# Waves on an elastic string

#### Prerequisites

The physics system we will use as our introduction to the mechanics of waves is the long elastic spring or string. A long spring is shown in the figure at the right. By an elastic string, we might mean a guitar or piano string. Although the latter are more interesting, since the speed of waves on guitar and piano strings are typically too fast for us to see with the naked brain, we'll focus on the spring: a system where we can actually watch what's going on in real time. Previously, we considered our spring simply as a source of force and neglected its mass. Now we'll consider its own motion.

## What happens?

Typically, the first thing we like to do when considering a new phenomenon (choosing a channel on cat television) is to observe it and see what happens. What do we know about it from our everyday experience, if anything? What I might expect is, that if I pull the spring taut and give it a flick with my hand, it will send a pulse — a "bump" running down the spring.

My first guess is that the bump will decrease in size as it goes, showing some kind of damping, and suggesting that we might have to put that into our very first models. Let's see what it looks like. The video below was taken at the University of Maryland by Michael Wittmann with a taut spring (and a fast camera). Run it through and then look at it again, stepping it a frame at a time. (You can step back and forth with the left and right arrow keys.) Click on it to open the video in a new window.

Surprised? There is no detectable change in the size or shape of the pulse, at least for a travel over this distance. That suggests that our first models can neglect damping. On typical tightly stretched springs or strings pulses can travel distances that are very many times bigger than their size before they change noticeably. So in our first model we will ignore damping.

## A model of the spring

It's rather hard to think about the motion of a stretched spring. Each part of the spring plays multiple roles — both exerting forces on the neighboring bits of spring and having mass that provides the inertia of the motion. In order to clarify how we can talk about what's going on, we'll make a toy model of the spring that consists of two kinds of objects — beads (masses that have inertia) and massless connecting springs (that provide the forces). Really, all parts of the spring play both roles, but this makes it much easier to talk about. Here's what it looks like:

The demonstrator is shown holding the first bead on the left. The ... on the right means the string goes on for a long time before it is tied down somewhere. The demonstrator is pulling on the bead so that the springs are all stretched a bit and there is tension. Every bead is pulled equally from both sides.

## Creating a pulse

If the demonstrator raises her hand the first bead will move up and the forces on the second bead will no longer be in the same direction so they will unbalance. The forces will look something like this:

The free-body diagram for the second bead is shown at the right. The resultant force will be nearly vertical.  For small displacements, we can show that:

• the main effect comes from the unbalanced directions, not from the extra stretching of the spring on the left and
• the sum of the two vectors is almost vertical.

We will assume that the motion of all the beads will be vertical and that we can treat the tension as constant even though there is a little bit of extra stretching going on. (Remember that the static position of the string is taut — all the springs are stretched even at rest.)

[Technical comment: These are results of the small angle approximation. The extra length of the spring comes from a factor of cosine so the extra length is proportional to the square of the angle. The same is true about the failure of the horizontal components to cancel — since the horizontal component goes like the cosine, the difference is proportional to the square of the angle. But the vertical component comes from the sine so the difference is proportional to the angle. For a small angle (measured in radians), $θ >> θ^2$. And these angles tend to be quite small — especially in a guitar string.]

If she now lowers her hand quickly, the first bead will pull the second one up, then down, and the second bead will pull the third one up, then down, etc. along the string. Each bead will reproduce what the hand is doing for the next bead along, passing the motion that was imposed on it to the next.

As we saw in the video, the shape of the pulse is passed along without change and appears to move at a constant velocity. As a result, as the pulse passes each bead, it will "ride" the shape of the pulse up and down, mimicking what the hand did to create the pulse in the first place.

In the follow-on pages, we will analyze the motion of a pulse in detail, seeing how to represent its motion mathematically, how to separate the motion of a bead from the motion of the pulse, and what governs the speed of the pulse.

Joe Redish 3/26/12

Article 683