Since velocity is defined as the rate of change of position over time, the slope of this plot should yield velocity. This puts this wave speed at 2.85 m/s, which is quite close to the theoretical prediction. I’m happy with that.
But what if I want to watch the speed of a wave in a giant metal chain, instead of a pearl necklace? I actually don’t have one of those things lying around – and I probably couldn’t move it anyway. So let’s build a computer model.
Here’s my idea: I’m going to let the chain be made up of a set of point masses connected by springs, like this:
A spring exerts a force 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 of each part using Newton’s second law: Freport = my. The problem with this spring force is that it is not constant. As the masses move, the stretch of each spring changes, as does the force. It’s not an easy problem. But there is a solution that uses a bit of magic.
Imagine that we are calculating the forces on each mass of this model series of springs. Now suppose we are just considering a very short time interval, like maybe 0.001 seconds. During this interval, the beads do move, but not that much. It’s not a stretch (pun intended) to assume that the spring forces don’t change. The shorter the time interval, the better this assumption.
If the force is constant, it is not too difficult to find the change in speed and position of each mass. However, by simplifying the problem, we just created more problems. In order to model the movement of the beaded string after only 1 second, I would need to calculate the movement for 1000 of these time intervals (1/0.001 = 1000). Nobody wants to do so many calculations, so we can just get a computer to do it. (This is the main idea behind a numerical calculation.)