Modeling an Elastic String

A Derivation of the Wave Equation

Colin Mitchell

We want to be able to model the action of an elastic string over time. Our general model will look like:

We want to apply Newton's law to this string. So first, we shall consider a tiny element of the string:

Here, we are looking at the interval at time t. We will define to be the mass density at the point x on the string, and we will also define to be the tension in the string at point . Finally, we will define to be the angle between the string and a horizontal at , and note that

We can note that the mass of this small piece of the string is

.

The forces that we have acting on this piece of the string are the tension to the right and the tension to the left, which have magnitudes and , respectively. Also, these forces act at angles and , respectively. So we can write these into Newton's Second Law of Motion. We know that

.

Also, the vertical forces are formed from the tension placed on the string, and so we use to find the vertical aspect of the forces, and combine them to get

,

because the left side of the string element wants to move downwards, and the right part of the string element wants to move upwards. So the vertical component of Newton's Law becomes

If we take the limit as , we notice that , and so the left-hand-side becomes

.

Also, we notice that

.

So the PDE becomes

For this model, we want to produce only small vibrations in the string, common in musical instruments. By small vibrations, we mean that the magnitude of the peaks of the vibrations are small, and so we realize that will also be quite small. This implies that , and hence , will also be quite small. From this, we can reduce the PDE by finding that

And so

.

Let's now consider the horizontal component of Newton's Law. We assume for our model that there are only transverse vibrations, and so the string does not move horizontally, but only vertically. So we know that the total horizontal force must be zero. We again note that the tension on the left of the string element wants to move leftwards, and vice versa, so we can express the horizontal forces as

Dividing by , and taking the limit as , yields

We again note here that since we are modeling only small vibrations, we find that is close to 1, and that is close to 0, because there will not be much stretching going on the in string, relatively. So what we find is that T will be a function only of time, as the overall tension will be governed by the tension put on the ends of the string, which is a function of t. Since we know this, we can further reduce the PDE to

For simplicity, we will consider the string to be homogenous in density in this model, and also that T will be constant. So it is obvious that the boundary conditions will be , and that the initial position condition will be , and we will need an initial velocity to take care of two constants of integration.