Formula & Calculator
Doppler Effect (Sound, Moving Source)
Calculates the frequency heard by a stationary observer when a sound source moves toward them at a given speed.
Interpretation
Doppler effect for moving source: f_obs = f_source · v_sound / (v_sound – v_source) when source approaches; denominator plus for receding. It changes observed frequency. Example: f=440 Hz, v_sound=343 m/s, v_source=30 m/s approaching → f_obs = 440×343/(313) ≈ 482 Hz.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| f_observed | Frequency heard by the observer | Hz |
| f_source | Frequency emitted by the source | Hz |
| v_sound | Speed of sound in the medium | m/s |
| v_source | Speed of the source toward the observer | m/s |
What it means
The Doppler effect describes the change in frequency of a wave when the source or observer is moving relative to the medium. For sound, when the source moves towards a stationary observer, the observed frequency is higher; when moving away, lower. The formula for a moving source (observer stationary) is f_obs = f_source (v_sound / (v_sound ∓ v_source)), with minus for approaching and plus for receding. This effect is used in radar and sonar to measure speed, in astronomy to measure radial velocities of stars (redshift/blueshift), and in medical ultrasound to detect blood flow. Understanding the Doppler effect is essential in acoustics, astrophysics, and many practical measurement techniques.
Worked example
Doppler Effect – Two Examples
Real‑World| Parameter | Value |
|---|---|
| f_source | 500 Hz |
| v_sound | 343 m/s |
| v_source | 20 m/s |
| Parameter | Value |
|---|---|
| f_source | 1000 Hz |
| v_source | 30 m/s |
Common mistakes
- Sign convention: This formula is for a moving source approaching a stationary observer. For receding, use v_source positive in denominator? Actually, use proper signs: f_obs = f_src (v_sound / (v_sound ∓ v_src)). Check the sign carefully.
- Sound speed v_sound: In the medium – depends on temperature, pressure.
- Source speed v_source: Must be less than v_sound – supersonic sources create shock waves (different physics).
- Stationary observer: This formula assumes the observer is at rest relative to the medium.
- Moving observer case: Different formula – do not confuse.
Applications
The Doppler effect for sound, f_observed = f_source · v_sound / (v_sound − v_source), describes the frequency change when the source moves relative to the observer. This effect is used in radar and lidar for speed measurement, in astronomy to measure stellar velocities, and in medical ultrasound for blood flow detection. Engineers apply it in traffic speed cameras, in weather radar to track precipitation, and in automobile collision avoidance systems. The formula is also used in acoustic diagnostics and to study the expansion of the universe. By understanding the Doppler effect, professionals can infer relative motion remotely, making it indispensable in many sensing and measurement systems.
- Traffic speed enforcement radar
- Weather radar and precipitation tracking
- Medical ultrasound for blood flow and cardiac motion
- Astronomical redshift and cosmic expansion studies
- Collision warning systems in autonomous vehicles
Frequently Asked Questions
When a sound source moves relative to a stationary observer, the observed frequency is shifted. If the source moves toward the observer: f_obs = f_source · (v_sound / (v_sound – v_source)). If moving away: f_obs = f_source · (v_sound / (v_sound + v_source)).
Using the wrong sign when the source moves away. Also, forgetting that it is the component of velocity along the line of sight that matters (radial velocity).
If the observer moves toward the source: f_obs = f_source · (v_sound + v_obs)/v_sound. If moving away: f_obs = f_source · (v_sound – v_obs)/v_sound.
For both source and observer moving: f_obs = f_source · (v_sound ± v_obs) / (v_sound ∓ v_source), with signs depending on direction.
As an ambulance approaches, the pitch is higher; as it recedes, the pitch is lower. This is a classic example.
For light, the formula includes relativistic effects: f_obs = f_source · √((1 ± β)/(1 ∓ β)), where β = v/c. This is used in astronomy to measure the motion of stars (redshift/blueshift).
The Mach number is the speed of the source relative to the speed of sound. When the source exceeds Mach 1, a shock wave forms (sonic boom). The Doppler formula becomes singular at v_source = v_sound.
Police radar and Doppler ultrasound measure the frequency shift of reflected waves to determine the speed of a target (car or blood flow).