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Re: [Phys-L] highway mirage



/arm waving ON/
Considering the finite physical extent of a wavefront, the portion exposed to a smaller index advances wrt the remainder.
/arm-waving OFF/

Brian Whatcott Altus OK

On 4/8/2014 12:44 PM, Carl Mungan wrote:
There seems to be a problem with typical textbook discussions of the highway mirage of a pool of water appearing on a hot road. If the index n monotonically decreases as we drop in altitude toward a hot road, shouldn't a ray of light from the sky bend toward 90 degrees (so that it is traveling parallel to the road)? What bends it upward?

One fix I have heard of is to say that once the ray gets close to being horizontal, it reaches the critical angle and totally internally reflects off the warmer layer of air below it.

Sounds good at first, but that explanation seems as flawed as the textbook discussions on closer thought. The equation for critical angle is ArcSin[n2/n1]. If n2 = n1-dn, then the critical angle approaches 90 degrees as dn -> 0 for a continuous function. There is no single well-defined critical angle unless you have layers of air with steps in the index between them.

So who has a better fix? I suppose we could get small discontinuous steps due to fluctuations.

Or perhaps one could say that even before it reaches the critical angle, more and more of the light begins to reflect as the angle approaches 90 degrees. Again this sounds reasonable, but I'd like to see it quantitatively checked before I accept it. It should be experimentally testable, because it suggests that a single incident ray will gradually reflect into a fan of rays.

Finally I suppose one could abandon ray optics and consider distortions of the actual wavefronts.

Discussion?