Explosive Lens shape

[I.N. Galidakis]

I am trying to determine the cone shape of the Baratol (slow) explosive placed
on the middle of the explosive lens, whose purpose is to focus the blast waves
emanating from the detonator, to the pit:

http://en.wikipedia.org/wiki/Fat_Man (C on diagram, below)

Now, it seems (and I say "it seems" because I haven't seen it documented
anywhere, yet) that the shape relies on the "reflection property" of the
hyperbola:

http://en.wikipedia.org/wiki/Hyperbola (section "Geometrical Properties")

The above property in optics, is equivalent to having a hyperbolic reflector and
a light source at the focus F1, whose imaginary image, based on the hyperbola's
reflection property, will be at F2 (inside the reflective hyperbola's wing).

F1 ------ < (Hyperbolic reflector) ---- F2.

The following reference:

http://misc.virtualcomposer2000.com/US4729318.pdf

solves the same problem, but instead of 2 foci, it creates a plane wave from a
single focus at the detonator. The shape given therein, is:

X = a0 + a1*sqrt(a2*(Y + a3)^2 + a4)

The above is a hyperbola:

(X - a0)/a1 = sqrt(a2*(Y + a3)^2 + a4) =>

[(X - a0)/a1]^2 - a2*(Y - (-a3))^2 = a4 =>

[(X - a0)/a1/sqrt(a4)]^2 - [(Y - (-a3))/(1/a2)/sqrt(a4)]^2 = 1

Questions: The refractive "dual" of a plane wave, using Snell's Law, would
require here a parabola, not a hyperbola, since after "focusing" the resultant
waves are plane waves. How come this case resolves to a hyperbola as well?

Is it possible to derive the two foci solution using Snell's Law?:

|----R1---|--|----------R2--------------|
F1 ------ <(*)------------------------- F2.

(*) Cone shaped Baratol

I.e., if the waves start at F1 and then meet the cone < which focuses them at
F2, how can I determine the shape of the surface <?

It seems to me that it should be a surface of revolution around the x-axis, with
the upper side (left) having the wanted shape and the lower side (right) being a
spherical surface of revolution of curvature R2. But the upper surface has me
stumped.

My guess is that the shape would also depend on the index of refraction n of the
Baratol cone:

n_{cone} = (Explosion speed of Comp B explosive)/(Explosion speed of Baratol)

Both speeds are known, hence n is known.

If anyone has any references or can derive the cone shape using Snell's Law
(which in principle *should* work even for blast waves) I'd appreciate the info.

Many thanks,
--
Ioannis

[Helmut Wabnig]

On the first glance I would rather see ellipsoids,
instead of hyperbolas,
did I miss something?

w.

[Bob May]

Dont think so. If you're dealing with finite point sources, the eelipse
fits the bill better than a parabola will as the source is an eextended
source. We're dealing basically6 with "undercorrected" parabolas here.
For transmission work, the hyperbola is indeed the correct surface tho but I
don't think that he's going to be putting the gasses through a lens system.

--
Bob May

rmay at nethere.com
http: slash /nav.to slash bobmay
http: slash /bobmay dot astronomy.net

[I.N. Galidakis]

Bob May wrote:
[snip]

> For transmission work, the hyperbola is indeed the correct surface tho but I
> don't think that he's going to be putting the gasses through a lens system.

It is indeed a hyperola. I think I was able to solve the resultant differential
equation with Maple 9. See:

http://tinyurl.com/yazb3wh
--
Ioannis

[Bob May]

Remember that for this discussion, transmissions are pretty much imaginary!

[I.N. Galidakis]

Bob May wrote:

> Remember that for this discussion, transmissions are pretty much imaginary!

Indeed. But note:

The solutions in the math posts are for the "reflective" case. Each "reflective"
case seems to have a "refractive" dual, which is obtained by "swapping" the
locations of the wings of the hyperbola.

Reflective:

F1-->---<--F2

Refractive:

F1--<--->--F2

It seems to me that a hyperbola opening towards (and close to F2, like that:

F1------<--F2

) will reflect waves from F1 in the radial direction opposite to F2 (creating an
imaginary image of F1 @ F2), but if you place the hyperbola at the position of
its dual "wing" (like that:

F1--<------F2 (*)

), and assume the contents of the hyperbola wing are transparent to whatever
waves are used, then it seems to me that the hyperbola will *focus* waves from
F1 onto F2.

For example, if the waves are light, then in (*) the entire hyperbola can
consist of glass, in which case it seems to me the glass will focus the light
source of F1 @ F2, with the image of F1@F2 being "real".
--
Ioannis