Example: Cancelling pulses

Understanding the situation

In order to get some practice with how pulses combine when they overlap, we'll consider a highly simplified toy-model problem: square pulses. For waves on a string, this doesn't actually work well because the string doesn't like to make sharp corners. (It actually works perfectly for digital signals.)

But we'll ignore those problems and use it to illustrate how the basic "algebraic addition of signals" idea works. Following the idea of using video frames as data will help you make a concrete matching of the physics to the graph.

Presenting a sample problem

Two almost rectangular pulses approach each from opposite directions on an elastic string. The pulses have the same symmetric shape, but the one on the right is reversed. A video tape of the string is taken of the motion of the pulses. A few frames before the pulses reach each other (frame number 1237), they look like shown in the figure below at the left. At the instant when the pulses are right on top of each other, a frame from the video tape (frame number 1287) looks like the figure below at the right. This frame shows that the they cancel perfectly at that instant.

The video tape keeps rolling. What do you think frame number 1337 looks like? Draw a sketch and explain why you think so. Make sure that your explanation includes an "analysis of the mechanism," that is, your explanation includes a discussion of how the various bits of string "know what to do" to get to the state you have described.

[Note: In actual fact, a string cannot support sharp edges. The rises and falls of the pulses will actually be somewhat rounded off. We have described the pulses as sharp-edged in order to simplify the analysis. If you prefer, you may assume that the pulses are slightly trapezoidal rather than rectangular, but assume that the width of the rise and the fall is narrow compared to the width of the overall pulse.]

Solving this problem

Since the pulses don't interfere with each other but just keep moving, frame 1337 will look like this:

That is, the individual pulses will just keep going — traveling on at the same speed, ignoring the other pulses that might be passing through. This is a "macro" description — on the level of the rules for the patterns and how they travel. To make a connection with a "micro" description — on the level of bits of string and how they each move in response to the forces they feel, we will have to go a little deeper. 

The difficult question is: If the string appears totally flat in frame 1237, why should it do anything? Why doesn't it just stay flat? The answer is that according to the rules of Newtonian mechanics, you can't tell what an object is going to do by looking at where it is. A single video frame of a thrown object won't tell you if the object is on its way up, on its way down, or is at the top of its trajectory. You have to look at more than one frame and extract a velocity. If we look at frame 1236 as well as 1237 we might see something like this:

This tells us that when the string looks flat, the "beads" (bits of string) at the ends of the combined pulses are not stationary. One end is moving up and the other end is moving down. The velocity pattern will look something like this: 

These are the same velocities that are seen at the beginning of each pulse as they travel through the medium alone so this tells us that this motion of these beads will regenerate the pulses ahead of them. (What happens in between is a bit more complicated, but we can see by symmetry that staying flat is plausible for the string in between.)

This can be proved mathematically by using the wave equation and showing that the general solutions are of the form of pulses traveling in two directions: $y(x,t) = f(x-v_0t) + g(x+v_0t)$ for arbitrary functions $f$ and $g$. This shows that the functions don't change when they happen to overlap — they just add.

Joe Redish 2/11/2002