A non-mathematical introduction to the wave equation

Waves pop up everywhere in physics. They’re most obvious at the beach, but waves are also used to describe light, pendulums, and all sorts of other things. Because waves can describe so much in physics, it’s important to know what it actually means when you talk about waves.

The medium is not the wave

When physicists talk about waves, they mean something very specific. They mean that some material is moving in a specific way. Waves at the beach are the best way to visualize this for me. Water waves have a very obvious medium: water. The waves are not the water, they’re the way the water moves up and down over time. And the water doesn’t just move up and down randomly; a peak on the water seems to travel towards the beach.

This is a very key point. Waves are just the way that some type of medium is moving. Light waves, for example, are just motion in the electromagnetic field.

So what causes the wave to move like it does? The answer to that relies on two separate ideas: energy input and strain in the medium.

Energy input

For most mediums (like water), if you leave it alone the waves will all die out and it’ll be still. There needs to be some transfer of energy into the medium in order for a wave to start. At the beach, the energy to start a wave often comes from wind. For light, the energy to start a light wave (photon) usually comes from electrons bouncing around.

Not all mediums are like this. In outer space, the electromagnetic field will keep a light wave going forever. That only works once the light wave gets started, which still takes energy input.

Strain

Strain in a medium is the tendency for it to return to it’s original position. In water, strain is provided by gravity. Because gravity pulls on all the water in the ocean, the water tries to keep an even level. If wind pushes some water up higher, gravity will try to pull it down. The energy in the peak will get transferred to nearby water molecules, and the wave will move.

The wave equation

The way that strain in a medium causes a wave to travel is described by the wave equation

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x}

This equation shows how the time rate of change of the wave (the left hand side) is related to the strain in the medium (the right hand side). This equation is what’s called a second-order differential equation, which just means we’re taking derivatives twice (the $\partial$ stuff).

To understand this equation, it helps to take a look at each side separately. To do this, we’re going to look at snapshots of the wave in a couple of different ways.

Localizing space

Let’s think about the left side of this equation first. The entire left side basically means “the way the medium changes with time”. To get a feel for this, I like to imagine I’m treading water at the beach. Maybe I’ve swum out a few leagues and I’m just bobbing up and down with the water. You could plot my height above sea level over time, and that would give you one view of the ocean waves. We’ve made a little movie of the wave for a single point in space, and ignored all the other points.

The left hand side of the wave equation represents how fast my height at that point is changing (which is equivalent to saying that it’s the second derivative of my height with respect to time).

Freezing time

The right hand side of this equation is another second derivative. This time it’s a derivative with respect to space, instead of time (x instead of t). To understand this, think of freezing time instead of localizing space. If we take a snapshot of an ocean wave at a given time, it would be like a bunch of troughs and valleys in the water that don’t change.

The second derivative on the right hand side of this equation represents the curvature of the water at any given point. That curvature is a pretty good measure of the strain the water is under. The spikier the water, the more the curvature, the more the strain.

Time and Space together

The wave equation is an assertion that curvature of a wave in space (the strain) is related to the way that the wave travels as time moves forward. The relation is given by the factor $c^2$ in the wave equation. The travel of some part of the wave through time is equal to the curvature of the wave at that point multiplied by speed of the wave in that medium.

If we’re talking about light waves, then $c$ is the trusty old 3*10^8 m/s. If we’re talking about sound waves or water waves, then $c$ is going to be different. The exact value of $c$ depends on what kind of medium we’re talking about.

Conclusions

So that’s the main thing that a wave is. A wave is just a way that strains in a medium move around, and you can describe that motion using a specific equation. The wave equation says that the curvature in the medium at a point is related to the rate at which the strain of the medium changes at that point.

If you know the properties of a medium and what the strain in the medium looks like now, you can calculate the curvature of the medium. That let’s you figure out what the medium looks like later. You may also need some extra information, like the velocity of the wave now or the energy being put into the wave.

Walking Through Walls

I think my favorite super power is probably the ability to walk through walls. I’ve always wanted to be able to break into any building, escape any pursuer, or drop through the floor instead of taking the stairs. It’s one of my main regrets that I’ll never be able to do it.

The next best thing to being able to do something is knowing as much as you can about it. In elementary school I’d learned that atoms were mostly empty space. You have the electrons and the nucleus, but it seemed to me that if you managed to squish two atoms into the same space then they might be able to pass through each other.

I was pretty excited to learn, when I got to middle school, why exactly objects couldn’t do that. For one thing, if you got the atoms occupying roughly the same space, then the electromagnetic forces would interfere with each other and the electrons of both atoms would be disrupted. This would severely mess up any chemistry that was going on with those atoms, and probably do very bad things to a person walking through a wall (and the wall itself).

