P****d offFirst a warning—LADIES, this post is not for you. You don’t have the problem, so move on to another post, there are plenty of interesting ones further down.
Right—I hate urinals! I refuse to use them unless there is nothing else and I am bursting to go. Why? Next time you need to use one, try this experiment. Do your business as usual and when you have zipped/buttoned up take one pace backwards. Now take both hands and rub them up the inside of your trouser legs from knee to groin. Now wash your hands because you certainly need to.
The problem is a fine mist splash-back, even with my late middle age feeble stream. I don’t know of any solution so I always use a cubicle. It is a shame that is uses far too much water; few have the water saving half flush as they think you are in there for “bigger things”.
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Fred again, Just a tip from a 65, don’t pee straight at the flat surface but rather at a very acute angle to it, preferably a side wall or try for the drain hole, practice makes perfect and at night, at home, sit down especially if much wine has been drained from the box/cask.
Yes, this makes an interested applied maths problem. As Fred has said, part of the problem is the angle of incidence. If the pee is hitting the service at a perpendicular angle, then:
a) There is a high likelihood for a direct perpendicular reflection
b) If this happens, the pee droplets have a minimum distance to cover before hitting the trouser-leg
c) The effect of gravity upon the pee droplets is minimal
With a more acute angle of incidence, not only does the reflected pee bounce downwards (like a snooker ball would), but it’s also got a greater momentum acting in same plane as acceleration due to gravity (I think you’d need to do x sin theta to work this component of the forces out).
As you’re married to a mathematician, it would have made an interesting A-Level question to pose to the pupils in the school, but may also have risked hastening her retirement!!
In the USA, of course they’ve already worked this out. The plumbing is built to withstand thermonuclear attack, and the urinals emerge forth from the wall so that a “trough” is located between the legs of the user. This makes possible an angle of incidence of zero – ie a vertical pee in the plane of any possible reflecting surface. Any reflected droplets have to work against acceleration due to gravity and would have to travel a good bit further anyway before encountering any fabric of the trousers…
Disregarding questions of lamina or turbulent flow (aerospace immersion for 35 years) with the more acute angle you get the inevitable problem of the permanent puddle in the bottom which creates splash – so no win there I’m afraid.