Doppler formula

f_obs = f_src × (v_wave + v_obs) / (v_wave − v_src)

Positive velocities indicate approach; negative velocities indicate recession. As the source approaches the observer the denominator decreases, raising the observed frequency. When the source recedes, the denominator increases and the observed frequency drops.

Why does the frequency change?

Sound travels as a series of pressure waves. When a source is stationary, those waves spread out evenly in all directions and an observer hears exactly the frequency being emitted. When the source moves toward the observer, each successive wave is emitted slightly closer than the last — the waves pile up, shortening the wavelength and raising the pitch. When the source moves away, the waves spread apart, lengthening the wavelength and lowering the pitch. The source's actual sound hasn't changed; only the spacing of the waves reaching the observer has.

Speed of sound in common media

Medium	Speed of sound	Notes
Air at 20°C	343 m/s (767 mph)	Decreases with altitude and cold
Air at 0°C	331 m/s (740 mph)	Standard reference condition
Fresh water (20°C)	1480 m/s	Used in sonar calculations
Human tissue	~1540 m/s	Basis for medical ultrasound imaging
Steel	5960 m/s	Used in non-destructive testing (NDT)

Worked example

An ambulance siren emits a tone at 700 Hz. The ambulance is approaching at 96 km/h (60 mph), which is approximately 26.7 m/s. The observer is stationary. Using the formula with the speed of sound in air at 20°C (343 m/s):

f_obs = 700 × (343 + 0) / (343 − 26.7) ≈ 700 × 1.084 ≈ 759 Hz

The observer hears a pitch about 59 Hz higher than the actual siren — roughly a minor third sharper. As the ambulance passes and recedes at the same speed, the observed frequency drops to about 648 Hz, a shift of 111 Hz from the approaching tone to the receding tone.

Real-world applications

Radar speed guns fire a microwave pulse at a moving vehicle and measure the frequency shift in the reflected signal to calculate speed. Weather Doppler radar does the same with precipitation — returning echoes from raindrops are shifted higher if the rain is moving toward the antenna and lower if moving away, allowing meteorologists to map wind velocity across a storm. Medical ultrasound uses the Doppler effect to measure blood flow velocity: the transducer emits pulses and detects the frequency shift in echoes from moving red blood cells. In astronomy, the same principle scaled to light rather than sound — called redshift and blueshift — lets astronomers measure how fast distant galaxies are moving toward or away from Earth.