Wave pulses


In this page, we consider the motion of a single pulse moving along a taut elastic string. This is the simplest example of wave phenomena and will allow us to clarify some of the basic concepts.

We will work with a model of the string as massive beads connected by stretched (but massless) springs. Although in the actual string every part of the string is playing both an inertial and an interaction role, the beaded spring model allows us to separate these functions and talk about them in a less confusing way. If you like, you might think of these "beads" as representations of the molecules of the string and the "springs" as visualizations of the attractive interaction force between the molecules. (Of course, then the picture would be messier and not so one-dimensional.)

The single pulse

Let's imagine that we have a demonstrator who has a long elastic string (or spring) connected to a fixed point far away. She then walks back from that fixed point, pulling the string until it's stretched tightly as in the figure below. We've put in a "..." to indicate that the wall on the right where the string is attached is very far away and we will not consider (for now) what happens when our pulse reaches the end of the string. We will also ignore the force of gravity.

[Technical comment: The effect of gravity is simply to curve the string downward in the middle slightly. The curvature adjusts so that the angles of the springs on the two sides of any bead are not exactly straight but adjust so that their sum cancels the force of gravity on the bead. Our considerations are then about the changes from this curve. Or, if you prefer, we can imagine we are looking from above at the string stretched on a frictionless table and all the motions are horizontal.]

Now imagine that the demonstrator quickly moves her hand up and down.

[Technical comment: Why quickly? Compared to what? What we are studying here is the propagation of signals along this string. If she moves the end of the string very slowly — slow compared to the time it takes a signal moving along the string to get to the end — then the whole string would move up and down in a straight line as if it were a rigid object. The whole issue here is about the propagation of deformations through an extended object. Even a rigid body actually responds to forces by creating deformations that propagate through it at a high speed (the speed of sound in the object). Calling an object "rigid" means that we are looking at it at time scales where signals propagate through it in a negligible time.]

What we know happens (from watching the video of the pulse on the stretched spring) is that a bump is formed in the string and it moves along the spring without changing its shape noticeably — a wave pulse.  Here's a picture of what the pulse will look like in the bead & massless springs model:

Each bead goes through the same motion as the hand did, just a little later in time, the farther it is down the string. Each bead responds to the net forces of the pair of springs to either side — and that's all! (See Newton's 0th law.)

Here's an interesting question: The pulse in this photo looks symmetric. If the pulse is moving to the right, then the bead immediately to the right of the peak should be moving up (it will be the peak next) and the bead immediately to the left of the peak should be moving down (the pulse is past it). But the forces that they feel are both down and, since we are ignoring the small left-right components of the forces, the forces are the same. Why should they move differently?

The answer, of course, is that the forces don't change position (create velocity), they change velocity (create acceleration). (See the "net force" discussion in Reading the Content in Newton's 2nd Law.) Since each of the beads are moving in different ways the effect of the forces on them is different. The bead to the right of the peak is moving up, and the force is slowing it down, while the bead to the left of the peak is moving down, and the force is speeding it up.

What's moving?

The situation here is way trickier than it looks. On the surface it looks simple. There's just a pulse moving along a string? What's hard? The issue is, what's moving? Each bead on the string just moves up and down — just like the bead held by the demonstrator's hand that instigated the motion. It looks like something is being carried down the string — and there is! But it's energy and momentum that is being transported, not mass.

Stadium crowd performing "the wave"
at the Confederations Cup 2005
in Frankfurt. Source: Wikemedia Commons.

The beads only move up and down. It's the pattern of motion that's moving down the string. It's like doing the wave in a stadium. You go up and down. The "wave" moves from one person to the next.

It's clear that energy is carried down the string. Initially the beads at the beginning are moving (have KE), finally the beads at the end are moving (have KE). So kinetic energy is transported down the string even though no matter moves down the string. What is even more peculiar is that momentum is also transported. It's actually a pattern of momentum since the total transported momentum is 0. But first a bead moves up, then down. And the momentum that is transported is transverse — perpendicular to the string and to the motion of the pulse down the string

Workout: Wave pulses


Joe Redish 3/27/12


Article 684
Last Modified: June 1, 2019