Doppler Effect (Sound, Moving Source)

As the source speed approaches the speed of sound, the denominator (v_s − v_src) approaches zero, and the Doppler factor diverges — the observed frequency tends to infinity. Physically, this means all the wavefronts emitted by the source pile up at the same position, creating a shock front of enormous pressure and density. When the source reaches v_s, this shock front reaches the observer as a sonic boom — a sudden, sharp pressure pulse. The moving-source Doppler formula does not apply at or above the speed of sound; instead, the source outruns its own wavefronts and creates a Mach cone trailing behind it, with half-angle θ = arcsin(1/M) where M is the Mach number. Behind the cone, the observer hears nothing until the shock arrives; inside the cone, they hear the sonic boom.

No. The Doppler frequency shift depends only on the relative velocity between source and observer along the line of sight, not on the distance between them. A source moving at 60 m/s toward you produces the same frequency shift whether it is 10 m or 10 km away. What does change with distance is the intensity of the sound (the inverse-square law) and the time delay for the sound to arrive. The Doppler shift is a kinematic effect, not a propagation effect. In real situations, the source is usually not moving exactly toward or away from the observer, but at some angle; only the component of velocity along the line of sight matters, and the shift goes to zero when the source moves perpendicular to the line of sight.

No. For sound, the medium (air, water) provides a preferred reference frame, and the formulas for a moving source and a moving observer are different. For light, there is no medium, and the special theory of relativity requires that the formulas for source motion and observer motion be identical — only the relative velocity matters. The relativistic Doppler formula is f_obs = f_src · √((1 − β)/(1 + β)) for a source receding along the line of sight, where β = v/c. For β ≪ 1, this reduces to the classical formula f_obs ≈ f_src(1 − β), but for large β the relativistic correction becomes important. The expanding-universe redshift of distant galaxies is a relativistic Doppler effect, as is the transverse Doppler shift observed in fast atomic beams.

For sound in a medium, the two formulas have a different mathematical structure. For a moving source at speed v_src toward a stationary observer: f_obs = f_src · v_s/(v_s − v_src). For a stationary source with an observer moving toward it at speed v_obs: f_obs = f_src · (v_s + v_obs)/v_s. The two formulas are not identical: moving the source changes the wavelength of the emitted wavefronts, while moving the observer changes how frequently the observer encounters those wavefronts. For small speeds compared to v_s, both formulas reduce to the same first-order approximation, but at high speeds the asymmetry becomes important. For light, relativity requires the two formulas to be the same.

The Doppler effect has a remarkable range of applications. Doppler radar measures the speed of cars (police radar), weather systems (Doppler weather radar detects precipitation motion), and baseball pitches. Medical ultrasound uses Doppler to measure blood flow velocity and detect blockages. Sonar uses Doppler to track submarines and measure ocean currents. Astronomy uses the Doppler shift of spectral lines to measure the velocities of stars, galaxies, and gas clouds, and to detect exoplanets by the wobble of their host stars. In everyday life, the passing siren and the race-car engine are familiar examples. The Doppler effect is also used in laser cooling of atoms, in GPS (relativistic corrections), in flow measurement (laser Doppler velocimetry), and in the search for gravitational waves (where the Doppler shift of pulsar signals provides a test of general relativity).

Yes. The audible range for humans is roughly 20 Hz to 20,000 Hz. A source with a frequency near the edge of this range can be shifted out of it by the Doppler effect. For example, a 15,000 Hz whistle approaching at 100 m/s in air (v_s = 343 m/s) produces an observed frequency of 15,000 × 343/(343 − 100) = 21,200 Hz — above the upper limit of hearing, so it becomes inaudible. Similarly, a 25 Hz rumble receding at 100 m/s drops to 25 × 343/(343 + 100) = 19.4 Hz — below the lower limit. Bat echolocation (20–100 kHz) and dolphin sonar (up to 150 kHz) operate above the human range, and Doppler shifts in these signals are used by the animals to judge the speed of their prey. Medical Doppler ultrasound also operates above 1 MHz, well above human hearing.