|Photo Credit Sam Russell, using my iphone in class.|
I've got this ancient "wave machine," which consists of a set of iron* bars connected to one another atop a wooden base. You can send a wave across the machine by pumping one of the bars up and down. These waves are visually exciting, to the extent that students can never resist playing with the machine before and after class. Good.
* Yes, iron, not some other metal: the rust due to storage in my leaky classroom for decades is a bit of a clue.
I use the machine to perform a quantitative demonstration with v=lf.
Step 1: I ask the class how I might measure the speed of waves on the machine. Someone will manage to suggest measuring the distance across the machine with a meterstick, and dividing that distance by the time measured on a stopwatch for a wave pulse to travel that distance. We discuss how that method assumes constant wave speed -- that's a good assumption, because wave speed depends on the material through which the wave moves, and that material is the same throughout the wave machine.*
* I do not use the term "homogeneous," but y'all reading know that's what I mean.
Okay, so we do the experiment: A student measures the wave machine to be 90 cm long. We let a wave pulse go back and forth, so it moves 180 cm; they measure about 4.2 s for the wave to travel the 180 cm. That gives a measured wave speed of 43 cm/s.
Step 2: I ask what I should do in order to put a wave with frequency 1 Hz onto the machine. It takes a moment of thought, but generally the class figure out that this means I push the end up and down once every second. Seems silly, but not everyone has internalized the meaning of a frequency. Here's an opportunity to translate "1 Hz" into "one wave per second."
Step 3: I ask the class to predict the wavelength of the 1 Hz wave on this machine. The first answer that someone always spews: "43." That student looks sheepish for a moment as he realizes he didn't put units on. He quickly computes the units: in his mind, the frequency was "1", so 43 cm/s divided by 1 = 43 cm/s.
Um, so a wavelength has units of cm/s now?
[Don't think I'm being silly or hyperbolic here. This happened in all three sections yesterday.]
So finally we all think carefully, and agree that 43 cm/s divided by 1 Hz gives 43 cm for a wavelength -- the units of seconds cancel in the improper fraction, and a wavelength has to be a length, anyway.
Step 4: What good is a prediction without verification? We must measure the wavelength.
For years, and in my first section yesterday, we've had a student use his hands to estimate a peak-to-peak distance between the waves, and then use a meterstick to measure that distance between his hands.
Yesterday, someone suggested taking a picture of the wave machine with a meterstick nearby. "Just read the peak-to-peak distance right off of the meterstick in the picture," he said. Certainly I've heard that suggestion before, and it's most definitely an appropriate method. But until now, I've never had the technology to get this done in class.
I whipped out the iphone, and... handed it to a student. What, you think that I can handle newfangled technology better than a whippersnapper? Bah.
I created the 1 Hz wave, and the student took the picture. What he actually did was take a video, then play back the video to capture a still shot when the wave was in a good place. He emailed the photo to me; I got the email, and put the photo up on the screen in my classroom. No more than about 3-4 minutes elapsed. That's still too long, but I'll bet with practice I can bring that time down.
The picture we took is at the top of the post. Problem: it's awfully blurry, so much so that we can't see the markers on the meter stick. The iphone camera is convenient, but not so, um, good. Nevertheless, we can make a quick estimate -- the peak to peak distance is clearly less than half a meter, but more than a quarter meter. That's as good a measurement as I used to get with a student holding his hands above the wave.
After class, I imported the picture into Microsoft Paint. I measured several times, to get about 680 pixels from one end of the meterstick to the other. I found the peak to peak wave distance (again, using several measurements) to average 270 pixels. Dividing the scale out, I get a wavelength of... 40 cm. Difference predicted to measured wavelength: 14%. Not bad for a blurred meterstick.