remaking echo 1 - stealth physics · pramit.gg
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back to the table<br>back to the table
how soviet math led to the first empirically stealth-optimized aircraft
I would like to preface this by saying I'm not an expert in this field by any means; I'm an average software engineer with an above-average interest in aerospace and physics. All visualizations were drawn/generated by me and code can be found in this project's github repo
black diamonds and bouncing photons
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Ever since I first delved into engineering, I became entranced with arguably the most captivating aircraft ever created, the SR-71 Blackbird. The technology behind this aircraft alone is mind-boggling, and if unfamiliar I highly recommend Real Engineering's youtube video on the topic.
The SR-71 was a stealthy aircraft, to be sure, but not in the current way we understand stealth. It had clever shaping and radar-absorbing paint, but its true defense was speed and altitude. Later aircraft, namely the F-117, however, strived to be completely invisible. To achieve this, it used a completely different technique, pioneered by physics rather than engineering.
(p.s. also highly recommend the video for the nighthawk for some slightly more intuitive visualizations)
The Soviet Paper
In 1971, Pyotr Ufimtsev published a paper1 describing how electromagnetic waves scatter off edges. This idea was largely ignored by Western engineers, who focused on practical application more than diffraction theory. However, one Lockheed Skunk Works engineer, Denys Overholser, realized that this theory could be used to create a plane that was able to minimize its radar cross section (RCS), and this very concept led to the development of the F-117. 2
In this article, I will attempt to unpack the math of stealth just enough to be intuitive, to prepare for a code implementation in the next post.
The Physics of Stealth
To understand stealth, one must understand radar. RCS is informally the effective area that reflects radar signals back to a source. What does this mean?
If you have a metallic sphere, and you fire a radar at it, some amount of energy will return to you, roughly increasing with the size of the sphere's area. A large RCS means a radar sees a large, bright object, while a small RCS produces a ghost. To give you an intuition of RCS numbers, and to emphasize the leaps this technology made, I made a small chart:
Object / AircraftRCS (m²)RCS (dBsm)NotesSmall bird3 0.01–20 dBsmHard to detectHuman4 10 dBsmBaseline for RCSF-15 Eagle (1972) ~12~11 dBsmConventional fighterSR-71 Blackbird (1964)5 ~10~10 dBsmSome passive RCS reductionF-117 Nighthawk (1981)6 0.01 – 0.1–20 to –10 dBsmTopological Optimization
Now, let's delve into a more formal definition, RCS is defined by the equation
σ=limr→∞4πr2∣EsEi∣2\sigma = \lim_{r \to \infty} 4\pi r^2 \left| \frac{E_s}{E_i} \right|^2σ=limr→∞4πr2EiEs2
where RRR is the distance from the target (imagine the radar being very far away)<br>en.wikipedia.org<br>. This formula might look intimidating, so let's break it down:
li]:relative [&>li]:pl-2 [&>li]:before:absolute [&>li]:before:-left-4 [&>li]:before:top-[0.6em] [&>li]:before:h-1.5 [&>li]:before:w-1.5 [&>li]:before:rounded-full [&>li]:before:bg-accent-orange/80 [&>li]:before:content-['']"><br>As R→∞R \to \inftyR→∞, the radar is in the far field, where electromagnetic waves can be treated like rays. The factor R2R^2R2 accounts for the fact that as you move away, the scattered wave spreads out over a sphere of area 4πR24\pi R^24πR2. Multiplying the scattered field intensity ∣E⃗s∣2|\vec{E}_s|^2∣Es∣2 by R2R^2R2 essentially measures the total power scattered in that direction.
The ratio ∣E⃗s∣2∣E⃗i∣2\frac{|\vec{E}_s|^2}{|\vec{E}_i|^2}∣Ei∣2∣Es∣2 compares the strength of the scattered wave to the incoming wave. If our object was a perfectly reflective sphere of cross-sectional area 1 m², this ratio (times 4πR24\pi R^24πR2) would come out to 1 in the exact back direction.
So σ\sigmaσ emerges as an effective area (in m²) that represents how strongly the target reflects radar energy back. If σ=1\sigma = 1σ=1 m², the target is as detectable as a 1 m² metal sphere. If σ=0.001\sigma = 0.001σ=0.001 m², it's as if the radar is seeing something the size of a marble (10 times the effective cross section of a B-2)!
So, to recap, magine wrapping the target in a huge bubble. measure the power bouncing back through a unit area, scale it by that bubble’s area ( (4\pi r^2) ), compare to how hard you hit it in the first place — out comes effective area.
From this equation we can also see that even a small change has an immense effect on detections; detection scales roughly by the fourth root of RCS, so cutting RCS by 100 cuts detection distance to 1/1004≈56%\sqrt[4]{1/100} \approx 56\%41/100≈56% of original. So how do we do this?
Calculating RCS
To empriically calculate the RCS of a complex shape like an aircraft, an exact solution require's solving...