The Physics Philes, lesson 97: Making Waves
Alrighty, people. It’s time to dig deeper into the mathematical description of a wave. Now we’re going to look at the different ways to use the wave function to graph a wave.
We can graph the wave function y(x,t) as a function of x for a specific time. It might look something like this:
only…you know…better than this. Notice that it has the familiar axes x and y. This gives the vertical displacement y of a particle from its equilibrium position. If this wave happened to be a transverse wave traveling through a string, this would be like a snapshot of the string at a particular moment in time. We can express the wave function at t = 0 as
But that’s not the only way to express a wave in graph form. You can also graph it as a function of time for a certain position x. Like this:
I know I’m super bad at drawing curves. But if you can look past that for a second, notice that the axes this time are t and y, rather than x and y. This is because this graph gives the vertical displacement of the particle at that coordinate as a function of time. In other words, this graph describes the motion of that particle, rather than the wave’s shape. This wave function, for x = 0, can be expressed
I know these graphs look really similar, but they aren’t. They are describing different things. So be careful whenever you come across the graph of a wave function in the wild so you know exactly what it’s trying to tell you.
So far we’ve been assuming that waves are traveling in the positive x direction, but that doesn’t have to be the case. Remember a couple of weeks ago, we discussed how to define a wave function at any time t by subtracting the time it takes for the wave to travel? Well we can do the same time if the wave is moving in the negative x direction by adding the time it takes for wave to travel. In this case, the displacement of x at a certain time is the same as the motion of x=0 at the later time t+x/v. So our equations now look like this:
That very last term, the (kx + ωt), is what we call a phase. It’s an angular quantity measured in radians. The value for any x and t determines what part of the sinusoidal cycle is occurring at any time. For example, let’s think of the cosine wave. At its crest, the phase could be 0, 2π, 4π, etc. At its trough, it could be π, 3π, 5π, etc. It just depends.
So that’s what it’s like to render a wave in graph form. Next week we’ll keep going with the wave function and find out all about particle velocity and acceleration.
Featured image credit: Creativity103 via Flickr