Doppler Effect Calculator — Frequency Shift

Doppler Effect Guide

What's the key thing to understand about The Doppler Formula?

f_observed = f_source × (v_sound + v_observer) / (v_sound + v_source). Sign conventions: v_observer positive when moving toward source, negative when moving away. v_source positive when moving away from observer, negative when approaching. Simplified for stationary observer and moving source: approaching: f_obs = f_s × v / (v - v_s). Receding: f_obs = f_s × v / (v + v_s). A 440 Hz source moving toward you at 30 m/s (v_sound = 343 m/s): f_obs = 440 × 343 / (343 - 30) = 440 × 1.096 = 482 Hz. Noticeably higher than the source's actual 440 Hz, which is exactly the pitch rise you hear as a vehicle approaches.

What should I know about Everyday Examples?

Ambulance siren: a siren at 700 Hz moving at 50 km/h (13.9 m/s). Approaching: f = 700 × 343/(343-13.9) = 730 Hz. Receding: f = 700 × 343/(343+13.9) = 672 Hz. Total shift heard as it passes = 730-672 = 58 Hz — clearly audible. Racing cars: Formula 1 cars at ~80 m/s. Approaching pitch at ~1.30× source, receding at ~0.81×. The characteristic 'nyeow' sound as a car passes. Supersonic aircraft: when v_source = v_sound, the formula breaks down — a sonic boom is produced (shock wave) rather than a frequency shift — a shock wave forms instead.

What's the key thing to understand about Doppler in Astronomy — Redshift?

For light, the relativistic Doppler formula applies: z = Δλ/λ = f_source/f_observed - 1. Recession velocity v ≈ z × c for v << c. Hubble's Law: galaxies are receding from us with velocity proportional to distance (v = H₀ × d). The cosmological redshift of distant galaxies provided the first evidence for the Big Bang. Edwin Hubble observed in 1929 that distant galaxies show redshift (light shifted to longer wavelengths) — implying the universe is expanding. The Cosmic Microwave Background is redshifted so dramatically by this same expansion that light originally emitted as visible or ultraviolet radiation is now detected as microwaves.

What's the key thing to understand about Medical Applications of Doppler?

Doppler ultrasound: ultrasound (2-18 MHz) reflects off moving red blood cells. The frequency shift of the reflected wave reveals blood flow speed and direction. Applications: measuring blood flow velocity in arteries and veins. Detecting deep vein thrombosis. Foetal heart monitoring. Echocardiography (heart valve function). Colour Doppler: blood flow toward the probe shown in red, away in blue. Doppler weather radar uses the same principle — the frequency shift of reflected microwave pulses from moving vehicles to measure their speed, exactly as Doppler ultrasound measures blood flow.

What's the difference between the Doppler Effect for Light and for Sound?

Sound travels through a physical medium (air), so its Doppler shift depends separately on whether the source or the observer is moving relative to that medium — giving slightly different formulas for each case, and a maximum effect when the source approaches the speed of sound itself. Light needs no medium, and Einstein's special relativity shows its Doppler shift depends only on the relative velocity between source and observer, with the same formula regardless of 'which one is moving.' For speeds much less than light speed, the relativistic redshift simplifies to the familiar Δf/f ≈ v/c, which is what astronomers use to measure how fast galaxies are receding.

What's the key thing to understand about Sonic Booms?

A sonic boom isn't a single event but a continuous shockwave cone trailing any object exceeding the local speed of sound. As the object accelerates past Mach 1, sound waves it emits can no longer outrun it, piling up into a cone-shaped shockwave with half-angle θ given by sinθ = 1/M (the Mach angle), where M is the Mach number. Anyone the cone sweeps past hears the boom at the instant it passes — not when the aircraft overhead first went supersonic — which is why a boom can seem to arrive 'out of nowhere' well after the plane has gone by.

What should I know about Doppler Radar and Speed Detection?

Police speed guns, weather radar, and traffic sensors all use the Doppler effect to measure velocity directly, exploiting the fact that a radio or microwave signal reflected off a moving object returns with a frequency shifted in proportion to that object's speed. The device emits a signal of known frequency at a target; the target (a car, a raindrop, an aircraft) reflects the wave back, and because the target is moving relative to the source, the reflected wave is Doppler-shifted twice — once as the target 'receives' the wave while approaching or receding, and again as it re-emits the reflection, which the stationary receiver also perceives with a Doppler shift. This double shift makes radar-based speed measurement about twice as sensitive as a simple one-way Doppler shift for the same relative speed, and the beat frequency between the emitted and received signals is directly proportional to the target's radial velocity (its speed directly toward or away from the radar, not its full 3D speed). Weather radar uses the same principle on a larger scale, measuring the velocity of raindrops and ice particles within storm systems to map wind patterns and detect the rotating air columns associated with tornadoes.

Worked Example: Frequency Shift of an Approaching and Receding Ambulance

An ambulance siren emits a steady tone at 700 Hz. The speed of sound in air is about 343 m/s. Using the Doppler formula for a moving source and a stationary observer, f_observed = f_source × v_sound / (v_sound ∓ v_source), the perceived pitch can be found for both approach and departure. While approaching at 30 m/s, the denominator uses subtraction: f_obs = 700 × 343 / (343 − 30) = 700 × 343 / 313 = 700 × 1.0958 ≈ 767 Hz. The siren sounds noticeably higher-pitched than its true 700 Hz. While receding at the same 30 m/s, the denominator uses addition: f_obs = 700 × 343 / (343 + 30) = 700 × 343 / 373 = 700 × 0.9196 ≈ 644 Hz, a lower pitch than the true tone. The total audible jump as the ambulance passes by — from about 767 Hz down to about 644 Hz, a drop of over 120 Hz in an instant — is the classic Doppler "pitch drop" everyone recognizes, and it happens because the source speed changes sign relative to the listener at the moment of passing, not because the siren itself changes.

Reference Table: Doppler Frequency Shift vs. Source Speed

The size of the Doppler shift depends on how fast the source moves relative to the speed of sound. For a 1,000 Hz source approaching a stationary listener through air (v_sound = 343 m/s), the percentage increase in perceived frequency grows faster than the speed itself:

Notice the shift is not linear with speed — doubling the source speed from 10 to 20 m/s roughly doubles the percentage shift, but pushing from 50 to 100 m/s more than doubles it (17.1% to 41.2%), because the formula's denominator (v_sound − v_source) shrinks faster as the source speed approaches the speed of sound itself. This is also why, as a source gets close to Mach 1, the Doppler-shifted frequency rises very steeply just before the sonic boom threshold is reached.

Doppler Shift for a Moving Observer, Stationary Source

The Doppler effect also occurs when the source is stationary and the listener is the one moving. The formula flips: f_observed = f_source × (v_sound ± v_observer) / v_sound, with addition when the observer moves toward the source and subtraction when moving away. For a 500 Hz stationary source and an observer walking toward it at 20 m/s (v_sound = 343 m/s): f_obs = 500 × (343 + 20) / 343 = 500 × 363/343 = 500 × 1.0583 ≈ 529 Hz. Moving away from the same source at 20 m/s: f_obs = 500 × (343 − 20)/343 = 500 × 323/343 = 500 × 0.9417 ≈ 471 Hz. Note that a moving observer produces a smaller shift than a moving source at the same speed — the source-moving case divides by (v_sound − v), while the observer-moving case only multiplies by (v_sound + v)/v_sound, which changes less steeply with speed.