Re: approximate Huyghens' construcion

[John Denker <jsd@MONMOUTH.COM>]

At 09:29 AM 9/25/99 -0500, Cliff Parker wrote:
> isn't the pattern of energy, and the lack there of,
> a result of not two, three or even four or five coherently derived waves but
> rather the result of virtually an infinite number of waves all converging at
> the same time?

1) It's OK to say there are an infinite number of contributions, but...

2) One shouldn't insist on infinity.  A big part of real-world physics (and
real-world life in general) revolves around making sensible approximations.

For Huyghens' construction in particular, once you have one or two
re-radiators per radian of wavelength across the slit, adding more makes
only small improvements in the accuracy of the result.

You can check this assertion by brute force:  Imagine a slit two
wavelengths across.  (That's about the smallest slit that is wide enough to
give a nice-looking diffraction pattern.)  Construct 5 re-radiators across
the slit.  Calculate the voltage and power radiated for various angles of
outgoing rays.  Redo the calculation with 10 re-radiators.  Observe that
the answer changes, but not very much.  Redo it with 20.  Observe
practically no change.

To save you the trouble of grinding out the numbers, I've prepared a handy
spreadsheet:
  http://www.monmouth.com/~jsd/physics/diffraction.xls
that leaves nothing to the imagination.

You can change the number in cell C20 to be 20 or any divisor thereof.
Watch what happens to the graph when you do.

Details:  In the big arrays, each row corresponds to a re-radiator
somewhere along the width of the slit.  Each column corresponds to an
outgoing ray angle.  Each cell is the contribution from that re-radiator to
that ray.  There are two arrays: one for the sine phasor component and one
for the cosine phasor component.

> When I describe something like
> this I do tell students that if they ever want to more fully work with these
> ideas they will have to learn the calculus

Again, I'm not going to say anything bad about calculus -- but calculus is
not necessary for understanding the physics of diffraction or Huyghens'
construction.

Qualitative understanding is more important than grinding out lots of
insignificant digits.

______________________________________________________________
John S. Denker                                jsd@monmouth.com