Luckily, my youthful attempts to pass through my bedroom wall were doomed to fail for a reason that didn’t involve all of my molecules coming apart. Originally, I thought this was due to electron repulsion. According to my high school physics teachers, atoms can’t get that close together because the electrons of the atoms repell each other. Just like magnets of the same pole, two electrons will stay as far apart as they can. You can’t make use of all that empty space within an atoms because the electrons form a kind of force field to keep other atoms out of their own territory.

It turns out that electron repulsion isn’t actually what prevents objects from passing through each other. It’s actually a quantum effect called electron degeneracy pressure, Basically, two electrons can’t possibly be in the same place at once (that’s part of the Pauli exclusion principle). When electrons get too close to each other, they must assume different energy levels. This means that to bring electrons close together, you need to add enough energy to put most of them into very high energy states. The closer objects come, the more energy you need. On the macro-level, that manifests as degeneracy pressure. That’s why objects feel solid.

Understanding this almost makes up for not being able to walk through walls.

Relativity and your smartphone

Einstein’s theory of general relativity has dramatically changed life on our planet. It’s used in a lot of different technologies, but perhaps the most surprising place to find the theory of relativity is in your smartphone. Smartphones account for general relativity in two different ways.

The place that it’s most commonly pointed out is in GPS. Your phone figures out where it is by calculating the distance to a number of satellites. It does this by measuring the time of flight of a signal broadcast by each satellite. Once the phone knows how far away different satellites are, it can do triangulation on the known positions of the satellites to figure out where you are. This location measurement can be pretty precise (on the order of a meter).

The precision of GPS is possible because your phone takes into account special relativity in the form of time dilation. Satellites are travelling very fast with respect to a stationary smartphone. That high speed means that time goes slower for the satellite, and the clock it uses to calculate time of flight is off. Your phone takes that into account when calculating how long it took the signal broadcast by the satellite to get to wherever you are.

General relativity comes into play because satellites are so much higher than your phone, which means that they experience less of Earth’s gravity than you do (note that this is different from microgravity). Since satellites experience less gravity than you do, time travels faster for them. So there are really two relativistic effects that need to be taken into account to actually figure out how fast time is travelling for the satellite, which can help tell how long it takes for a radio signal to travel from the satellite to your phone.

The second way that a smartphone takes general relativity into account is far simpler. Your phone has an accelerometer in it that measures acceleration on the phone. This is how your phone knows which way you’re holding it. It’s also how it makes those cool light-saber sounds when you swing your phone around.

When you’re having a light-saber duel, your phone is measuring the acceleration applied by your wild jabs and lunges. No relativity there. However, when the phone is stationary and it detects which way it’s oriented, it’s measuring gravity. Gravity isn’t acceleration, but it is indistinguishable under the theory of general relativity. It’s only through the effects described by general relativity that your phone works the way that it does.

Science! It’s closer than you think!

Entropy and Externalities

There’s a concept in economics called the externality that my environmentalist friends like to talk a lot about. An externality is a cost that exists for some enterprise, but it’s a cost on somebody other than the enterprise itself. The classic environmentalist example is that environmental damage is an externality for oil companies. Oil companies get a lot of money for extracting oil, and they sometimes don’t bother to take care of the environment as they do that. This is because environmental damage affects the local community, but not the oil company’s profits.

In many ways, it seems to me that an externality in economics is similar to entropy in physics. Entropy in a closed system never decreases, it’s only by ignoring some part of the system that you can say that you’re increasing order. So too with externalities. Those costs created by the enterprise still exist and still need to be paid for. The only reason that a company (or person, or government) can console themselves about not paying for those costs is that they’re not a part of the closed system that is the company and its customers and suppliers.

As the concept of externalities has come to be better understood by governments, there have been attempts to make destructive companies take responsibility for their actions. This seems like what I used to do in my physics classes by redrawing system boundaries to account for entropy. Redrawing system boundaries for economic externalities is usually done by creating laws that require companies to pay for any damage that they may create. One good example of this is Montana, where mining companies have been required to create trusts that are responsible for cleaning up after them.

What’s interesting to me is that the owners and CEOs of possibly damaging companies sometimes realize that they live inside the wider system that encompasses whatever damage is caused by their company. One example of that is Sunoco, which is the only oil company to sign on to the Ceres Principle.

I wonder if a better understanding of physics would cause people to realize the impact of such externalities to other parts of their lives. Even companies that work to mitigate externalities don’t do all that they could. Perhaps CEOs of potentially harmful companies should be required to take a course in thermodynamics to get a good understanding of entropy and system boundaries.