# Waves: A Simple Harmonic Notion

In my opinion, one of joys of physics is recognizing patterns in the world. When you see patterns in the universe, it’s a sign that there is some underlying connection between things that isn’t immediately obvious. And those underlying connections are cool; they show us that there’s order and beauty in the universe. And when it comes to beautiful patterns in the nature, there’s probably no better example than waves. Waves occur pretty much everywhere in the world, from the eddies and tides of our oceans to traffic patterns (so you might have to expand your definition of “beauty” to include traffic patterns now). So accordingly, all of physics is full of them too. Quantum mechanics? Yeah, that’s actually just all about waves. Electricity and magnetism? You better believe we’re largely still talking about waves. Relativity? Nuclear physics? Astronomy? Waves, waves, waves. So let’s talk about waves!

So what is a wave? Maybe that sounds like a simple question. Unfortunately, It’s not. There’s a temptation to just start listing different kinds of waves, but at this stage that will be confusing. An intuitive (but not perfect) definition might be that a wave is a disturbance propagating in a medium. There’s some jargon to unpack there: a “disturbance” actually just means anything of interest, anything out of the ordinary. “Propagating” just means moving, and a “medium” is pretty much any material or substance in which a wave could travel. So that is to say, all it really means: a wave is something strange that moves across the material it’s made of. At least it appeals to our common sense. We’ve all seen waves moving in water, so we’re familiar with the idea that waves come in different shapes and sizes, and travel at different speeds. A lot of the time, waves repeat over and over again, but they can also be a single disturbance (think of a tidal wave) and that isn’t repetitive, so can we say it’s the same thing? Unfortunately, this intuitive definition doesn’t give us much of an idea what a wave “is” and why it appears all over the universe in so many different forms.

We can take a step closer to the answer by considering a useful example of something that a wave is not. When we talk about wave motion, we are generally not talking about the motion of individual things. A water wave might travel across the ocean, but that does not mean that the actual water molecules are moving across the ocean. If you see a people doing the wave at a sports game, it’s not like the individual people are moving around the whole stadium. They can just stand up and sit back down without spilling their soda (well, sometimes), while the “wave” travels across the stadium. These types of people, or water molecules are what physicists might call “particles” within the bounds of the situations we just described. Ok, that’s what a wave isn’t, so that’s technically progress, right?

If you ask a physicist who is in a bad mood, they might tell you “Well, a wave is a solution to the wave equation.” Nice. Real helpful. Unfortunately, this definition, however snarky, is the one that cuts to the core of the issue. A “wave” is just a name that we give to something that fits our model of “what waves do”, and the language used to describe our model is math. So we have to delve into some math to understand “what waves do.” If you haven’t taken a few calculus classes recently, the equation might look scary at first, but I promise it’s not beyond you. I’ll walk you through what it means without a whole course on partial differential equations. Alright, behold the Wave Equation:

$\frac{\partial^2 }{\partial t^2}\psi(x,t) = v^2 \frac{\partial^2}{\partial x^2} \psi(x,t)$

So let’s unpack this. There are essentially only four “things” here that we need to interpret in this equation (listed from left to right):

• $\frac{\partial^2 }{\partial x^2}$,
• $\psi (x,t)$,
• $v^2$,
• and finally $\frac{\partial^2 }{\partial x^2}$.

Let’s start with $\psi(x,t)$, (the lower case greek letter psi). $\psi(x,t)$ is the symbol that here represents the quantity that does the waving. It could be the height of the surface in a water wave, or it could be the density of air in a sound wave. Each side of this equation is “doing something” to $\psi(x,t)$. So the next question is what each side of the equation is “doing to” $\psi(xt)$.

The symbols $\frac{\partial^2 }{\partial t^2}$ and $\frac{\partial^2 }{\partial x^2}$ are formally known as second order partial derivative operators. All you really need to know right now about these operators is that do their “operation” on the function to their right and they tell you about how it changes when you vary one of the variables that it depends on. (In case you’re interested, a first order partial derivative operator tells you how much the function changes when you vary one of the variables, and a second order partial derivative operator tells you how the function changes, i.e. if the change is accelerating: speeding up or slowing down). The letter in the denominator lets you know what variable you are varying, and it acts on whatever function is to its right. In this case, $\frac{\partial^2 }{\partial t^2} \psi(x,t)$ refers to how the function $\psi(x,t)$ changes with respect to time (t) and $\frac{\partial^2 }{\partial x^2} \psi(x,t)$ refers to how $\psi(x,t)$ changes with respect to space (x). So what this equation is really saying is that the way that $\psi(x,t)$ changes over space is linked to the way it changes over time. The way that it changes in space and time are coupled by a constant of proportionality that is here labeled v. The constant v is known as the wave speed. For sound, that speed is about 340 meters per second. For light waves, the speed is a little bit higher, 299 million meters per second. For people doing the wave in a stadium, it probably depends on the amount of caffeine they’ve all had.

The solutions to this equations are what we call waves. There are a lot of solutions to this equation (by solutions we just mean different functions of $\psi(x,t)$ that don’t violate the wave equation). In fact, there are INFINITELY MANY solutions. You should probably raise your eyebrows when somebody talks about infinitely many of anything, so let me explain. If you have two different solutions to the wave equation, and you add them together, the result is also a solution to the wave equation! This property is very important in characterizing waves, so it’s given a fancy name: the principle of superposition. It’s given this name because it amounts to positioning waves on top of each other (super-imposed on each other, one might say). You might have heard about wave interference, which describes what happens when two waves solutions are added together, and their combined form changes. For example, this is how noise cancelling headphones work. Any given sound wave, and any wave in general, can be cancelled by superimposing a specific wave that is the opposite of the original wave. Wherever there is a peak in the original wave, the “anti-wave” should have a minimum. This is called destructive interference. It might sound like science fiction but that’s actually how active noise cancelling headphones work: by playing the “anti-sound” for whatever sound is present. Wild.

There are an awful lot of other types of waves out there. We’ve got earthquakes (which both compress the ground like air waves and also shake it up and down like water waves), gravitational waves (definitely an upcoming post of its own), waves of cars moving and stopping on the interstate (aka traffic), and many many more phenomena, all of which can be described using the same mathematical tools, and in some way relate back to the wave equation. That’s what makes it so cool and also powerful. Unfortunately there’s still an elephant in the room: quantum mechanics. It turns out that if you look carefully enough, even the particles, which we were saying are definitely not waves, yeah, they’re waves too. Whoops. But that’s going to have to wait for another post! But rest assured, waves are pretty much everywhere, and now you know what they’re all about!