Since speed is defined as the speed of change of position, the slope of this graph should give the speed. This places this wave speed at 2.85 m / s, which is very 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 chain of beads? In fact, I don’t have any of these things out there, and I probably couldn’t move it anyway. So let’s build a computational model.
Here’s my idea: I’ll 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 stretching (or compression). This makes them very useful. I can now look at the positions of all the masses in this model and determine how much each connecting spring stretches. With this, it is a fairly simple step to calculate the net strength of each mass.
Of course, with clean force I can find the acceleration of each piece using Newton’s second law: Fnet = ma. The problem with this spring force is that it is not constant. As the masses move, the stretch of each spring changes and so does the strength. 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 we consider only a very short time interval, such as perhaps 0.001 seconds. During this interval, the pearls do move, but not so much. It is not a big deal (pun) to assume that the forces of the spring do not 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, making the problem easier, we just created more problems. In order to model the motion of the bead string after only 1 second, you should calculate the motion for 1,000 of these time intervals (1 / 0.001 = 1,000). No one wants to do so many calculations, so we can only make one computer do it. (This is the main idea behind a numerical calculation.)