482.1725
Hz
42.1725
Hz
9.58
%
0.779545
m
0.711364
m
0.0875
0
1.095847
482.1725
Hz
42.1725
Hz
9.58
%
0.779545
m
0.711364
m
0.0875
0
1.095847
The Doppler Effect Calculator computes the observed frequency when a sound source and observer are in relative motion. Named after Austrian physicist Christian Doppler (1842), this effect explains why an ambulance siren sounds higher-pitched as it approaches and lower-pitched as it recedes. The general formula is:
$$f' = f_0 \frac{v \pm v_o}{v \mp v_s}$$
where $$f_0$$ is the source frequency, $$v$$ is the speed of sound, $$v_o$$ is the observer's velocity, and $$v_s$$ is the source's velocity. The upper signs apply when source and observer approach each other; the lower signs when they recede.
The Doppler effect has far-reaching applications beyond everyday experience. Police radar guns use the Doppler shift of reflected microwaves to measure vehicle speed. Medical Doppler ultrasound measures blood flow velocity. Astronomers use the cosmological redshift — the Doppler effect for light — to determine how fast distant galaxies are receding, providing evidence for the expanding universe.
This calculator handles four scenarios: source approaching, source receding, observer approaching, and observer receding. It computes the observed frequency, frequency shift, percentage change, and the apparent wavelength heard by the observer.
The classical Doppler formula for sound waves in a medium is:
$$f' = f_0 \left(\frac{v + v_o}{v - v_s}\right) \quad \text{(approaching)}$$
$$f' = f_0 \left(\frac{v - v_o}{v + v_s}\right) \quad \text{(receding)}$$
Convention: $$v_o$$ is positive when the observer moves toward the source. $$v_s$$ is positive when the source moves toward the observer.
The frequency shift is $$\Delta f = f' - f_0$$. A positive shift means the observer hears a higher pitch; a negative shift means a lower pitch.
The Mach number $$M = v_s / v$$ indicates how fast the source moves relative to the speed of sound. At $$M = 1$$ (sonic barrier), the denominator approaches zero and the observed frequency theoretically diverges — this corresponds to the formation of a shock wave (sonic boom).
Important: This formula applies to mechanical waves (sound) where there is a physical medium. For electromagnetic waves (light), the relativistic Doppler formula is needed, which accounts for time dilation.
An approaching source compresses the wavefronts ahead of it, reducing the wavelength and increasing the observed frequency. A receding source stretches the wavefronts, increasing wavelength and decreasing frequency. The asymmetry is notable: approaching from Mach 0.5 doubles the frequency (100% increase), but receding at Mach 0.5 only reduces it by one-third (33% decrease).
Inputs
Results
An ambulance with a 700 Hz siren approaching at 30 m/s (108 km/h) sounds about 67 Hz higher — a noticeable pitch increase of nearly 10%.
Inputs
Results
A train horn at 500 Hz moving away at 40 m/s (144 km/h) sounds about 52 Hz lower — the familiar drop in pitch as a train passes by.
The formula is asymmetric: approaching compresses wavefronts by a factor of $$(1 - v_s/v)$$ while receding stretches them by $$(1 + v_s/v)$$. This means the frequency increase for approaching is always larger than the frequency decrease for receding at the same speed. The pitch jump at the moment of passing is therefore perceived as sudden.
When $$v_s = v$$, the source moves as fast as its own wavefronts. The waves pile up into a cone-shaped shock wave (Mach cone), producing a sonic boom. The Doppler formula gives an infinite observed frequency at this point, indicating the classical formula breaks down — the wave behavior transitions to a shock regime.
Yes, but light requires the relativistic Doppler formula: $$f' = f_0 \sqrt{(1 + \beta)/(1 - \beta)}$$, where $$\beta = v/c$$. Unlike sound, light has no medium, so only relative velocity matters. The astronomical redshift ($$z = \Delta\lambda/\lambda$$) is the Doppler effect applied to starlight.
A radar gun emits a microwave beam at a known frequency. The beam reflects off a moving vehicle and returns with a Doppler-shifted frequency. The frequency shift is proportional to the vehicle's speed: $$v = c \Delta f / (2f_0)$$. The factor of 2 accounts for the double Doppler shift (outgoing and reflected waves).
Medical Doppler ultrasound sends sound waves into the body and measures the frequency shift of echoes reflected from moving blood cells. This reveals blood flow velocity and direction, helping diagnose conditions like deep vein thrombosis, heart valve problems, and arterial blockages — all non-invasively.
Yes. The general formula accounts for both: $$f' = f_0 (v \pm v_o) / (v \mp v_s)$$. This calculator handles one motion at a time for clarity, but in real situations (two cars passing on a highway, for example), both velocities contribute to the total frequency shift.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
