Since the velocity is defined as the time-rate of change of position, the slope of this plot should give the velocity. That puts this wave speed at 2.85 m/s, which is pretty close to the theoretical prediction. I’m happy with that.

But what if I want to look at the speed of a wave in a giant metal chain, instead of a string of beads? I actually don’t have one of these things lying around—and I probably couldn’t move it anyway. So let’s build a computational model.

Here’s my idea: I’m going to let the chain be made of a bunch of point masses connected by springs, like this:

A spring exerts a force that is proportional to the amount of stretch (or compression). This makes them very useful. Now I can look at the positions of all the masses in this model and determine how much each connecting spring is stretched. With that, it’s a fairly simple step to calculate the net force of each mass.

Of course, with the net force I can find the acceleration for each piece using Newton’s second law: F_{net} = ma. The problem with this spring force is that it’s not constant. As the masses move, the stretch of each spring changes and so does the force. It’s not an easy problem. But there is a solution that uses a bit of magic.

Imagine that we calculate the forces on each mass of this modeled series of springs. Now suppose that we just consider a very short interval of time, like maybe 0.001 seconds. During this interval, the beads do indeed move—but not that much. It’s not a huge stretch (pun intended) to assume that the spring forces don’t change. The shorter the time interval, the better this assumption becomes.

If the force is constant, it’s not too difficult to find the change in velocity and position of each mass. However, by making the problem simpler, we’ve just made more problems. In order to model the motion of the beaded string after just 1 second, I would need to calculate the motion for 1,000 of these time intervals (1/0.001 = 1,000). No one wants to do that many calculations—so we can just make a computer do it. (This is the main idea behind a numerical calculation.)