When an ambulance approaches, its siren sounds high-pitched; when it moves away, it sounds low-pitched. This is the Doppler effect.
Basic Knowledge
Doppler effect: A phenomenon in which the frequency heard by an observer changes when the source or observer moves.
Formula for the Doppler effect (sound speed V, source frequency f₀):
f = f₀ × (V + v_o) / (V − v_s)
・v_o: speed of the observer (+ when approaching the source, − when moving away)
・v_s: speed of the source (+ when approaching the observer, − when moving away)
※Sign in the numerator: Direction in which the observer approaches the source → +
※Sign in the denominator: Direction in which the source approaches the observer → − (the denominator becomes smaller, so f increases)
Qualitative rule:
・Source approaches → observed frequency increases (sound is higher)
・Source moves away → observed frequency decreases (sound is lower)
Doppler effect (Doppler effect)（Observed frequency changes because of the relative motion of the source and observer）
Observed frequency f（The frequency actually heard by the observer）
Source frequency f₀（The original frequency produced by the source）
Mach number（An object's speed divided by the speed of sound. Mach 1 = the speed of sound）
Shock wave (shock wave)（A wavefront produced when a source exceeds the speed of sound）
Redshift（The Doppler effect for light. The frequency of light from a receding celestial body decreases）
Further Study (Background and Significance)
Understanding the derivation of the formula makes memorization unnecessary. When a source moves toward an observer at speed v_s, the wave pattern emitted per second is compressed in front of the source to wavelength λ' = (V − v_s)/f₀. Converting this into the frequency received by the observer gives the formula above.
When the source speed equals the sound speed V, the denominator becomes 0 and the formula diverges. In reality, waves accumulate in front of the source and a shock wave forms. This produces the “sonic boom” of a supersonic jet.
The Doppler effect also applies to light. Light from a receding celestial body is shifted toward red, while light from an approaching body is shifted toward blue. Hubble observed this and discovered that the universe is expanding (Hubble's law).
The Doppler effect is widely used in modern technology, including police speed guns, weather radar, and medical blood-flow measurement.
- Approaching → higher sound (frequency increases)
- Moving away → lower sound (frequency decreases)
- Formula: f = f₀(V + v_o)/(V − v_s)
- Sign: Use + in the approaching direction
- A shock wave forms when the sound speed is exceeded
- Also applies to light: redshift and blueshift
- Applications: speed guns and weather radar
Important Notes (Common Confusions)
① In the formula, the numerator is the observer's speed and the denominator is the source's speed. Do not reverse them. ② It is less confusing to consistently take the approaching direction as +. ③ Even when the source moves, the speed of sound does not change. Sound speed is determined by the medium (air). ④ The Doppler effect is a change in wavelength; the energy of the sound (amplitude) does not change.
Practice
- When the speed of sound is 340 m/s, a source with a frequency of 500 Hz approaches an observer at 34 m/s. Find the frequency heard by the observer.
- When an ambulance moves away, does its siren sound higher or lower?
- In the Doppler-effect formula f = f₀(V + v_o)/(V − v_s), what is the sign of v_o when the observer moves away from the source?