Chain windshield washing, where the spray of a car's
wiper jets splatters the car behind's
windshield forcing that driver to use their wiper jets, and so on, and so on, can be best observed on a road with moderate
spray but definitely a lot of intense
sunshine.
Bright sunlight causes any marks on your windshield to appear completely opaque. The headlights of oncoming cars at night also causes this. Suffice to say, it's very annoying and also dangerous to drive with an obscured view. Note that marks on the inside can also turn opaque, but those are harder to remove and not part of this node.
The problem is, the few flecks of water will just splatter your view, and a flick of the wipers will smear it everywhere. The only solution is to provide enough moisture for the wipers to do their job, and that means using your own jets. Unfortunately, the speed of your car and force of the jets means a few flecks will miss, and...
There are ways to break the chain, or avoid starting it altogether. If you have a newer car, your wipers might well be automatic, meaning you push a switch and it sprays then wipes three or four times. There's very little you can do to avoid a massive spray in this case, it's down to the enthusiasm of your washer system. If your car is a bit older, the jets will likely only spray as long as you hold the switch. A quick push of the switch, then, should only emit a small amount of washer fluid, which, if you wait, should flow up your windshield. As it nears the top, flick the wipers on... with any luck, your view will be cleared, but no liquid will have travelled beyond the windshield.
Another solution is to avoid the spray! Whether this means changing lanes (be careful!) or leaving more space between you and the spraying car in front is up to you. If you're really nice, you could conceivably move so there's no-one behind you to catch any errant droplets.
This phenomenon does not affect rear-window wiper jets, as the airflow tends not to catch the spray. However, any station wagon driver can tell you that driving in the wet means a constantly obscured rear window, and a filthy tailgate.