crucial to everything in our universe, including you. Look at what has happened. No physical object has moved from left to right — at the beginning, before your friend shakes the rope, the rope is stretched across the room, and at the end, after your own end of the rope shakes and the wave is gone, the rope is still stretched across the room exactly as before. And yet! Energy, and information, has moved across the room. The wave, as it travels, is carrying the energy that your friend expended when he shook the rope — and carrying the information, in its shape, about how many times and how quickly he shook it — to you, where it made your hand shake, and (in this case — though not true for all waves) it even shook your hand the same number of times and in the same pattern. Wow! No physical object moved across the room, but still, energy and information did.
Or wait. Should we think of the wave as a physical object? Just as physical as the rope?
With that extraordinarily deep question in mind (and if its profundity isn’t clear to you yet, do not fear — it will be soon enough) let’s now turn to the little bit of math we need to describe what a wave looks like and how it behaves, and then the slightly more math we need to write down the equations whose solutions are waves. This is analogous to what we did for the classical ball on the spring; if you haven’t read that article recently you may wish to review it.
A Formula for an Infinite Wave at a Specific Time
The reason this set of articles goes directly from the subject of a ball on a spring — an oscillator — to the topic of waves is this: a wave is sort of double oscillator — oscillating both in time and in space. We’ll refer to time using the variable “t”, and space using “x”.
Look at Figure 3 below. It illustrates a wave that extends on in both directions for a great distance, with a huge number of crests and troughs. This is a bit different from the wave we saw in Figure 2, which only had a few crests and troughs. The difference is inconsequential: in Figure 2 I was interested in illustrating something for which the precise shape of the wave didn’t matter; but now we’re going to focus on the math formula for waves, and that’s a lot simpler for waves that have a huge number of equal-height crests and troughs. And this case will also turn out to be very useful for understanding how quantum mechanics changes how waves behave.
If you click on Figure 3, it will show you an animation of the wave moving to the right — all the crests and troughs moving together.
Our first task is to define some language and write a formula that describes the motion and shape of the wave in Figure 3, just as we did for the ball on the spring.
If you don’t click on Figure 3, so you just look at the static, unanimated figure, you notice that I’ve labeled a few things on the plot.
The plot shows the size of the wave Z as a function of space, at a particular value of time, t=t0: we write this as Z(x,t0). As one traces it across space, it oscillates back and forth, with Z increasing and decreasing repeatedly; at any fixed time, it is an oscillator in space.
Note that Z may or may not itself be related to a physical distance; it might be something like the height of a rope, as in Figure 2, or it might be something rather different, such as the temperature of the air at a certain point in space and time, or the orientation of a magnetic atom at a certain location inside a magnet. But x really does represent a physical distance, and t is really time.
The snapshot of the wave, Z(x,t0) has three interesting properties, the first two of which are shared by the ball on the spring.
- First, there is the equilibrium value Z0, which lies halfway between the largest value of Z at each crest and the smallest value of Z at each trough. (Most of the time we’ll just study waves with Z0 = 0, because often the value of Z0 is unimportant — but not always.)
- Next, there is the amplitude A, the change in Z from its equilibrium value to the top of each crest, or (it’s the same) the change in Z from its equilibrium value to the bottom of each trough.
- Finally there’s the wavelength of the wave — the distance λ between adjacent crests, or (it’s the same) between adjacent troughs, or (it’s the same) twice the distance between each crest and the adjacent trough. This characterizes the back and forth wiggling in space, just the way the period (=1/frequency) characterized the back and forth wiggling in time of the ball on a spring.
