What is the Doppler Effect?
The Doppler Effect is the change in observed frequency of a wave when the source or observer is in relative motion. When they approach each other, frequency increases (blue shift); when they recede, frequency decreases (red shift). It applies to sound, light, and all wave phenomena.
The General Doppler Formula
| Symbol | Quantity | Unit | Sign Convention |
|---|
| f' | Observed (apparent) frequency | Hz | Result |
| f | Source (actual) frequency | Hz | Must be > 0 |
| v | Speed of sound in the medium | m/s | Always positive |
| vo | Speed of observer | m/s | + when moving towards source |
| vs | Speed of source | m/s | - when moving towards observer |
Physical Interpretation
When the source moves towards the observer, the wavefronts compress, reducing the wavelength and increasing observed frequency. Conversely, a receding source stretches wavefronts, lowering the pitch. This is the principle behind SONAR, Doppler radar, medical ultrasound, and the redshift of distant galaxies used to prove cosmic expansion.
The effect is asymmetric: it matters whether the source moves or the observer moves, even if the relative speed is the same, because sound propagates relative to the medium, not relative to the observer (unlike light in special relativity).
Special Cases and Derived Formulas
Case 1 — Source Moving, Observer at Rest
f' = f × v ÷ ( v ∓ vs )
Use the minus sign when the source approaches (f' > f) and the plus sign when it recedes (f' < f).
Case 2 — Observer Moving, Source at Rest
f' = f × ( v ± vo ) ÷ v
Use the plus sign when the observer approaches (f' > f) and the minus sign when the observer recedes (f' < f).
Case 3 — Both Moving in the Same Direction
f' = f × ( v - vo ) ÷ ( v - vs ) [Source and observer both moving in same direction]
Case 4 — Sonic Boom (vs = v)
When a source travels at exactly the speed of sound (Mach 1), the denominator becomes zero and f' approaches infinity. All wavefronts pile up at a single point, creating a shock wave or sonic boom. The calculator correctly flags this as a physically impossible input.
The Mach Number
Mach Number (M) = vs ÷ v
Mach 1 = speed of sound; Mach 2 = twice the speed of sound. Supersonic motion (M > 1) produces a Mach cone with half-angle sin(θ) = 1/M.
Solved Examples (NCERT / JEE Level)
Example 01 — Source Approaching Observer
A train sounds its horn at 800 Hz while approaching a station platform at 72 km/h. If the speed of sound is 340 m/s, what is the frequency heard by a person standing on the platform?
Given: f = 800 Hz, vs = 72 km/h = 72 × (1000/3600) = 20 m/s
v = 340 m/s, vo = 0 (observer is stationary)
Formula: f' = f × v ÷ (v - vs) [source approaching]
Substituting: f' = 800 × 340 ÷ (340 - 20)
f' = 800 × 340 ÷ 320
f' = 800 × 1.0625
f' = 850 Hz | Blue Shift — Frequency increased by 50 Hz
Example 02 — Observer Moving Towards a Stationary Source
A cyclist moves at 18 km/h towards a stationary factory siren of frequency 1000 Hz. Speed of sound = 340 m/s. Find the apparent frequency.
Given: f = 1000 Hz, vo = 18 km/h = 18 × (1/3.6) = 5 m/s
v = 340 m/s, vs = 0
Formula: f' = f × (v + vo) ÷ v [observer approaching]
Substituting: f' = 1000 × (340 + 5) ÷ 340
f' = 1000 × 345 ÷ 340
f' = 1000 × 1.01471
f' = 1014.71 Hz | Blue Shift — Frequency increased by 14.71 Hz
Example 03 — Source Receding, Observer Also Receding
A source of frequency 600 Hz moves away from an observer at 10 m/s. The observer also moves away from the source at 5 m/s. Speed of sound = 340 m/s. Find the observed frequency.
Given: f = 600 Hz, vs = 10 m/s (away), vo = 5 m/s (away)
v = 340 m/s
Formula: f' = f × (v - vo) ÷ (v + vs) [both receding]
Substituting: f' = 600 × (340 - 5) ÷ (340 + 10)
f' = 600 × 335 ÷ 350
f' = 600 × 0.9571
f' = 574.29 Hz | Red Shift — Frequency decreased by 25.71 Hz
Frequently Asked Questions (FAQs)
What is the Doppler Effect in simple terms?
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The Doppler Effect is the perceived change in pitch (frequency) of a sound when the source and listener are moving relative to each other. A classic real-life example is an ambulance siren: as it approaches you, the pitch sounds higher than it actually is; as it moves away, the pitch sounds lower. This happens because motion compresses or stretches the sound waves reaching your ears.
Does the Doppler Effect apply to light?
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Yes. For electromagnetic waves including light, the relativistic Doppler Effect governs the frequency shift. When stars or galaxies move away from Earth, their light shifts towards the red end of the spectrum (redshift). This was used by Edwin Hubble to discover that the universe is expanding. When objects approach, their light undergoes blueshift. Unlike sound, the relativistic Doppler formula is symmetric since light speed is absolute.
Why is the Doppler Effect asymmetric for sound?
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For sound, the same relative speed between source and observer gives a different frequency shift depending on who is moving, because sound propagates relative to the medium (air, water, etc.), not relative to either party. If the source moves at 20 m/s towards a stationary observer, the result differs from an observer moving at 20 m/s towards a stationary source, even though the relative speed is identical. This asymmetry does not exist for light, where only relative motion matters.
What is the sign convention in the Doppler formula?
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The NCERT/JEE standard convention: In the numerator, use v + vo when the observer moves towards the source, and v − vo when moving away. In the denominator, use v − vs when the source moves towards the observer, and v + vs when moving away. A helpful memory device: the sign favoring approach goes in the direction that increases the overall fraction (and hence the frequency).
What are the real-life applications of the Doppler Effect?
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The Doppler Effect has extensive real-world applications: (1) Medical Ultrasound — measures blood flow velocity in arteries. (2) Doppler Radar — used by meteorologists to track storm speed and direction. (3) Speed Guns (Radar Guns) — police use microwave Doppler to measure vehicle speed. (4) SONAR — submarines use Doppler-shifted echoes to detect and range other vessels. (5) Astronomy — determines stellar velocities and planetary rotation rates.