+ "theory": "The Doppler Effect is the change in frequency and wavelength of waves when the source or observer moves relative to the medium. This is one of the most important phenomena in physics, affecting sound, light, radar, medical imaging, and astronomy.\n\n>> FUNDAMENTAL PRINCIPLE:\nWaves propagate at a constant speed (v) through their medium. The frequency observed depends on the relative motion between source and observer.\n\nKey relationship: v = f × λ. If v is constant but the source moves, then f and λ must change.\n\n>> DETAILED MECHANISM - THREE REGIONS:\n\n[1] IN FRONT OF MOVING SOURCE (Compression Side):\n• The source chases its own wavefronts\n• Wavefronts get compressed together\n• Distance between crests decreases: λ' < λ\n• Observer measures higher frequency: f' > f\n• Formula: f_observed = f × [v / (v - v_source)]\n• Example: ambulance siren sounds higher-pitched approaching\n\n[2] BEHIND THE SOURCE (Stretching Side):\n• Source moves away from previous wavefronts\n• Wavefronts get pulled apart\n• Distance between crests increases: λ' > λ\n• Observer measures lower frequency: f' < f\n• Formula: f_observed = f × [v / (v + v_source)]\n• Example: ambulance siren sounds lower-pitched receding\n\n[3] PERPENDICULAR MOTION:\n• At points to the side, motion is purely tangential\n• Minimal frequency shift compared to front/back\n• Creates characteristic 'drop' as vehicle passes\n\n>> ENERGY REDISTRIBUTION:\n• Front: Waves compressed, amplitude increases\n• Back: Waves stretched, amplitude decreases\n• Total energy increases (source does work against wavefronts)\n\n>> MACH REGIMES:\n\n[SUBSONIC] (v_source < v_wave, Mach < 1):\n• Source slower than waves\n• Wavefronts spread ahead of source\n• Normal Doppler effect\n• Observer has warning before source arrives\n\n[TRANSONIC] (v_source = v_wave, Mach = 1):\n• Source exactly matches wave speed\n• Wavefronts cannot escape ahead\n• Massive amplitude buildup at source (Sound Barrier)\n\n[SUPERSONIC] (v_source > v_wave, Mach > 1):\n• Source outruns its own disturbances\n• Forms conical shock wave behind (Mach cone)\n• Cone angle: sin(θ) = 1 / Mach_number\n• Concentrated energy creates sonic boom\n• Zone of silence outside cone (waves haven't arrived)\n\n>> MACH CONE GEOMETRY:\nCone half-angle θ determined by:\n• M = 1.0 -> θ = 90° (flat shock, sound barrier)\n• M = 2.0 -> θ = 30°\n• M = 3.0 -> θ = 19.5°\n• Higher speed = narrower cone\n• Energy concentrated on cone surface\n\n>> REAL-WORLD APPLICATIONS:\n\n1. ASTRONOMY: Redshift/Blueshift of distant galaxies (cosmic expansion)\n2. MEDICAL IMAGING: Doppler ultrasound measures blood velocity\n3. RADAR: Police speed guns use Doppler shift of radio waves\n4. WEATHER: Doppler radar detects storm motion and wind velocity\n5. MUSIC: Moving sources affect pitch; Doppler used in electronic music\n6. PARTICLE PHYSICS: Relativistic particles create frequency shifts\n\n>> DEEP INSIGHTS:\nDoppler effect is consequence of finite wave speed. Different media (air vs water vs vacuum) show different shifts. For light at relativistic speeds, special relativity required for accurate calculations. Asymmetric frequency shift between approaching/receding at extreme velocities.\n",
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