Doppler Effect

Overview

The Doppler effect is the change in observed frequency (or wavelength) of a wave when there is relative motion between the source and observer.

Basic Concept

• When source approaches observer: frequency increases (higher pitch)
• When source moves away: frequency decreases (lower pitch)
• Same applies when observer moves toward or away from source

General Formula

$$f' = f \times \frac{v \pm v_o}{v \mp v_s}$$

Where:

• $f'$ = observed frequency
• $f$ = source frequency
• $v$ = wave speed in medium
• $v_o$ = observer velocity
• $v_s$ = source velocity

Sign Convention

• Upper signs: approaching
• Lower signs: receding
• Observer moving toward source: + in numerator
• Source moving toward observer: − in denominator

Special Cases

Stationary Observer, Moving Source

Source approaching:

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

Source receding:

$$f' = f \times \frac{v}{v + v_s}$$

Stationary Source, Moving Observer

Observer approaching:

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

Observer receding:

$$f' = f \times \frac{v - v_o}{v}$$

Both Moving (Same Direction)

Source following observer:

$$f' = f \times \frac{v - v_o}{v - v_s}$$

Source ahead, observer behind:

$$f' = f \times \frac{v + v_o}{v + v_s}$$

Wavelength Changes

The observed wavelength:

$$\lambda' = \frac{v}{f'} = \lambda \times \frac{v \mp v_s}{v \pm v_o}$$

For stationary observer, moving source:

$$\lambda' = \lambda\frac{v - v_s}{v} \quad \text{(approaching)}$$ $$\lambda' = \lambda\frac{v + v_s}{v} \quad \text{(receding)}$$

Doppler Effect for Light

For electromagnetic waves (where $v = c$):

Relativistic Formula

$$f' = f \times \sqrt{\frac{c - v_{\text{rel}}}{c + v_{\text{rel}}}}$$

For $v \ll c$ (non-relativistic approximation):

$$f' \approx f \times \left(1 - \frac{v_{\text{rel}}}{c}\right) \quad \text{(receding)}$$ $$f' \approx f \times \left(1 + \frac{v_{\text{rel}}}{c}\right) \quad \text{(approaching)}$$

Redshift and Blueshift

• Redshift: Source moving away → lower frequency, longer wavelength
• Blueshift: Source approaching → higher frequency, shorter wavelength

Redshift Parameter

$$z = \frac{\lambda_{\text{observed}} - \lambda_{\text{emitted}}}{\lambda_{\text{emitted}}} = \frac{\Delta\lambda}{\lambda}$$

For receding source:

$$z = \frac{v}{c} \quad \text{(for } v \ll c\text{)}$$

Shock Waves

When source moves faster than wave speed ($v_s > v$):

Mach Number

$$M = \frac{v_s}{v}$$

Shock Wave Angle

$$\sin(\theta) = \frac{v}{v_s} = \frac{1}{M}$$

• $M < 1$: Subsonic
• $M = 1$: Sonic (sound barrier)
• $M > 1$: Supersonic (sonic boom)

Examples

Example 1: Car Horn

A car horn ($f = 400$ Hz) approaches at 30 m/s. Find observed frequency ($v = 343$ m/s).

$$f' = f \times \frac{v}{v - v_s} = 400 \times \frac{343}{343 - 30} = 400 \times \frac{343}{313} = 438 \text{ Hz}$$

Example 2: Police Siren

A police car (siren at 1000 Hz) moves at 40 m/s toward a stationary observer, then passes.

Approaching:

$$f' = 1000 \times \frac{343}{343 - 40} = 1133 \text{ Hz}$$

Receding:

$$f' = 1000 \times \frac{343}{343 + 40} = 896 \text{ Hz}$$

Frequency change: $1133 - 896 = 237$ Hz

Example 3: Moving Observer

A person runs at 5 m/s toward a stationary 500 Hz source.

$$f' = 500 \times \frac{343 + 5}{343} = 507 \text{ Hz}$$

Example 4: Both Moving

A train (500 Hz whistle) moves at 20 m/s toward a car moving at 15 m/s toward the train.

$$f' = 500 \times \frac{343 + 15}{343 - 20} = 500 \times \frac{358}{323} = 554 \text{ Hz}$$

Example 5: Sonic Boom

A jet flies at Mach 2.5. Find the shock wave angle.

$$\sin(\theta) = \frac{1}{M} = \frac{1}{2.5} = 0.4$$ $$\theta = \sin^{-1}(0.4) = 23.6°$$

Example 6: Galaxy Redshift

A galaxy shows hydrogen line at 680 nm (lab value: 656 nm). Find recession velocity.

$$z = \frac{\Delta\lambda}{\lambda} = \frac{680 - 656}{656} = 0.0366$$ $$v = zc = 0.0366 \times 3 \times 10^8 = 1.1 \times 10^7 \text{ m/s}$$

Applications

• Radar speed guns: Police use Doppler radar to measure vehicle speed
• Medical ultrasound: Doppler imaging for blood flow
• Astronomy: Measuring stellar velocities and cosmic expansion
• Weather radar: Tracking storm movements