9853

Typical Wavefronts in 2D and 3D

This Demonstration shows all "generic" wavefronts and boundary wavefronts in the 2D plane and in 3D space. It also shows all generic intersections of such wavefronts. You can observe their shapes from any direction by rotating the 3D images, which will help your understanding of the structure of generic wavefronts.

THINGS TO TRY

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Wavefronts are defined by projected images of Legendrian submanifolds on Legendrian bundles. By Darboux's theorem, all Legendrian bundles are locally contact diffeomorphic to the canonical Legendrian bundle: for some , where is the 1-jet bundle of functions in variables with the contact structure defined by the canonical one-form . Thus, it is enough to consider the canonical Legendrian bundle for local situations. An -dimensional submanifold in is a Legendrian submanifold if vanishes on . The wavefront of is defined by . The theory of wavefronts describes stabilities of wavefronts generated by a hypersurface without boundary in a smooth manifold (see [2]).
The author constructs the theory of reticular Legendrian singularities that describes stabilities and genericities of wavefronts generated by a hypersurface with a boundary, a corner, or an -corner in a smooth manifold (see [3]). Let and for each
σ⊂Ir
. Let represent a germ of the union of for all . We call a map germ a reticular Legendrian map if there exists a contact diffeomorphism germ on such that . The wavefront of is defined by the union of for all .
For the case , the hypersurface that has a boundary, the wavefront of is . The sets and correspond to the wavefronts generated by the hypersurface and its boundary, respectively. A smooth function germ defined on is a generating family of if is non-degenerate and
, .
Then the wavefronts of are given by
.
You can observe behaviors of wavefronts in a plane in Bifurcation of Boundary Wavefronts for Some Graphs.
Typical wavefronts are defined by generic reticular Legendrian maps and their generating families are stably reticular --equivalent to one of the following:
In the case :
,
,
,
,
,
,
,
,
,
In the case :
,
,
,
,
,
.
References
[1] V. I. Arnold, Singularities of Caustics and Wave Fronts, Dordrecht: Kluwer Academic Publishers, 1990.
[2] V. I. Arnold, S. M. Gusein–Zade, and A. N. Varchenko, Singularities of Differential Maps I, Basel: Birkhäuser, 1985.
[3] T. Tsukada, "Genericity of Caustics and Wavefronts on an r-Corner", 14(3), The Asian Journal of Mathematics, 2010 pp. 335–358.
[4] T. Tsukada, "Bifurcations of Wavefronts on r-Corners: Semi-Local Classification," 18(3) Methods and Applications of Analysis, 2011 pp. 303–334.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+