Properties of StealthStea

By: Jae Park, Johanna Hoyt, and Maria Verdel.

[ http://www.ntskeptics.org/1994/1994may/stealth.gif]

Stealth technology is the ability to bypass radar waves. We researched the properties and characteristics of the aircrafts. Most of our results are based on Pyotr Ufimstev’s groundwork of stealth technology, more specifically, his article “Methods and Principles in reducing RCS.” The stealth aircraft is composed of flat surfaces and sharp angles, while the commercial aircraft is composed of curved surfaces. Our results show that flat surfaces and sharp angles reflect radar waves away from the radar antenna. The significance of these results is that, if we can minimize radar exposure, we can figure out how to maximize it to make air travel safer.

In 1960 the United States began researching ways to create a military aircraft that would be undetectable by radar. This research is known as “stealth technology.” Radar is an active sensing device, which transmits electromagnetic waves, and listens to the returns (which are the radio waves bouncing off the objects and traveling back toward the transmitter, used to detect range and movement of a craft). The most critical characteristics that contribute to an aircraft’s stealth ability are its shape and its ability to absorb radar waves. In order to help an aircraft achieve radar invisibility, scientists can coat the surface of the aircraft with radar-absorbent paint. Also, sharp angles on an aircraft refract radar waves away from the source, which is a key factor in minimizing an aircraft’s ability to be detected by radar.

Stealth provides aircraft with invisibility from radar, creating an advantage over an enemy. This allows pilots to fly missions safely resulting in fewer casualties. Although stealth is in widely used within the United States Air Force, there is much discrepancy regarding how stealth technology actually works. By analyzing the works of Pyotr Ufimtsev, we are able to learn how stealth works, and how to represent the details simply using mathematics, specifically, calculus. Stealth technology is not only for war machines. A better understanding of how this technology works can provide insight on how to improve radar visibility, for safer commercial flights. For our project, we would like to find out how stealth really works?

According to research, there seems to be a tradeoff between aerodynamics and stealth. The more aerodynamic (curved) a craft is, the worse its stealth ability is, appearing bigger on the radar screen

In a two dimensional explanation, the more faces a shape has, the better it is detected by radar. The ultimate goal in achieving stealth is to reduce the amount of faces exposed. The reason for this is because when a wave hits at a perpendicular, or head on, it is reflected directly to the source. This is so because the “law of reflection holds true: the waves will always reflect in such a way that the angle at which they approach the barrier equals the angle at which they reflect off the barrier.[1]”

[2]

**If the incident ray hits the barrier at a perpendicular, there is no angle, therefore is reflected back to the source.

A circle has an infinite number of tangent lines that a radar wave could hit perpendicularly (refer to Figure 3). Line XP is tangent to circle O, with line XO as the perpendicular to line XP. Line XO represents a radar signal from any arbitrary source. As you can see, the source can be anywhere around the circle, because there are infinite points of tangency on the circle. Any signal that is sent the circle will be reflected back (due to the law of reflection), since for every line of tangency, there is also a perpendicular line.

If we observe triangle ABC (refer to Figure 4), there are only three surfaces that allows perpendicular lines, compared to the infinitely many perpendiculars a circle may offer. With this, we are able to reach a conclusion that the more closer to a circle a shape is, the worse it would perform stealth wise, although a curved surface would prove better aerodynamically. (This is so because the more curved a surface is, the less friction it provides for the aircraft, allowing it to achieve and maintain flight with less work.)

For more in depth analysis, we turn to calculus in order to give us a better understanding of what happens in real life. We analyze and graph to show how a radar signal behaves with regular and stealth aircraft.

In our first analysis, we establish the curve y= - √(x) as the underside a regular aircraft’s nose, since the shapes are quite similar.

[3] [4]

The graph y=x² represents a parabolic radar reflector, mainly due to the face that “most common types of radar antenna are parabolic reflectors.[5]”

Also, please note that we have simplified these equations and have not placed any x or y intercepts pertaining to position on the coordinate plane because they are not significant. This analysis is just to show that there are indeed tangent lines that are parallel, and share a common perpendicular line, due to the Parallel Transversal Theorem. This theorem states that in a plane, if two lines are perpendicular to the same line then they are parallel; therefore the reverse is true as well.

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Using calculus, we differentiate to find the equations of the tangent lines for each curve. If we find the tangent lines, we will be able to show that a perpendicular exists between these lines, showing the behavior of the radar beam throughout the curves. For the function y= - √(x), we arrive at the slope m¹, while the function y=x² yields a slope m².

m¹ = 2x'

m² = -1 / [ 2√(x) ]

We assume that the two slopes are equal…

\\> m¹ = m²

\\> 2x' = -1 / [ 2√(x) ] Dom: [ x > 0 ; x' < 0 ]

Now we are able to solve for equal slopes…

Example: x'=4

\\> 2x' = 2(4) = 8

\\> (4,8)

\\> We plug into the original equation m¹ = m²,

8 = -1 / [ 2√(x) ]

We then solve for x and arrive at x = 1/ (16²)

So at these coordinates, the slopes are equal, allowing the perpendicular to be shared by both parallels. This shows the radar signal, or the perpendicular line, interacting with the underside of an aircraft (-1 / [ 2√(x) ]). We now know that there are more than one point throughout the curves where a similar behavior occurs like that shown in Figure A.

For our second analysis, we want to see how a stealth aircraft differs from a regular aircraft. Stealth aircraft use triangular shapes and faces to maximize efficiency, as shown previously with the 2-Dimensional analysis.

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For this analysis, we have decided to use a combination of two lines, of the form y=ax and a y=b line to represent the nose section of the F-117 aircraft (a and b being any real integer).

[8]

The parabola y=x² will still represent the parabolic radar antenna. The radar signal will not be reflected back to the antenna. There is much less of a chance of detection with a plane that has flat surfaces with sharp edges and angles. This aircraft has a surface created specifically so that radar signals do not hit perpendicular. As a result of this design, when the radar wave hits a side of the F-117, the wave will be reflected away from the antenna, or not even come into perpendicular contact.

Using calculus, we differentiate to find the equations of the tangent lines for each function. For the function y=ax, we arrive at a slope a, therefore the tangent line is the line itself. For the function y=b, the slope is 0, once again the tangent line is the line itself. It is virtually impossible to have the two functions y=ax and y=b to have the same tangent slope as the parabola y=x², therefore they cannot share a perpendicular line. This in turn, means that there is no perpendicular surface for the radar wave to hit, preventing it from returning to the source, resulting in radar invisibility or stealth.

As a result of all the analyses, we are able to conclude and prove that there is indeed a tradeoff between aerodynamics and stealth ability. The rounder a surface is, the more probable it is for the plane to come into direct contact with a radar signal. This is quite interesting that we are able to prove such physical characteristics through simple coordinate geometry and Calculus. Now that we know how to minimize appearance on radar, we are now able to figure out ways to improve radar visibility to improve air safety for commercial jet liners and space shuttles.