최초의 스텔스기를 만든 소련의 수학 이론

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 1752308995975_bzq2ct.png 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 paper 1 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 / Aircraft RCS (m²) RCS (dBsm) Notes Small bird 3 0.01 –20 dBsm Hard to detect Human 4 1 0 dBsm Baseline for RCS F-15 Eagle (1972) ~12 ~11 dBsm Conventional fighter SR-71 Blackbird (1964) 5 ~10 ~10 dBsm Some passive RCS reduction F-117 Nighthawk (1981) 6 0.01 – 0.1 –20 to –10 dBsm Topological Optimization Now, let's delve into a more formal definition, RCS is defined by the equation σ = lim ⁡ r → ∞ 4 π r 2 ∣ E s E i ∣ 2 \sigma = \lim_{r \to \infty} 4\pi r^2 \left| \frac{E_s}{E_i} \right|^2 σ = lim r → ∞ ​ 4 π r 2 ​ E i ​ E s ​ ​ ​ 2 where R R R is the distance from the target (imagine the radar being very far away) en.wikipedia.org . This formula might look intimidating, so let's break it down: As R → ∞ R \to \infty R → ∞ , the radar is in the far field, where electromagnetic waves can be treated like rays. The factor R 2 R^2 R 2 accounts for the fact that as you move away, the scattered wave spreads out over a sphere of area 4 π R 2 4\pi R^2 4 π R 2 . Multiplying the scattered field intensity ∣ E ⃗ s ∣ 2 |\vec{E}_s|^2 ∣ E s ​ ∣ 2 by R 2 R^2 R 2 essentially measures the total power scattered in that direction. The ratio ∣ E ⃗ s ∣ 2 ∣ E ⃗ i ∣ 2 \frac{|\vec{E}_s|^2}{|\vec{E}_i|^2} ∣ E i ​ ∣ 2 ∣ E s ​ ∣ 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 4 π R 2 4\pi R^2 4 π R 2 ) would come out to 1 in the exact back direction. So σ \sigma σ emerges as an effective area (in m²) that represents how strongly the target reflects radar energy back. If σ = 1 \sigma = 1 σ = 1 m², the target is as detectable as a 1 m² metal sphere. If σ = 0.001 \sigma = 0.001 σ = 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 / 100 4 ≈ 56 % \sqrt[4]{1/100} \approx 56\% 4 1/100 ​ ≈ 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 Maxwell's equations with appropriate boundary conditions; this is because you are fundamentally solving how EM waves scatter off a shape. This is extremely challenging, because this requires you to: discretize your aircraft's surface into N N N patches set up a system of equations to model interactions between all pairs of the patches to compute a total scattered field This effect is governed by Green's function (a ( 3 N × 3 N ) (3N\times3N) ( 3 N × 3 N ) linear system). For a mesh of 1,000 vertices ( N = 1000 N=1000 N = 1000 ), the Green's evaluations scales to 3000 ∗ 3000 3000*3000 3000 ∗ 3000 equations. In the 1970s, just storing those equations would be a problem at around 100MB, let alone solving them; the best computers at the time had 1MB of memory. Enter PO (Physical Optics) As many great engineering tricks are born, so is PO by using one simple trick: simplifying assumptions. The surface is a PEC (Perfect Electrical Conductor): If you assume the surface is a perfect conductor, the tangential electric field on the surface is zero, so you no longer need to model material properties, and you can treat the EM action exclusively as an integral over the surface 1752312039402_0227on.png Only the illuminated (front-facing side contributes : If a facet is not on the side fo the object directly hit by the radar wave, completely ignore it. This cuts computation into half (or more), and you no longer need to model complex interactions or reflections where a wave creeps onto the other side of the obect. 1752310428680_qyt61x.png Fields scatter by specular reflection: reflected waves always bounce off a surface at an angle equal to the incidence angle, like a mirror. This way you never need to model multiple bounces, diffractions, scattering, or anything else messy. All of these assumptions compromise accuracy. But they vastly simplify the computations, and it turns out that (at least for the time) these simplified models were more than enough to drastically decrease RCS. Now, let's get into the math and see how these assumptions work for us: For a perfectly conducting surface, the induced surface current density under PO is given (in vector form) by: J ⃗ s ( r ⃗ ) = 2 n ^ ( r ⃗ ) × H ⃗ i ( r ⃗ ) , \vec{J}_s(\vec{r}) = 2 \hat{n}(\vec{r}) \times \vec{H}_i(\vec{r}), J s ​ ( r ) = 2 n ^ ( r ) × H i ​ ( r ) , where n ^ ( r ⃗ ) \hat{n}(\vec{r}) n ^ ( r ) is the outward-pointing unit normal at the surface point r ⃗ \vec{r} r , and H ⃗ i \vec{H}_i H i ​ is the incident magnetic field at that point. This formula basically says the surface current flows in such a way as to cancel the tangential magnetic field on the surface, and the factor of 2 comes from the reflected field adding to the incident field just at the surface. Now that we know what the surface current is, we can get the scattered field in some arbitrary direction k ^ s \hat{k}_s k ^ s ​ by just integrating those currents over the surface S S S : E ⃗ s ( k ^ s ) = j k η 4 π ∫ S lit [ k ^ s × ( k ^ s × J ⃗ s ( r ⃗ ) ) ] e j k ( k ^ i − k ^ s ) ⋅ r ⃗ d S . \vec{E}_s(\hat{k}_s) = \frac{jk\eta}{4\pi} \int