Singularities of an Ellipsoidal Wave Front

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration shows the appearance of singularities on an elliptical wave front. The singularities are semicubical cusp edges and swallowtails.

Contributed by: Andrzej Kozlowski (September 2012)
Open content licensed under CC BY-NC-SA



A "wave front" begins with a smooth, closed surface in space (or a curve in the plane) that propagates inward so that each point moves with the same speed in the direction of the inward normal to the initial surface. The position of the front at time is the equidistant surface of the initial surface. For a generic initial surface, singularities of the semicubical cusp and swallowtail type appear.


[1] V. I. Arnold, The Theory of Singularities and Its Applications, Cambridge, UK: Cambridge University Press, 1993.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.