Now, what does the shape shown in Figure 3 look like? It looks like the graph of a sine or cosine function; see Figure 4, where cos(w) [that is, cosine of the number w] is graphed versus w. Now cos(w), is an oscillating function that clearly has equilibrium position at zero, amplitude 1, and wavelength 2π. How do we go from Figure 4 to a formula for the wave in Figure 3? First we multiply cos(w) by A to make the amplitude equal to A; then we add Z0 to the whole thing to shift to the correct the equilibrium value (so that if A = 0 there’s no wave and the whole thing sits at Z=Z0); and finally we replace w with 2πx/λ, because cos w has crests at w=0 and w=2π , so cos 2πx/λ has crests at x = 0 and x = λ. Altogether this gives us
- Z(x,t0) = Z0 + A cos (2π x/λ)
This is almost the same formula that describes how a ball on a spring bounces in time, which was z(t) = z0 + A cos (2π ν t) = z0 + A cos (2π t / T), where ν is the frequency of oscillation and T = 1/ν is the period of oscillation. You see the analogy: Period is to time as wavelength is to space.
Let me make one last remark before we go on. I could just as well have written
- Z(x,t0) = Z0 + A cos (-2π x/λ)
That’s because cos[w] = cos[-w] . The fact that we are free to put a minus sign into the formula for the shape of the wave in Figure 3 will be important later.
A Formula for an Infinite Wave at a Specific Position
Now let’s ask a different question: let’s look at how the wave changes in time, but follow a particular point on the rope, to see where it goes and how it moves. This is shown in Figure 5; you see that I’ve circled a particular point x0 that happens to be at a crest at time t0, and if you click on Figure 5 you’ll see an animation that both shows the wave as it moves to the right and also follows the size of the wave Z at the point x0 as it varies in time: Z(x0,t). And you’ll immediately observe that the height of the wave at this particular point behaves just like a ball bouncing on a spring! And it therefore has the same formula as a ball on a spring, as a function of the frequency ν of the wave, or the period T = 1/ν, where T is the time between the moment when the wave at x0 is a crest and the moment when it is next at a crest, after being at a trough just once.
- Z(x0,t) = Z0 + A cos (2π ν t) = Z0 + A cos (2π t / T)
[I've written it both in terms of frequency and in terms of period because physicists usually use frequency, but the period is more similar to the wavelength, so you may prefer that. Either way is fine.]
The Complete Formula for the Infinite Wave
Now we need a formula for Z(x,t), which describes the wave that we saw in Figures 3 and 5 (or any similar wave) at all points x at all times t. And the right answer is
- Z(x,t) = Z0 + A cos (2π [ν t - x/λ]) = Z0 + A cos (2π [t/T - x/λ])
which incorporates both of our formulas for a fixed point in time and for a fixed point in space.
One thing to notice is the minus sign that appears in front of x. I mentioned earlier that we were free to put a minus sign into our formula for Z(x,t0). With a minus sign in front of x and a plus sign in front of t, the formula describes a wave moving to the right, as in the animations in Figures 3 and 5. To see this, note that when t/T – x/λ = 0, the wave is at a crest, because cos=1. So at t=0, there’s a crest at x=0; but if t moves forward a bit, say to T/10, then there’s now a crest at x = λ/10, a little bit to the right of where it was at t=0 — so therefore the crest (along with the rest of the wave) is moving to the right.
What would be different if we put a plus sign instead of a minus sign into our formula for Z(x,t)? Well then there would be a crest at t/T + x/λ = 0, so in that case, at the time t = T/10, there would then be a crest at x = -λ/10, a little bit to the left of where it was at t=0 — and so now the wave is moving to the left (Figure 6).
Waves that are functions of x and t can move in either direction, so we just have to choose the right formula for the given wave. More generally, when we work with waves that can move not just along one spatial direction x but in any of our our three coordinates x, y and z, then those waves can move in any direction, and we have to choose the right formula based on the direction that the wave is moving.
Fine point: we could also put a minus sign in front of t instead of in front of x. But +t, +x is the same as -t, -x, because that’s the same as multiplying the whole formula inside the cosine by a minus sign, and cos[w]=cos[-w]. So +t, +x and -t, -x both give a wave that moves to the left, and +t, -x and -t, +x both give a wave that moves to the right.
Equation of Motion for Waves
Now, just as for the ball on the spring, where we first figured out the formula for the oscillatory motion of the ball, and then looked at the equation of motion for which that formula was a solution, we want to do the same here. We’ve found a formula for the shape and motion of a wave. What equation of motion has such a formula among its solutions? Click here for the next article.