Re: [Phys-l] Sun's image due to reflecton by not a pinhole.

[John Denker <jsd@av8n.com>]

On 04/30/2009 03:34 AM, chuck britton wrote:
> I had students outside on a sunny day with round and/or rectangular  
> mirrors reflecting sunlight onto the brick wall.
>
> Some wondered why square mirrors made a round spot of light.
>
> Some didn't care (of coarse).
>
> If the 'screen' is CLOSE to the mirror - the 'spot' of light is square.

OK.

> Is this 'correctly' viewed as near-field and far-field domains??

I wouldn't have said that.

It seems much simpler to describe it in terms of a convolution.
If this were a spectrometer, we would say that the "intrinsic
lineshape" (round) is convolved with the "instrumental lineshape"
(square).  The concept of convolution is so very broadly useful 
that it is worth spending some time on.

There are lots of pedagogical models you can use to help get the
students up to speed.  For example, compare a square array of
small round things to a round array of small square things.  

You can also collect additional partially-independent data by
using other instrumental shapes.  For example, use tape to put 
a "black belt" on some of the square mirrors:

          M M k k M M
          M M k k M M
          M M k k M M
          M M k k M M
          M M k k M M

where "M" stands for mirror and "k" stands for black.  The idea
is to emphasize the difference between the intrinsic shape and
the instrumental shape.

Also be sure to do the intermediate case, not just the asymptotic
near case and the asymptotic far case.

In a non-calculus course you can define convolution without 
mentioning integrals, but if any of the students have heard of 
integrals you should make that connection.

If they've done any AC circuits you should mention that the
input/output relationship of a simple RC circuit is a convolution.

Another simple, familiar example: the potential due to given distribution 
of charges is a convolution.  Some guy named Green had something to say 
about this, several decades before Maxwell wrote down the Maxwell equations.
And the importance of Green functions continues to this day, with
innumerable applications from quantum field theory to civil engineering.
I get multiple hits from
  http://www.google.com/search?q=%22green+function%22+site%3Anobelprize.org