The Physics Philes, lesson 101: Wave Energy and Intensity
Over the past several weeks we’ve learned a lot about waves. We’ve learned what waves are, we’ve learned about wave functions, and we’ve learned how to find the velocity and acceleration of particles in a wave. But, if you’re like me, you may be wondering about the energy of a wave. I mean, the energy released by the waves we see in real life can be incredibly destructive. (Think earthquakes and tsunamis.) So…what about energy?
I’m glad you asked.
All waves have energy associated with it. In order to produce a wave, a force must be applied to some portion of the wave medium. Work is done on the system when the point at which the force is applied moves. The energy moves through wave as the medium exerts a force and does work on an adjacent portion of the medium. This is how energy is transported from one region to another.
To make this easier to visualize this energy transportation, let’s think again about a transverse wave on a string traveling from left to right. Here is a portion of that string:
Think about that point a. The string to left of that point exerts a force on the string to the right of the point. Like this:
As you can see, the force exerted at a is represented by components F and Fy. Remember that Fy/F is equal to the negative of the slope of the string at a. We can also represent that as the partial derivative of y with respect to x. Working our math magic, we can combine these two equations to get
We need for that F on the right side to be negative because Fy is negative when the slope is positive.
When point a moves up and down, the force in the y direction does work on the point, hence transfers energy to the string to the right of the point a. That means that the rate work is being done at point a (otherwise known as the power P) is the force in the y direction times the transverse velocity at the same point. If the transverse velocity is equal to the partial derivative of y with respect to x, then we can express the power of the wave like this:
This is the instantaneous power, or the instantaneous rate at which energy is transferred along the string. Energy is only being transferred at point where the string’s slope and velocity are not zero. This needs to be so there is a transverse component of the force and so the transverse force can do work.
This equation is valid for any wave. But for a sinusoidal wave, we can describe it with this series of equations:
If you’ll remember, that’s the sinusoidal wave function. Next are the slope and velocity, respectively:
But you knew these already. Let’s get to the power of a sinusoidal wave:
Now we can use the relationships ω= vk and v^2=F/μ to express the power equation slightly differently:
Since sine squared is never negative, the instantaneous power of a sinusoidal wave is only either positive or zero. That means that energy either flows in the positive x direction or there is no energy transfer at all. Energy is never transferred in the direction opposite of the wave propagation.
From this equation we can even figure out the maximum value for instantaneous power. Since the maximum value for sin is 1, that means the max value for sin^2 is also 1. So the max power will be:
But this is all instantaneous power. We can find the average power by thinking about the average of sine squared, which is 1/2. That means the average power can be expressed as:
So the average power is just half of the maximum instantaneous power. It’s also proportional to the square of the amplitude and the square of the frequency. Meaning that if the frequency or the amplitude is doubled, the power will quadruple. This is true for all type of mechanical waves, but not for other types of waves. The power of electromagnetic waves is independent of the frequency, so the average power in those types of waves are proportional to the square of the amplitude only.
This has been a pretty equation-heavy post, but there is one more thing I’d like to cover before I go. Let’s talk for a minute about wave intensity and the inverse square law.
So far we’ve been talking about a transverse wave on a string, which is a wave in one dimension. But there are waves that travel in three dimensions, like seismic waves. For those types of waves, we talk about their intensity I. Intensity is is the time average rate at which energy is transported by the wave, per unit area, across a surface perpendicular to the direction of propagation. Whoa, that’s a mouthful. Think of it as just the power per unit area, measured in watts per square meter.
OK, so if the waves spread out equally in all directions from its source, the intensity at distance r from the source is inversely proportional to r squared. We can get this from energy conservation. Say we have power output P from the source. Then the average intensity I1 through a sphere of radius r1 and surface area 4πr1^2 is:
The average intensity I2 through a sphere with radius r2 is given by a similar expression. If no energy is absorbed between the two spheres, the power must be the same for both:
After we do some cancelling, we find that:
And that, my friends, is the inverse square law. The intensity at any distance from the source is inversely proportional to the square of the distance.
Looooong post is long, but I feel like we covered a lot of good physics this week. You should be proud of yourself. Now it’s time to relax, put your feet up, and mentally prepare yourself for next week when we’ll wave interference and superposition.
Featured image credit: UN-CECAR via Flickr