Doppler Effect

The Doppler Effect: An Overview

The Doppler Effect is the change in Apparent Frequency of a wave when the source and observer are in Relative Motion.

$$f_{obs} \neq f_{source}$$

The observed frequency deviates from the source frequency due to motion.

$f_{obs}$=Frequency heard by observer(Hz)

$f_{source}$=Frequency emitted by source(Hz)

Frequency-Pitch Link: In sound, this is perceived as a change in Pitch.

Wave Fronts: Motion causes Wave Fronts to bunch up or spread out in the medium.

Key Distinction: The source frequency $f$ never changes — only the observed frequency $f'$ shifts due to motion.

Medium Dependence: The Doppler effect for sound depends on motion relative to the medium (air), not just relative motion between source and observer.

Case 1: The Observer is Moving

When an observer moves towards a Stationary Source, they encounter wave crests more frequently, increasing the Modified Frequency.

$$f_A = f \left( \frac{v \pm u_o}{v} \right)$$

The relative speed of the waves changes as the observer moves through the medium.

$v$=Speed of sound in medium(m/s)

$u_o$=Velocity of the observer(m/s)

$f$=Source frequency(Hz)

$u_o$ towards source

→

Use $+$ (Higher frequency)

$u_o$ away from source

→

Use $-$ (Lower frequency)

Wavelength Stability: Unlike a moving source, the physical wavelength in the medium remains constant.

Relative Velocity: The observer perceives a higher wave speed ($v + u_o$) when moving toward the source.

Proportionality: The fractional frequency shift is $\Delta f / f = u_o / v$ — directly proportional to observer speed.

Limiting Case: When $u_o = 0$, $f' = f$ — no shift. When $u_o = v$ (observer at speed of sound towards source), $f' = 2f$ — frequency doubles.

Case 2: The Source is Moving

A moving source physically alters the wavelength in the medium, causing Wavelength Compression ahead and expansion behind.

$$f_C = f \left( \frac{v}{v \mp u_s} \right)$$

The distance between wave crests changes because the source 'chases' its own emitted waves.

$u_s$=[Source Velocity](m/s)

$v$=Speed of sound(m/s)

$u_s$ towards observer

→

Use $-$ in denominator (Increases frequency)

$u_s$ away from observer

→

Use $+$ in denominator (Decreases frequency)

Doppler Shift: The change in wavelength is calculated as $\Delta \lambda = u_s / f$.

Wave Speed: The speed of waves ($v$) relative to the medium is unaffected by the source's speed.

Asymmetry: Source approaching gives $f' = f \cdot v/(v - u_s)$ and receding gives $f' = f \cdot v/(v + u_s)$. The approaching shift is always larger than the receding shift for the same speed.

Limiting Case: As $u_s \to v$, the denominator $(v - u_s) \to 0$ and $f' \to \infty$ — the source catches up to its own waves, creating a Shock Wave (sonic boom).

Case 3: Both Source and Observer Moving

When both the source and observer are in motion, both effects combine into a single general formula for the Modified Frequency.

$$f' = f \left( \frac{v \pm u_o}{v \mp u_s} \right)$$

The numerator handles observer motion (changes how fast the observer encounters waves) and the denominator handles source motion (changes the physical wavelength).

$v$=Speed of sound in medium(m/s)

$u_o$=Observer velocity(m/s)

$u_s$=Source velocity(m/s)

Both approaching each other

→

$f' = f(v + u_o)/(v - u_s)$ — maximum frequency increase

Both receding from each other

→

$f' = f(v - u_o)/(v + u_s)$ — maximum frequency decrease

Same speed, same direction

→

$f' = f$ — no shift (relative velocity is zero)

Sign Convention: Use $+u_o$ when observer moves towards source, $-u_o$ when away. Use $-u_s$ when source moves towards observer, $+u_s$ when away.

Memory Aid: Think 'towards = higher frequency'. In the formula: numerator increase or denominator decrease both raise $f'$.

Echo Problems: When sound reflects off a surface (wall, cliff), treat it as two Doppler shifts — the reflecting surface acts as both receiver and re-emitter.

Applications in Technology and Astronomy

The Doppler Effect is a fundamental tool for measuring velocity in Radar, Sonar, and medical imaging.

$$\Delta \lambda = \lambda_{obs} - \lambda_{source}$$

The Doppler shift in wavelength indicates the direction and speed of relative motion.

$\Delta \lambda$=Change in wavelength(m (or nm for light))

$\lambda_{obs}$=Wavelength measured by observer

$\lambda_{source}$=Wavelength emitted at rest

$\Delta \lambda > 0$

→

Red Shift — source moving away (wavelength increases)

$\Delta \lambda < 0$

→

Blue Shift — source approaching (wavelength decreases)

Red Shift: Distant galaxies moving away exhibit a shift toward longer (red) wavelengths — key evidence for the expanding universe.

Blue Shift: Stars or galaxies approaching Earth have spectra shifted toward shorter (blue) wavelengths.

Radar Speed Trap: Microwaves reflected off a moving car are Doppler-shifted; the shift is doubled because the signal bounces back.

Sonar: Underwater echo-ranging uses sound waves. Doppler detection measures target speed via frequency shift of echoes — used for submarine detection and depth measurement.

Echolocation: Bats and dolphins use Doppler shifts of reflected ultrasound to determine the speed of moving prey.

Blood Flow: Medical ultrasound (5–10 MHz) directed at arteries detects backscattered signal shifts to measure blood flow velocity.