Mathematics 309  spring 2003  assignment #3
This assignment is about rainbows.
It is due by class on Friday, February 14.
The programms should be handed in by email,
the mathematics by paper or email.
A ray of light entering a drop of water is first refracted, then reflected,,
finally refracted again as it leaves the drop,
as you saw in Assignment #2.
The acute angle between the entering and exiting rays is
called the deflection angle, which I'll call d.
It depends entirely on the incidence angle i
of the ray where it hits the drop, or equivalently on
h = sin i, the relative distance of the ray
from the center of the drop.
The problems
(1) Find a formula for d
as a function of i,
also of h. Let n
be the index of refraction.
Hint:
first find a formula for d
in terms of the refraction angle r,
then plug in a formula for r.
Plot the graph of d versus h
for red light, in which case n = 1.33.
You should do the graph in a PostScript program.
(See the sample program below.)
Extra bonus for explaining the formula clearly.
Note: you should understand well why this formula works,
because sooner or later I will ask you
to do a similar calculation quickly.
(2)
As i
goes from 0
to 90^{o},
the deflection goes from 0 up
to a maximum
and then down again.
Use calculus to find a formula for this maximum.
Find it explicitly when n = 1.33.
(3)
If an observer on the ground watches rainfall
opposite a sunset, he will see a rainbow. Why?
The rays of light entering his eye will
come from raindrops with different deflection.
He is not seeing
the drops, but the rays that have come through the drops.
In any particular direction,
he is seeing the rays with a given deflection. Illustrate
that the great preponderance of the rays he sees are those
whose deflection is close to the maximum, by drawing on
a circle around the observer the drops
that he sees at a given angle,
in the following way.
Choose an integer N, and divide up
the interval [0, 1]
into N intervals, corresponding to N+1
rays entering the drop. Calculate the deflection d
of each of these, and plot on the circle
at angle d a small dot at that angle.
Then draw the radius from that dot to the observer.
Do this for N = 10, 100, 1,000.
All in PostScript. Make the pictures fairly large.
How to draw function graphs in PostScript
The only new ingredient is defining the function to
be graphed as a procedure with
one argument x that leaves f(x) on the stack.
Click on image to see PostScript source.
Internet references on rainbows
None is quite so clear as it should be.
