Since the rate is outlined because the time-rate of change of place, the slope of this plot ought to give the rate. That places this wave pace at 2.85 m/s, which is fairly near the theoretical prediction. I’m proud of that.
But what if I need to take a look at the pace of a wave in an enormous metallic chain, as an alternative of a string of beads? I truly do not have considered one of these items mendacity round—and I most likely could not transfer it anyway. So let’s construct a computational mannequin.
Here’s my concept: I’m going to let the chain be manufactured from a bunch of level plenty linked by springs, like this:
A spring exerts a pressure that’s proportional to the quantity of stretch (or compression). This makes them very helpful. Now I can take a look at the positions of all of the plenty on this mannequin and decide how a lot every connecting spring is stretched. With that, it is a pretty easy step to calculate the online pressure of every mass.
Of course, with the online pressure I can discover the acceleration for each bit utilizing Newton’s second legislation: Finternet = ma. The downside with this spring pressure is that it is not fixed. As the plenty transfer, the stretch of every spring modifications and so does the pressure. It’s not a straightforward downside. But there’s a resolution that makes use of a little bit of magic.
Imagine that we calculate the forces on every mass of this modeled collection of springs. Now suppose that we simply take into account a really quick interval of time, like perhaps 0.001 seconds. During this interval, the beads do certainly transfer—however not that a lot. It’s not an enormous stretch (pun meant) to imagine that the spring forces do not change. The shorter the time interval, the higher this assumption turns into.
If the pressure is fixed, it is not too tough to seek out the change in velocity and place of every mass. However, by making the issue easier, we have simply made extra issues. In order to mannequin the movement of the beaded string after simply 1 second, I would wish to calculate the movement for 1,000 of those time intervals (1/0.001 = 1,000). No one needs to try this many calculations—so we are able to simply make a pc do it. (This is the principle concept behind a numerical calculation.)