wendt@arizona.UUCP (Alan Wendt) (10/03/83)
I thought somebody might like to see some physics in this group
for a change. This is the analytical developement of specular
trapezoidal holes, ignoring reflection off the sides of the
trapezoid. The reflective case will be forthcoming shortly; this
case is presented as an introduction to the techniques and an example
of the use of macsyma.
Let a trapezoidal hole with large aperture 1, small aperture "a",
in a wall of thickness "h" and angle theta be oriented on a
Cartesian plane, large aperture facing down, centered at the origin,
with the edge of the thick side of the wall along the x-axis (better
draw yourself a picture).
In this discussion I'm going to assume when needed that theta = 45%
because it makes things much simpler. If theta = 45% then h will
equal (1 - a) / 2.
In order to compare the probability of "random" trajectories passing
in either direction, I'll compute for each side the integral
for x from -1/2 to 1/2 of (total angles passing through/(2*pi)). The
division by 2 pi normalizes the angles to fractions of circles.
The result for each side will be the probability of a particle
randomly placed on (-1/2, 1/2) and traveling in a random direction
of passing through the hole.
This integral for the narrow side is easy; its a/2.
The integral for the wide side is somewhat more complicated.
Particles can pass directly through or they can be reflected.
An accurate analysis for the general case requires taking arbitrary
numbers of reflections into account. Fortunately if theta=45% only
one reflection needs to be accounted for. This can be verified by
checking a few incident angles and reflections; if a particle has
reflected twice it's heading out of the system and need not be considered.
I'm going to to the non-reflective integral here. To illustrate,
Draw the diagram I've described and put a point x on the
x-axis about halfway in between -1/2 and -a/2. Draw the angle subtended
by the narrow side of the funnel; ie draw lines between x and (-a/2,h)
and between x and (a/2,h). Draw the vertical line from x up to the line
(x = h). The angle subtended can be seen to be
a a
x + - x - -
2 2
(d1) atan(-----) - atan(-----)
h h
Substituting in the value for h:
(c2) subst((1-a)/2,h,d1);
a a
2 (x + -) 2 (x - -)
2 2
(d2) atan(---------) - atan(---------)
1 - a 1 - a
(c3) radcan(%);
2 x - a 2 x + a
(d3) atan(-------) - atan(-------)
a - 1 a - 1
Integrating over the large aperture:
(c6) integrate(%,x,-1/2,1/2);
2 2
(d6) - ((2 a - 2) log(2 a - 4 a + 2) + (2 - 2 a) log(2 a + 2)
a + 1
+ (4 a + 4) atan(-----) - %pi a + %pi)/4
a - 1
Isn't macsyma wonderful?
Dividing into a*pi to express the result as the ratio of the two
probabilities (and scale the integral to 2 pi)
(c13) a*%pi/d6;
2 2
(d13) - 4 %pi a/((2 a - 2) log(2 a - 4 a + 2) + (2 - 2 a) log(2 a + 2)
a + 1
+ (4 a + 4) atan(-----) - %pi a + %pi)
a - 1
Evaluations at some small values of a. This seems to approach 2 as a limit
for small values of a. Remember the large aperture is normalized to 1.
(c16) plot(d13,a,0.05,0.95,19);
0.05 1.937305840557515
0.1 1.876551970837062
0.15 1.81773369236794
0.2 1.760830375585575
0.25 1.705805277138637
0.3 1.652605602312768
0.35 1.601162687365323
0.4 1.551392129016621
0.45 1.503193632624162
0.5 1.456450272053093
0.55 1.411026723834166
0.6 1.366765797549657
0.65 1.323482109128977
0.7 1.280950744382355
0.75 1.23888648916742
0.8 1.196903468859297
0.85 1.154428251570369
0.9 1.110478132810438
0.95 1.062887997606364
(c18) ev(d13,a = 0.0001,numer);
(d18) 1.999872679906421
(c21) ev(d13,a = 1e-08,numer);
(d21) 1.999999981136749
(c22) ev(d13,a = 1e-13,numer);
(d22) 1.99925468572737
I don't know whether this last is a numerical fluke or what. Macsyma
claims that the limit of this function as a->0 is 0 (it looks a whole
lot like 2 to me), so perhaps the function nosedives very very close
to zero.
Anyway, assuming the math is right, this completes the non-reflecting
case, showing that I can get advantages of up to a factor of two,
with constant theta=45%, by varying the small aperture size (and
consequently the width of the wall). The one-reflection case should
be forthcoming soon. If anybody manages this for the reflective
case of arbitrary theta, where you can get any number of reflections,
that would be an accomplishment.
Alan Wendt