Doppler Effect Equations Calculator
Doppler effect equation
Receiver Approaching
Observed frequency increases when receiver approaches source.
f′ = f₀(1 + uᵣ/v)
Receiver Receding
Observed frequency decreases when receiver moves away.
f′ = f₀(1 − uᵣ/v)
Source Approaching
Wavelengths compress when source approaches.
λᶠ = (v − uₛ) / f₀
Source Receding
Wavelengths stretch when source recedes.
λᵇ = (v + uₛ) / f₀
New Freq Approaching
Frequency from compressed wavelength.
f′ = v / λᶠ
New Freq Receding
Frequency from stretched wavelength.
f′ = v / λᵇ
How It Works
The Doppler effect describes how observed frequency changes when source or receiver is moving. Approaching raises frequency; moving apart lowers it. Applies to sound, radar, and light.
Example Problem
An ambulance siren at 700 Hz approaches at 30 m/s. Speed of sound is 343 m/s.
- Identify the knowns. Emitted source frequency f₀ = 700 Hz, source speed uₛ = 30 m/s (toward the stationary observer), and wave speed in air v = 343 m/s.
- Identify what we're solving for. We want the observed frequency f′ heard by a stationary listener as the ambulance approaches.
- Write the source-approaching formulas. The compressed wavelength reaching the observer is λᶠ = (v − uₛ) / f₀, and the new frequency is f′ = v / λᶠ.
- Substitute the known values into the wavelength step: λᶠ = (343 m/s − 30 m/s) / 700 Hz.
- Simplify the wavelength: λᶠ = 313 m/s / 700 Hz = 0.4471 m.
- Convert wavelength back to frequency: f′ = 343 m/s / 0.4471 m ≈ 767 Hz. **Observed frequency f′ ≈ 767 Hz**, about a 67 Hz upward shift — the higher pitch you hear as the ambulance approaches.
Higher pitch as the ambulance approaches.
When to Use Each Variable
- Solve for Observed Frequency — when you know the source frequency, receiver speed, and wave speed — e.g., predicting the pitch an observer hears from a passing ambulance.
- Solve for Wavelength — when a moving source emits a known frequency and you need the compressed or stretched wavelength — e.g., radar gun design.
- Solve for New Frequency — when you already know the shifted wavelength and wave speed — e.g., converting a measured wavelength back to frequency in spectroscopy.
Key Concepts
The Doppler effect arises because relative motion between source and observer compresses or stretches wave crests. For sound, the wave velocity is fixed by the medium, so motion changes the observed frequency and wavelength. For light, relativistic effects must be included. The shift magnitude depends on the ratio of object speed to wave speed — faster motion produces a larger frequency change.
Applications
- Emergency services: explaining why a siren's pitch rises as an ambulance approaches and drops as it passes
- Medical imaging: Doppler ultrasound measures blood flow velocity by detecting frequency shifts of reflected sound
- Astronomy: red-shift and blue-shift of starlight reveal whether celestial objects are moving toward or away from Earth
- Speed enforcement: radar and lidar guns use the Doppler shift of reflected microwaves to calculate vehicle speed
- Weather radar: Doppler weather stations detect wind speed and rotation inside storm cells
Common Mistakes
- Confusing source-moving and receiver-moving equations — the formulas are different because only relative motion matters for the wave medium
- Forgetting that the wave velocity v is relative to the medium, not to the source or receiver
- Applying the classical Doppler equations to light — electromagnetic waves require the relativistic Doppler formula
- Using the wrong sign convention — approaching objects increase frequency, receding objects decrease it
Frequently Asked Questions
Why does a siren sound higher when approaching?
Wave crests compress in front of the moving source, shortening the wavelength reaching a stationary listener. Because the wave still travels at the medium's fixed speed, a shorter wavelength means a higher observed frequency — that's the rising pitch you hear as the siren approaches.
Does the Doppler effect work for light?
Yes — astronomers see red-shift for receding stars and blue-shift for approaching ones. For light the classical formulas only approximate the true shift; the full result needs the relativistic Doppler equation f' = f₀ × √((1 − β) / (1 + β)) once the source velocity becomes a noticeable fraction of c.
What is the Doppler effect used for in medicine?
Doppler ultrasound transmits a known frequency into tissue and measures the shift in echoes returning from moving red blood cells. The frequency offset is proportional to blood velocity, which is how cardiologists noninvasively map flow through valves and arteries.
What is the difference between source motion and receiver motion?
When the source moves, the wavelength in the medium changes because successive crests are emitted from progressively different positions. When the receiver moves, the wavelength in the medium is unchanged but the observer encounters crests at a different rate. The two cases obey different formulas and give slightly different shifts at the same speed.
How fast does an object have to move for the Doppler shift to be noticeable?
For sound in air (v = 343 m/s), a 30 m/s ambulance produces about a 9% shift — easy to hear. For visible light (v = c ≈ 3×10⁸ m/s), you need orbital or galactic speeds for the shift to matter, which is why redshift is mostly a tool of astronomy rather than everyday measurement.
When do the classical Doppler formulas break down?
