Doppler Effect Calculator — Calculate Observed Frequency for Moving Sources

The Doppler Effect Formula

The general Doppler formula for sound (or any mechanical wave in a medium) is:

f₀ = fₛ × (v + vₒ) / (v − vₛ)

This formula uses the sign convention where positive velocities are directed toward the opposite party:

• vₒ is positive when the observer moves toward the source
• vₒ is negative when the observer moves away from the source
• vₛ is positive when the source moves toward the observer
• vₛ is negative when the source moves away from the observer

When both source and observer are stationary (vₒ = 0, vₛ = 0): f₀ = fₛ × v/v = fₛ — no shift.

Variables in the Doppler Calculator

How to Use the Doppler Effect Calculator

1. Open the Doppler effect calculator.
2. Enter the source frequency (fₛ) — the natural frequency when stationary.
3. Enter the wave speed (v) — use 343 m/s for sound in air at 20°C, or the actual medium speed.
4. Enter the source velocity (vₛ) — positive if moving toward the observer, negative if moving away. Use 0 if stationary.
5. Enter the observer velocity (vₒ) — positive if moving toward the source, negative if moving away. Use 0 if stationary.
6. Click Calculate. The tool returns observed frequency (f₀) and the frequency shift (f₀ − fₛ).

Common Scenarios and Their Effects

Worked Example: Ambulance Siren

An ambulance emits a siren at 700 Hz and is travelling at 30 m/s toward a stationary pedestrian. The speed of sound in air is 343 m/s.

Using: f₀ = fₛ × (v + vₒ) / (v − vₛ) = 700 × (343 + 0) / (343 − 30)

f₀ = 700 × 343/313 ≈ 700 × 1.096 ≈ 767 Hz

As the ambulance passes and moves away: vₛ = −30 m/s (now moving away).

f₀ = 700 × 343 / (343 − (−30)) = 700 × 343/373 ≈ 700 × 0.920 ≈ 644 Hz

The siren drops from ~767 Hz to ~644 Hz as it passes — a shift of 123 Hz, clearly audible as the characteristic pitch drop of a passing emergency vehicle.

Speed of Sound in Different Media

The Doppler formula depends on the wave speed v, which varies with the medium:

• Air at 0°C: 331 m/s
• Air at 20°C: 343 m/s (standard for most problems)
• Air at 30°C: 349 m/s
• Water at 25°C: ~1,480 m/s
• Steel: ~5,100 m/s
• Concrete: ~3,400 m/s

Sound travels faster in denser, stiffer media. Temperature also increases sound speed in gases: v ≈ 331 + 0.6T m/s, where T is temperature in Celsius.

Real-World Applications

The Doppler Effect for Light (Relativistic)

For electromagnetic radiation (light, radio waves, microwaves), the classical Doppler formula does not apply exactly because light travels at a fixed speed (c ≈ 3 × 10⁸ m/s) regardless of the motion of source or observer — a consequence of special relativity.

The relativistic Doppler formula is:

f₀ = fₛ × √((c + v)/(c − v)) (when source and observer approach each other)

For speeds much less than c (v << c), the relativistic and classical formulas give nearly identical results. Police radar guns, weather radar, and most engineering applications use the classical approximation because vehicle speeds are tiny fractions of the speed of light.

Astronomical applications (measuring galaxy recession speeds, gravitational redshift from black holes) require the relativistic formula. The famous "redshift" of distant galaxies — where light from galaxies moving away from us is shifted toward longer, redder wavelengths — underpins the Big Bang theory and Hubble's law.

The Mach Number and Sonic Booms

When a source moves faster than the speed of sound (Mach 1 or greater), the classical Doppler formula produces a denominator of zero or negative — no finite observed frequency makes sense. What actually happens is that the wave fronts pile up ahead of the source in a Mach cone (shock wave). The shock wave, when it reaches a stationary observer, is experienced as a sonic boom — a sudden pressure discontinuity rather than a sustained frequency shift.

The Mach number M = vₛ/v (source speed divided by wave speed). Mach 1 is the sound barrier; Mach 2 means the source moves at twice the speed of sound. The Doppler calculator assumes subsonic speeds (M < 1); for supersonic sources, the physics is qualitatively different.

Measuring Source Speed from Frequency Shift

Police radar guns measure the Doppler shift of reflected microwave signals to determine vehicle speed. For a stationary radar gun reflecting off a moving vehicle:

The vehicle acts as both a moving observer (approaching the radar wave) and a moving source (reflecting the wave back). The frequency shift is doubled compared to a one-way Doppler measurement. The radar computer calculates vehicle speed from this doubled shift. This is why radar guns do not need to time a vehicle over a fixed distance — a single measurement of frequency shift is sufficient.

Common Questions

Does the Doppler effect change the amplitude (loudness) of sound?

The classical Doppler formula changes frequency, not amplitude. However, in practice, an approaching sound source also gets louder as it gets closer (inverse-square law — intensity decreases with the square of distance). The two effects are separate: Doppler changes pitch; distance changes loudness.

Does the Doppler effect work in all directions?

The formula f₀ = fₛ(v + vₒ)/(v − vₛ) assumes the source and observer are moving directly toward or away from each other (along the line connecting them). When the source moves at an angle to the line connecting it to the observer, you must use the component of velocity along that line. At 90° (perpendicular motion), there is a transverse Doppler effect for light (purely relativistic), but for sound, the frequency shift is negligible.

Why does the pitch drop as a vehicle passes rather than changing gradually?

As a vehicle approaches, the observed frequency is consistently higher than the source frequency. As it passes and begins moving away, it is consistently lower. The pitch does not change gradually during approach — it stays at the elevated "approaching" frequency until the vehicle is at its closest point, then drops to the lower "receding" frequency. This abrupt shift at the moment of passing is why it sounds like a sudden drop.