Wolfram Archive

# Mathematica in Furtive Fighters and Squashed Sidelobes

Published September 14, 1998

#### Mathematica Facilitates Precision Coating Technology

September 14, 1998–You’ve decided to paint red and yellow flames on the side of your motorcycle. You want to blend the colors along each flare according to a mathematical formula involving hyperbolic trigonometric functions. Anything less than perfection would render the whole paint job useless. What’s the best approach?

Leading aerospace engineers faced a similar problem in applying a surface coating to fighter aircraft. The critical feature in this case is not color, but the electrical conductivity at the surface. This physical property determines how an electromagnetic wave will scatter when it hits the aircraft. But different parts of the aircraft have different levels of surface conductivity, and there’s the catch–if there are sharp changes in conductivity from one area to another, the incoming wave scatters in a way enemy radar can detect, thus giving away the fighter’s position.

To avoid this problem, airframers would spray a conductive surface coating of varying thickness onto the aircraft’s surface, blending the edges so that no sudden transitions occur in surface resistance. Although the mathematical properties of the ideal blending pattern are known–they’re a direct consequence of the laws of electrodynamics–until recently, applying the conductive coating properly still required a technician, gradually blending areas with a spray gun exactly the way an airbrush artist does when painting flames on a motorcycle.

At least, that’s how airframers used to do it until they added Mathematica to their process. Mathematica, the world’s only fully integrated technical computing system, is no stranger to either higher math or the creation of sophisticated graphic images.

The process now goes like this. Engineers use Mathematica‘s numerical power to define precisely the ideal coating pattern for a given surface. Next, using a very high resolution PostScript printer, technicians print out a “phototool,” an optical mask, from Mathematica. This phototool is then used in a photochemical etching process to place the conductive coating exactly where it’s needed with exactly the right thickness.

For some fighter parts, the ideal coating varies along the surface according to the hyperbolic cosine function. Because Mathematica is as familiar with hyperbolic trigonometric functions as it is with literally hundreds of other special mathematical functions, it handles the task easily. The tight integration among Mathematica‘s algebraic, numeric, and graphics capabilities makes the path from initial mathematical model to final phototool a single, smooth development process.

“Any number of programs can generate PostScript graphics, and some of these are even acceptable for CARTs,” said Matthew M. Thomas, Technology and Define Processes, The Boeing Company. “But few of them offer the ease of use Mathematica offers, and none of them offers the mathematical sophistication Mathematica offers. Ease of use is why we use Mathematica with CARTs today. Mathematical sophistication is why we’ll use Mathematica with CARTs in the future.”

The Boeing Company sees applications of this new surface coating technology–called CART for “Computer-Aided Resistive Taper”–in commercial products and is seeking to license it to communications-related manufacturers. For example, a similar process can prevent electromagnetic scattering within a parabolic antenna from creating large sidelobes that degrade the antenna’s performance.

To highlight CART technology, Boeing will share a booth with Wolfram Research, makers of Mathematica, during Wescon/IC Expo 98, North America’s largest technology expo for OEM electronics professionals, from September 15 to 17 in Anaheim, California.