A stationary observer hears 680 Hz as a siren approaches and 612 Hz after it passes. Take sound speed as 340 m s⁻¹ and model the source speed as constant, with no wind. Derive expressions for the two heard frequencies, determine source speed and emitted frequency, and explain how a steady wind from source toward observer would affect a naive calculation using ground-frame speed 340 m s⁻¹.

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For a moving source, f_app = f c/(c−u) and f_rec = f c/(c+u). The data give u = 17.9 m s⁻¹ and f = 644 Hz. A moving source compresses wavelength ahead to λ_app = (c−u)/f and stretches it behind to λ_rec = (c+u)/f. A stationary observer measures c divided by these wavelengths. Thus f_app/f_rec = (c+u)/(c−u). Solving gives u = c(f_app−f_rec)/(f_app+f_rec). Substitution gives u = 340(680−612)/(680+612) = 17.9 m s⁻¹. Using either branch, f = f_app(c−u)/c = 680(340−17.9)/340 = 644 Hz; the receding branch provides the same value as a check. Wind changes wave speed relative to the ground and can advect both source-generated wavefronts and observer. The simple still-air ground formula may bias inferred u; a consistent derivation must use velocities relative to the medium rather than replacing c in only one place.

Infer source speed from two Doppler readings

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