Prismatic Spray

It was raining as we were driving into Boston the other day, and the cars in front of us would send up blinding sprays of water droplets on the highway.  It got me thinking about the fact that, by themselves, water and air are each basically transparent, but mixing them into a mist makes them nearly opaque.  This is caused by refraction – when light crosses the boundary between two materials, it bends, so sending it through thousands of small droplets bends it many times, and spreads it out.

I tried to figure out an exact form for the degree of bending for a small drop, but that ended up being too difficult, so I used a geometry program called GeoGebra to make the construction I was looking for.  I collected a couple points and fit a parabola:

The x-axis is the distance off-center that the light hits the drop, and the y-axis is the degree of bending as a fraction of pi.  The equation for the parabola is

so for a large beam of parallel light entering a drop, the exit path looks like this:

 

Averaging over all y values gives a bend of 0.11π.  

If we assume the air/water mixture was about 10% water, then we have one 1 mm^3 drop in every 10 mm^3 of space, or approximately one drop every 4 mm.  If some light enters a drop, given our average bending it will spread out over the 4 mm to cover approximately 3 times the original space.

To get an idea of what this looks like, I took an image,

and divided it into a 5x5 grid.  Each grid square I divided into 3x3 and expanded the middle square to fill the larger square.  Mixing this with the original gives

This is a pretty coarse approximation of what's really going on, but it shows how quickly this process can make an image indistinct.