Rain, Rain, Go Away

I've been trying to get this post done for the past couple days, but I've been feeling some of the fatigue I was warned about with the radiation, so I was a bit delayed.

It was another rainy trip in and out of Boston Thursday.  Steve's car has the interesting feature of wipers that automatically adjust their speed, using a sensor along the top of the windshield.  Watching them speed and slow with the rain got me thinking: the faster you drive, the more rain you'll hit, but the extra speed also drives rain up the windshield and into the sensor.  Do the two balance each other out, or will the wipers end up going too fast or too slow?

The most difficult part of this will be figuring out how fast the rain slides up the windshield, so we'll handle that first.  Once the rain hits the windshield, it's going the same velocity as the car, and it's the oncoming wind that will push it up.  The force of the wind drag is given by

where ρ is the density of air, v is the velocity of the water drop relative to the wind (basically the car's velocity), C is the drag coefficient, and A is the cross-sectional area of the drop.  To simplify things, we'll assume gravity is much weaker than this and take it to be the only force on the drop.  Then the distance it moves up the windshield after a time t is

where m is the mass of the drop.  If the drops are uniformly spread over the windshield, then on average they'll need to travel up half of it to get to the sensor.  This gives a rate of

where d is the length of the windshield.

Suppose the number of drops that have entered the sensor after a time t is n(t).  Then the rate of drops is

where rr is the rate of rainfall directly into the sensor.  The density of rain in the air (drops per volume) is

where vr is the velocity of the falling rain and As is the area of the sensor.  Using this, we can find the rate drops fall on the windshield

where θ is the tilt of the windshield.

Comparing to the rate the drops are entering the sensor, we can see that both are linear in v with a constant offset, meaning that for sufficiently high velocities, the relationship should work out evenly.  Clearly those Volvo engineers knew what they were doing...