wendt@arizona.UUCP (09/22/83)
In regard to Bill Hughes refutation of my claims about trapezoidal holes: The number of trajectories emerging from the wide side of the funnel is the same as the number emerging from the narrow side, but their paths are different. Trajectories emerging from the narrow side are fairly well distributed in angle throughout 180 degrees. Those emerging from the wide side are more bunched together. Assuming random distribution of directions at some distance from the hole, it will be more likely that a particle on the narrow side is on a trajectory through the hole that it will be that a particle on the wide side is on a trajectory through the hole. For example, a particle which is close to the wall on the wide side of the hole can't possibly be on a trajectory which will send it through the hole. Whereas a particle anywhere on the narrow side can be on a trajectory through the funnel. True, the particle on the wide side mentioned above could collide with a wall on the opposite side of the box and be reflected through, but so could a particle on the narrow side. Anyway the model assumes random distributions at some distance (mean free path) from the hole. The intent of the model is to show that random distributions at this distance map into non-random distributions at closer distances. It's invalid to *assume* random distribution of particle directions closer to the funnel than mean free path. By way of more explanation: _____________________________________________________________________________ | |####| | | |####| | | (this one can get through). |####| . (this particle cant) | | |####| | | |####| | | |####/ | | |###/ | | |##/ | | |#/ | | |/ | | | | |\ | | |#\ | | (not to scale) |##\ | | |###\ | | |####| | | |####| | | |####| | | |####| | | |####| | ----------------------------------------------------------------------------- Think of this as a pool table with a couple of wedges set up in the middle. Forty-five degree angles are fine. We ran simulations starting from some fixed distance from both sides of the wedge, aiming in random directions, and counted the ones that bounced back in and the ones that got through the funnel. We got the ratio as high as three-to-one by tweaking the geometry. We didn't consider reflections off the back or sides of the pool table, and we considered the particle to have zero size. We did consider reflections off the sides of the funnel, and assumed angle i = angle r. What happens is this: coming from the narrow side, any particle that hits the aperture at any angle from 0 to 180 gets through. So the "effective width" of the narrow side is the narrow aperture diameter times 180 degrees. The "effective width" going the other way is the angle made by the funnel times the narrow aperture -- it has to get through both. The funnel sides raise the effective width of the wide side somewhat by sometimes bouncing particles toward the aperture, but not nearly enough to compensate -- most of the time that a particle hits a side of the funnel, instead of going through it gets bounced out. If you try to imagine a particle getting bounced through the aperture, you'll see how glancing a shot it would have to be in order to get bounced through. Lew Mammel posted an article purporting to refute this claim but I didn't understand it. Perhaps he would explicate. Alan Wendt