They break down when source or receiver speed approaches the wave speed. For sound, supersonic sources (uₛ ≥ v) produce shock waves rather than a Doppler-shifted tone. For light, the classical formula starts deviating from the relativistic one above a few percent of c — for accurate astronomical work you must use the relativistic Doppler equation including time dilation.
Reference:
Tipler, Paul A. 1995. Physics For Scientists and Engineers. Worth Publishers. 3rd ed.
Worked Examples
Emergency Vehicles
What wavelength does an ambulance siren produce as it approaches a curb-side observer?
An ambulance siren emits a 700 Hz tone while the ambulance drives toward a stationary listener at 25 m/s. Air temperature is 20 °C so the speed of sound is taken as 343 m/s. Find the compressed wavelength reaching the observer.
- Knowns: v = 343 m/s, u_s = 25 m/s, f₀ = 700 Hz
- Formula (source approaching): λ_f = (v − u_s) / f₀
- λ_f = (343 − 25) / 700
- λ_f = 318 / 700
λ_f ≈ 0.454 m
The compressed wavelength is shorter than the rest-frame wavelength v/f₀ = 0.490 m, which is why the siren sounds sharper while the ambulance is approaching.
Music Acoustics
What frequency does a runner hear from a stationary 440 Hz tuning fork?
A musician sprints at 5 m/s straight toward a stationary 440 Hz A4 tuning fork through 20 °C air. Find the pitch the runner actually hears as the observer-moving Doppler shift.
- Knowns: f₀ = 440 Hz, u_r = 5 m/s, v = 343 m/s
- Formula (receiver approaching): f' = f₀ × (1 + u_r / v)
- f' = 440 × (1 + 5 / 343)
- f' = 440 × 1.01458
f' ≈ 446.4 Hz
A 6 Hz upward shift from A4 is roughly a quarter-semitone — easily audible to a trained musician and a good test of pitch perception during physical motion.
Wave Physics
What pitch does a stationary observer hear when an approaching wave has a 0.45 m forward wavelength?
A moving source has compressed the wavelength of its emitted sound to 0.45 m in front of itself (the forward-going λ_f). For a stationary observer in 20 °C air, the speed of sound is 343 m/s. Find the observed frequency.
- Knowns: v = 343 m/s, λ_f = 0.45 m
- Formula (new frequency from approaching source's wavelength): f' = v / λ_f
- f' = 343 / 0.45
f' ≈ 762.2 Hz
This is the second step of the source-Doppler pipeline: once the moving source has reset the wavelength in the direction of motion, the stationary medium still propagates the wave at v, so a stationary observer reads off f' from v and the new λ.
Doppler Effect Formulas
Three pairs of equations cover the classical (non-relativistic) Doppler effect for a wave traveling through a medium. Each pair distinguishes motion toward versus motion away:
f′ = f₀ × (1 ± ur / v)Receiver moving (+ approaching, − receding)
λ = (v ∓ us) / f₀Source moving (− approaching, + receding)
f′ = v / λObserved frequency from shifted wavelength
Where:
- f₀ — emitted (source) frequency, in hertz (Hz)
- f′ — observed frequency at the receiver, in hertz (Hz)
- ur — receiver speed relative to the medium, in m/s
- us — source speed relative to the medium, in m/s
- v — propagation speed of the wave in the medium (e.g., 343 m/s for sound in 20 °C air), in m/s
- λ — wavelength of the wave reaching the observer, in meters (m)
These formulas assume the source and receiver move along the line connecting them and that speeds are small compared with the wave speed v. They apply to sound in air or water, ultrasound in tissue, and ripples on a surface. For light or radio waves the source/receiver speeds can approach the wave speed (c), so the relativistic Doppler equation should be used instead — see the FAQ for the relativistic form.
Speed of Sound in Common Media
The wave velocity v in the Doppler equations is the speed of sound (or any wave) through the medium between source and receiver. It is fastest through stiff, dense solids like aluminum and diamond, and slowest through soft or low-density media like rubber and air. Click a row to load that speed into the wave-velocity field above, keeping your frequency, wavelength, and source/receiver-velocity inputs and your current scenario (if you are solving for the wave velocity, the calculator switches to solving for the observed frequency or wavelength so the looked-up speed drives a result).
| Medium | Speed of sound (m/s) |
|---|---|
| Rubber | 60 |
| Air (20 °C) | 343 |
| Helium (20 °C) | 1,007 |
| Lead | 1,210 |
| Water (25 °C) | 1,493 |
| Seawater | 1,531 |
| Ice | 3,200 |
| Gold | 3,240 |
| Oak (wood) | 3,850 |
| Copper | 4,600 |
| Glass | 5,640 |
| Steel | 5,960 |
| Aluminum | 6,420 |
| Diamond | 12,000 |
Values at typical conditions (gases ~20 °C). The speed of sound rises with temperature in gases and varies with composition, density, and stiffness in liquids and solids, so treat these as representative figures. Synthesized from standard references (CRC Handbook of Chemistry and Physics; NIST; Engineering ToolBox). For more detail see the sound wave speed calculator.
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