Doppler Effect — Physica

Equivalent forms

f' = f \frac{v}{v - v_s}

f' = f \frac{v + v_o}{v}

\frac{\Delta f}{f} = \frac{v_s}{v}

A single fraction captures how relative motion reshapes frequency — the same physics behind radar guns and redshifted galaxies.

Symbol	Name	SI unit	Dimension	Range
$f_prime$	Observed frequencyoutput Frequency perceived by the observer	Hz	T^-1	1 – 2000
$f$	Source frequency Frequency emitted by the source at rest	Hz	T^-1	1 – 2000
$v$	Wave speed in medium Speed of the wave in the medium (e.g., speed of sound in air)	m/s	LT^-1	100 – 1500
$v_o$	Observer velocity Speed of the observer toward the source (positive = approaching)	m/s	LT^-1	-100 – 100
$v_s$	Source velocity Speed of the source toward the observer (positive = approaching)	m/s	LT^-1	-100 – 100

Unit systems

Symbol	SI	CGS	Imperial
$f_prime$	Hz	Hz	Hz
$f$	Hz	Hz	Hz
$v$	m/s	cm/s	ft/s
$v_o$	m/s	cm/s	ft/s
$v_s$	m/s	cm/s	ft/s

Where it holds

Valid for mechanical waves when source speed < wave speed (subsonic). For electromagnetic waves, use the relativistic Doppler formula instead.

Dimensional analysis

f' \to [T^{-1}]

f\cdot (v + v_o)/(v - v_s) \to [T^{-1}]\cdot [LT^{-1}]/[LT^{-1}] = [T^{-1}]

Both sides are

[T^{-1}] = Hz

.

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Discovery

Christian Doppler · 1842

Doppler predicted the effect for light from binary stars. Buys Ballot confirmed it in 1845 using musicians on a moving train.

Try this

Why does an ambulance siren sound higher-pitched as it approaches and lower as it drives away?

An ambulance siren emits at 700 Hz, approaching at 30 m/s. Find the perceived frequency using f' = f * v/(v - v_s) with v = 343 m/s.

Research status: stable

Real-world applications

Radar speed guns: police measure vehicle speed from the frequency shift of reflected microwaves.
Medical ultrasound: Doppler imaging measures blood flow velocity in arteries.
Astronomical redshift: recession velocity of galaxies reveals the expansion of the universe.
Weather radar: Doppler radar detects wind speed and rotation in storm cells.

Common misconceptions

The Doppler effect changes the speed of sound — it changes the observed frequency and wavelength, not the wave speed in the medium.
It only applies to sound — it applies to all waves, including light (redshift/blueshift of galaxies).
Source and observer motion are equivalent — for mechanical waves they are not: the medium defines a preferred frame. Only for light (relativistic Doppler) are they symmetric.

Experimental verification

Buys Ballot (1845) placed trumpeters on a train to verify pitch shifts. Modern: police radar guns use the electromagnetic Doppler effect, and Doppler ultrasound measures blood flow velocity in medical imaging.

Derivation

A source emitting at frequency f produces wavefronts separated

by \lambda = v/f

.

If the source moves at v_s toward the observer, each successive wavefront is emitted v_s/f closer, compressing the effective wavelength

to \lambda ' = (v - v_s)/f

.

The observer moving at v_o toward the source encounters wavefronts at rate

f' = (v + v_o)/\lambda ' = f\cdot (v + v_o)/(v - v_s)

.

Limiting cases

v_s \to 0

,

v_o \to 0

⟶

f' = f

No relative motion means no frequency shift.

v_s \to v

⟶

f' \to \infty

Source at wave speed: wavefronts pile up into a sonic boom (Mach 1).

v_s \to -v

(receding at wave speed)⟶

f' \to f/2

Source receding at wave speed halves the observed frequency.

What if…

What if

v_s = v

(source at wave speed)?

Denominator $\to 0$ , $f' \to \infty$ . All wavefronts pile up into a shock wave — the sonic boom at Mach 1.

What if both source and observer move toward each other?

Both effects compound: $f' = f\cdot (v + v_o)/(v - v_s)$ . The frequency shift is larger than either motion alone.

What if applied to light?

For electromagnetic waves, use the relativistic Doppler formula: $f' = f\cdot \surd ((1+\beta )/(1-\beta ))$ where $\beta = v/c$ . There is no medium, so only relative velocity matters.

1

Approaching ambulance siren

Given ·

f:: 700
v:: 343
v s:: 30
v o:: 0

Find · f_prime

Steps

Observer stationary $(v_o = 0)$ , source approaching $(v_s = 30\,\mathrm{m/s})$
$f' = f \times (v + v_o)/(v - v_s) = 700 \times (343 + 0)/(343 - 30)$
$f' = 700 \times 343/313$
$f' = 767.1\,\mathrm{Hz}$ — pitch sounds higher

Result ·

f' = f \times v/(v - v_s) = 700 \times 343/(343 - 30) = 700 \times 343/313 = 767.1\,\mathrm{Hz}

2

Receding ambulance siren

Given ·

f:: 700
v:: 343
v s:: -30
v o:: 0

Find · f_prime

Steps

Source receding: $v_s = -30$ (moving away)
$f' = 700 \times 343/(343 - (-30)) = 700 \times 343/373$
$f' = 643.7\,\mathrm{Hz}$ — pitch sounds lower
Frequency drop: $767.1 \to 643.7\,\mathrm{Hz}$ as ambulance passes

Result ·

f' = 700 \times 343/(343 + 30) = 700 \times 343/373 = 643.7\,\mathrm{Hz}

Wave Speed Equation→