ok,.. so the wave equation comes from local forces (2nd derivative) having to cancel out? I prefer thinking in terms of symmetries or conservations instead of equations but I guess it's the same thing?No, it has become trivial! Once you know it's a wave equation everything follows very simply! All the results about the bottom, gravity, etc. Had it been a diffusion equation, or elliptic, then the bottom would have fallen (out of this brain teaser) immediately!
The wave equation has to be solved, as outrun says, to find out what happens where all the action is. In the solution I found it looked like it was initially not fully compressed at the top. I didn't believe my result but if your observation are correct maybe it's not as bad as I thought!Hmmm... The strange thing is that the actual frame-by-frame dynamics are more complex than just a simple collapse from the top down. During the first 1/3 to 1/2 of the drop, the top portion is only partially compressed. It's only at about frame #36 that the top of the slinky is in a fully compressed state that is rushing down collapsing all the loops below it.
Good to know. This is not the same as your post of July 23.. where u state ..argh.. you're talking about the balls now, right? Paul is applying the wave eq to the slinky.
The track is not a Brachistochrone
There are *two* brainteasers in this thread. That in itself turns out to too complex for some to handle!Good to know. This is not the same as your post of July 23.. where u state ..argh.. you're talking about the balls now, right? Paul is applying the wave eq to the slinky.
The track is not a Brachistochrone
Yes, the question is: why is the wavy one faster then the flat one?
Have the requirements changed?
I think we see a different coordinate that x? x is the number of cycles? It need to be converted to stretched factor (=force) and then integrated to get actual distances between points?image1.JPG
The angled lines are the characteristics, think of them as lines along which information flows.
Yes, x is the distance measured in hoops. So 0.5 means paint the middle hoop red and follow its path. It won't, of course, remain in the "middle"! The distance of the point x=0.5 from the point at which the top was being held is d(0.5,t) and it's d(x,t) that you solve for. No integrating as such.I think we see a different coordinate that x? x is the number of cycles? It need to be converted to stretched factor (=force) and then integrated to get actual distances between points?image1.JPG
The angled lines are the characteristics, think of them as lines along which information flows.