§ 20. Doppler Effect & Wavefront Deformation (Grade 9–11)
Compression of wavefront crests ahead of a moving source, frequency shift & Mach cone. A moving source crowds wavefront crests ahead of itself, shortening wavelength λ′ and raising the observed frequency.
Theory Summary & Scientific Mechanism
The Doppler effect (Christian Doppler, 1842) is the shift in observed frequency ν and wavelength λ produced when a wave source moves relative to the propagation medium. The 3D wavefront geometry makes the mechanism transparent: each successive spherical wave crest is emitted from a newly displaced source position separated by vₛ T₀.
Ahead of a source moving at speed vₛ < cₛ, wave crests crowd together so the wavelength shortens to λ_front = (cₛ - vₛ)T₀ = λ₀ ≤ft(1 - (vₛ)/(cₛ) ) and the observed frequency rises to ν_front = (ν₀)/(1 - vₛ/cₛ). Behind the source, wavefronts stretch out to λ_back = λ₀ ≤ft(1 + (vₛ)/(cₛ) ) and the frequency drops to ν_back = (ν₀)/(1 + vₛ/cₛ).
The ratio M = vₛ / cₛ is the Mach number. As M → 1, forward wavefronts pile up constructively into a shock front (sonic boom and Mach cone with half-angle sin μ = 1/M). Doppler shifts are foundational to Doppler ultrasound imaging, weather radar, and cosmological redshift measurements.
Everyday Analogy
Ever heard an ambulance siren or race car pass by with a classic “NEEEE-YOWWW” pitch drop? Moving forward, the siren crowds its own sound waves together (higher pitch); behind it, the waves stretch out (lower pitch)!
Interactive Question & Derivation
A monochromatic wave source moves to the right at speed vₛ = 0.5 v, where v is the wavefront propagation speed in the medium. How do the crest spacing λ₊ and observed frequency f₊ change for a stationary observer in front of the source?
- A: Neither wavelength nor frequency changes because wave speed in the medium is constant
- B: Wavefronts ahead of the source are compressed by half (λ₊ = 0.5λ₀), so the observer measures twice the frequency f₊ = 2 f₀
- C: Wavelength ahead of the source stretches and frequency drops by half
Study Hint: What happens to the wavelength λ_front = λ₀(1 - vₛ/v) and observed frequency f in front of the moving source compared to behind it?
Proof & Explanation: During one emission period T₀ = 1/f₀, a wavefront travels v T₀ = λ₀ while the source advances by vₛ T₀ in the same direction. Thus the spacing between successive wavefronts ahead of the source shrinks to λ₊ = (v - vₛ)T₀ = λ₀(1 - M), raising the observed frequency to f₊ = (f₀)/(1 - M) (at M = 0.5, f₊ = 2f₀).
Key Equations & 3D Simulation Parameters
f₊ = (f₀)/(1 - vₛ/v), f₋ = (f₀)/(1 + vₛ/v), sin θ_Mach = (v)/(vₛ)
- Source Mach ratio M = vₛ / v: 0 … 1.25 (default: 0.55)
- Wavefront speed v: 2.2 … 5 m/s (default: 3.6 m/s)
- Source frequency f₀: 1 … 3.5 Hz (default: 2 Hz)
- Wavefront crest intensity I₀: 0.5 … 1.5 u. (default: 1 u.)