The Physics Philes, lesson 96: Surf the Wave Function
After a brief break, I’m back this week with more about mechanical waves. For the next couple of weeks I’ll attempt to explain something called a wave function and how we can use it to understand a lot of the attributes of waves.
A wave function is a mathematical function that describes the position of any particle in the medium through which the wave is traveling at any time. We’re going to focus on sinusoidal waves, so each particle will undergo simple harmonic motion from its equilibrium position. The wave we’ll be talking about looks something like this:
It’s a transverse wave (which, if you’ll remember, means that the particles in the medium move up and down around their equilibrium position). Notice that I placed the wave in a coordinate plane, with +x to the right and +y up. The y position of a particle depends on what particle we’re talking about – where it is on the x-axis – and also on the time, which we will denote with a t. So, y = (x, t), which we read as “y is a function of x and t.” This is the wave function that describes the wave. If we know the function, we can find out the velocity and acceleration of any particle and basically anything else we could possible want to know about the medium’s behavior at any time.
Now, we’re focusing in sinusoidal waves, which means that each particle oscillates in simple harmonic motion, but each particle is not necessarily in step with each other. Let’s look at a picture.
Notice that particle 2 isn’t quite in the same place as particle 1. For any two particles, the motion of the particle on the right lags behind the particle on the left by an amount proportional to the distance between them. The particles are out of step with each other by a fraction of a cycle. These differences are called phase differences. The phase of the motion is different at different points.
OK, we know what a wave function is, but what does one look like? I’m glad you asked. Let’s look at the wave function for a sinusoidal wave.
Let’s say that the displacement of a particle at x=0 is given by this wave function:
The particle oscillates in simple harmonic motion with an amplitude A, frequency f, and angular frequency ω=2πf. The form y(x=0, t) indicates that that this is a special case of the wave function y(x, t). At time t=0, the particle at x=0 is at it’s maximum positive displacement and instantaneously at rest. The wave travels from x=0 to the right. The amount of time it takes to do so is
where v is the wave speed. The motion of x at time t is the same as the motion of the point x=0 at the earlier time t – x/v. So we can find the displacement of point x at time t by replacing t with t-x/v. Now our wave function looks like this:
We can use the property of cosine that says that cos(-Θ) = cosΘ to rewrite the function in this way:
We can also express this wave function in terms of the period T and the wavelength Λ. Remember that the period is 1/f and the wavelength is v/f. In this case, the wave function is now
But wait! There’s more! We can find something called a wave number and use that to express the wave function in yet another way. A wave number found by diving 2π by the wavelength. If we substitute Λ=2π/k and f=ω/2π into the wavelength-frequency relationship v=Λf, we get ω=vk. That means we can rewrite the wave function equation like this:
Wow. That’s a lot of equations, isn’t it? And a little free of context. Don’t worry, though. Over the next couple of weeks I’ll explain how we can use the wave function to graph a wave and to find the particle velocity and acceleration.
Featured image credit: Mike Baird via Flickr