Waves & Oscillationsf' = f(v ± v_o)/(v ∓ v_s)

Doppler Effect Simulator

Visualize sound waves from a moving source. Adjust source speed and observe how frequency shifts for approaching and receding observers.

Parameters

Source frequency f₀

Source speed v_s

m/s

Computed

v_sound343 m/s

Mach0.233

f_ahead573.8 Hz

f_behind356.8 Hz

f₀440 Hz

v_s80 m/s

f_ahead0.0 Hz

f_behind0.0 Hz

Mach0.000

About the Doppler Effect

The Doppler effect is the change in frequency of a wave observed by someone moving relative to the wave source. When the source approaches an observer, wavefronts pile up and the observed frequency is higher. When the source recedes, wavefronts spread out and the observed frequency is lower.

ℹNamed after Austrian physicist Christian Doppler (1842). Applications include radar speed guns, medical ultrasound, weather forecasting, and measuring galactic recession (redshift).

Key Variables

Symbol	Name	Unit	Description
f₀	Source frequency	Hz	Frequency emitted by the source at rest
f_obs	Observed frequency	Hz	Frequency detected by the observer
v_s	Source speed	m/s	Speed of the source (0 to <343 m/s)
v_o	Observer speed	m/s	Speed of observer (0 in this simulator)
v	Speed of sound	m/s	343 m/s in air at 20°C
f_ahead	Frequency ahead	Hz	Frequency for observer in front of moving source
f_behind	Frequency behind	Hz	Frequency for observer behind moving source

How Wavefronts Shift

The source emits one wavefront per period T = 1/f₀. If the source moves at speed v_s toward an observer, each successive wavefront is emitted from a position closer to that observer. The apparent wavelength ahead is compressed; behind, it is stretched.

λ_{ah e a d} = \frac{v - v _{s}}{f _{0}} λ_{b e hin d} = \frac{v + v _{s}}{f _{0}}

Worked Example

A fire engine siren at f₀ = 800 Hz moves at v_s = 30 m/s (speed of sound v = 343 m/s):

f_{ah e a d} = f_{0} \cdot \frac{v}{v - v _{s}} = 800 \times \frac{343}{343 - 30} = 800 \times 1.096 \approx 877 Hz

f_{b e hin d} = f_{0} \cdot \frac{v}{v + v _{s}} = 800 \times \frac{343}{343 + 30} = 800 \times 0.920 \approx 736 Hz

As the engine passes, the observed pitch drops from 877 Hz to 736 Hz — a drop of 141 Hz, clearly audible.

The Sonic Boom

When the source speed equals the speed of sound (Mach 1), all wavefronts pile up at the front of the source. The denominator v − v_s → 0, so f_ahead → ∞. Exceeding Mach 1 creates a shock wave (sonic boom) — the source outruns its own wavefronts.

⚠The Doppler formula breaks down at and above the speed of sound. The simulator limits source speed to 300 m/s (Mach 0.87) to stay in the subsonic regime.

💡Watch the wavefront circles in the animation. Ahead of the source they are tightly packed (higher frequency); behind they are widely spaced (lower frequency).

Key Formulas

Doppler Frequency (General)

f_{o b s} = f_{0} \cdot \frac{v + v _{o}}{v - v _{s}}

Sign convention: v_o is positive when observer moves toward source; v_s is positive when source moves toward observer.

Stationary Observer — Source Moving Toward

f_{ah e a d} = f_{0} \cdot \frac{v}{v - v _{s}}

Stationary Observer — Source Moving Away

f_{b e hin d} = f_{0} \cdot \frac{v}{v + v _{s}}

Apparent Wavelengths

λ_{ah e a d} = \frac{v - v _{s}}{f _{0}} λ_{b e hin d} = \frac{v + v _{s}}{f _{0}}

Mach Number

M = \frac{v _{s}}{v _{so u n d}}

Formula	Description	Notes
f_obs = f₀(v + v_o)/(v − v_s)	General Doppler formula	All speeds relative to medium
f_ahead = f₀ v/(v − v_s)	Observer ahead (source approaching)	f > f₀; higher pitch
f_behind = f₀ v/(v + v_s)	Observer behind (source receding)	f < f₀; lower pitch
λ = v/f	Apparent wavelength	Compressed ahead, stretched behind
M = v_s/v	Mach number	M < 1: subsonic; M = 1: sonic boom
v_sound = 343 m/s	Speed of sound in air at 20°C	Increases with temperature: v ≈ 331 + 0.6T

ℹThe Doppler effect applies to all waves. For light, the relativistic Doppler formula applies: f_obs = f₀ √((1 − β)/(1 + β)) where β = v/c. Galaxies receding from us show redshift (f < f₀).

Frequently Asked Questions

Why does a passing ambulance's siren change pitch?

As the ambulance approaches, its speed compresses wavefronts ahead, raising the observed frequency (higher pitch). Once it passes and recedes, wavefronts stretch behind it, lowering the observed frequency (lower pitch). The drop in pitch you hear corresponds to the difference f_ahead − f_behind.

Does the Doppler effect apply to light?

Yes. Light from stars moving away from us is redshifted (longer wavelength, lower frequency). Light from approaching stars is blueshifted. This is how astronomers measure galactic recession and expansion of the universe. For light, the relativistic Doppler formula must be used.

What happens at Mach 1 (source speed equals sound speed)?

At Mach 1, the formula f_ahead = f₀ v/(v − v_s) → ∞ because the denominator is zero. All wavefronts emitted by the source pile up at the same point. Exceeding Mach 1 creates a conical shock wave (Mach cone) and a sonic boom.

How does a radar speed gun use the Doppler effect?

The gun emits microwave pulses at a known frequency. These reflect off a moving car and return with a Doppler-shifted frequency. The frequency difference Δf = f_obs − f₀ is proportional to the car's speed, giving an accurate speed measurement.

How is Doppler ultrasound used in medicine?

Ultrasound waves are sent into the body and reflect off moving blood cells. The frequency shift of the reflected signal reveals the blood flow speed and direction. This is used to detect blockages, measure heart valve function, and monitor fetal blood flow.