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Electromagnetic wave equation
Heinrich Hertz
The electromagnetic wave equation is the foundational PDE for classical electromagnetism. Maxwell derived it mathematically; Hertz verified it physically — not as a vision of wireless communication, but as proof that light is electromagnetic vibration. It works precisely where fields are classical and sources are macroscopic. It fails where quanta matter. Its value lies not in novelty but in necessity: every radio, radar, and optical system still obeys it.