What’s In a Wave?

by rob

Waves are all around us. At this moment your body is encountering many different types of waves from a vast array of sources. Some bounce right off your surroundings in to your iris, being seen as light. Some transfer their energy to tiny hairs in your ears, registering as sound. Many pass right through you without you even noticing, but are noticed by devices which use them to communicate, like the cell phone in your pocket. Some even originated at stars in distant galaxies millions of years ago. The only thing as diverse as physical waves are the ways in which we’ve figured out how to use them. Given their importance in science, mathematics, and industry it behooves us to understand them better.

How do we represent a wave? Here’s something you’ve probably seen before: a sine wave, or F(x) = A*sin(theta + phi). Here amplitude is 1 and phase is 0 so we just have sin(x).

It may not look like much given the fanfare above, but it’s important to distinguish the idea of a wave from ‘real’ waves. This is merely one method of representing a wave, and a poor one at that. The sine wave extends to infinity in both directions. Unfortunately a width of infinity is not supported by your monitor (or brain). Also this is only a 2-dimensional plot, we live in 4 dimensions (3 spatial plus time), so you have a right to be disappointed. What’s a better way to represent a wave? How can we represent any wave without losing information about it or requiring an infinitely large monitor? Differential equations to the rescue!

\nabla^2{\vec A} = \frac{1}{v^2}\frac{\partial^2 \vec A}{\partial t^2}

Don’t fear the notation! This is the wave function and it says something quite simple. Let’s call our wave “A”, arbitrarily so. More specifically, “A” is a measure of how much our system varies from equilibrium. The wave is oscillating like a spring. In fact the wave equation can be derived from Hooke’s Law. We’re describing how much and how quickly it oscillates as well as how fast it travels. The left side is the ‘Laplacian’, or, a description of how our wave changes in whichever universe or coordinate system we choose, for example, cartesian (3D):

\nabla^2{\vec A} = \frac{\partial^2A_x}{\partial x^2} + \frac{\partial^2 A_y}{\partial y^2} + \frac{\partial^2 A_z}{\partial z^2}

The curved ‘d’ symbol is a partial derivative, or, the value of the derivative of a term assuming the other terms stay constant. If you need a refresher, a derivative is essentially a description of how quickly a value changes based on its input. The derivative of position is velocity. The derivative of velocity (or, second derivative of position) is acceleration. If position is changing rapidly, the velocity is high. Now, let’s deal with the right hand side. First, ‘v’ is just the speed at which the wave propagates, or travels. The last term is the second partial derivative of our wave “A” with respect to time. Notice both sides are second order. So, we are talking about how our wave changes in time, while the left side talks about how it changes in space. Simple, right?

Now that we’ve discussed the idea of a wave, let’s talk about some real ones. I’m going to pick Electromagnetic waves because we understand them quite well. Electromagnetism can largely be described by Maxwell’s equations. Electricity and magnetism often appear distinct (what’s interesting is this is entirely unique to a 4th dimensional treatment) but they are really the same force. Take two of Maxwell’s (there are only 4 total) equations:

\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}

\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}

Again, notation is not your friend, but these equations rather elegantly describe how electricity and magnetism are related. By convention, E always refers to the electric field, and B the magnetic field. The first, the Maxwell-Faraday equation, basically says a changing magnetic field produces electric current. This is why when you swipe your (magnetic) credit card through a Square card reader, the audio jack can decode it based on the amount of electric current received, without needing to be powered itself.

The second, the Ampere-Maxwell equation, just says the opposite. Electric current produces a magnetic field. It’s okay if you don’t fully understand these relationships, just remember they depend on the two values: \mu_0, or ‘permeability’, a universal constant describing how strong a magnetic field is in free space,  and \epsilon_0 or ‘permittivity’, describing the same for electric fields. Here’s where it gets interesting. Recognize the \nabla \times operator? It’s called the ‘curl’. It’s just a mathematical operator, so what happens when we apply it again to both sides of each equation? Remember applying the same operation to both sides of an equation doesn’t change what it says at all. Now, I’ll spare you the complex process, but they simplify like so:

\nabla^2 \vec E = \mu_0\epsilon_0\frac{\partial^2 \vec E}{\partial t^2}

\nabla^2 \vec B = \mu_0\epsilon_0\frac{\partial^2 \vec B}{\partial t^2}

Don’t those look a bit familiar? It’s the wave equation!! These are the differential equations describing electromagnetic waves. Something’s a bit off though. If you look back to the wave equation you’ll see the propagation velocity is different, or is it? What happens when we equate the two?

\frac{1}{v^2} = \mu_0\epsilon_0

or equivalently,

v = \sqrt{\frac{1}{\mu_0\epsilon_0}}

Remember \mu_0 and \epsilon_0 are constants, so we can calculate a value for ‘v’.

v = \sqrt{\frac{1}{[4pi \times 10^{-7} m kg/C^2][8.8541878 \times 10^{-12} C^2 s^2 /kg m^3]}}

v = \sqrt{8.987552 \times 10^{16} m^2 / s^2} = 2.9979 \times 10^8 m/s

That’s 299,790 km / s. It’s the speed of light! Of course, electromagnetic waves travel at the speed of light. Light is an electromagnetic wave. This begs the question, though. Are permittivity and permeability emergent properties of the speed of light? Or, is the speed of light an emergent property of permittivity and permeability? Perhaps they’re both based off of a lower level property of physics?

There’s much, much more to waves than this. Did you know particles and matter can also be described as waves? Take a look at the famous Schrodinger equation and see if anything looks familiar.

i\hbar\frac{\partial\psi}{\partial t} = \frac{\hbar^2}{2m}\nabla^2\psi + V(\mathbf{r})\psi

Again, the notation hurts, but it’s a relationship of incredible simplicity and beauty once you understand it. Clearly waves are even more fundamental and important than we originally thought. Can you imagine if our eyes could detect more than such a tiny sliver of the electromagnetic spectrum? The universe is much more fantastic and beautiful through the unconstrained eyes of physics and mathematics. More to come…

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