Doppler Effect

The Doppler effect is the change between the frequency at which light or sound waves depart a source and the frequency at which they arrive at an observer, which is brought on by the observer's and the wave source's relative motion. This phenomenon is used in observing star motion, double star detection, radar, and modern navigation. Christian Doppler, an Austrian physicist, discovered it first in 1842.

Doppler Effect Definition

In physics, the way to define Doppler effect is the rise or fall in frequency of sound, light, or other waves when the source and observer move closer to or farther apart. A source's waves are compressed as they move in the direction of the observer. 

Understanding this concept will be much easier by conducting a simple Doppler effect experiment.

Doppler Effect Formula

The general formula for calculating the observed frequency (ƒ') in the Doppler Effect is:

$f^{\prime }=f\times \frac{(v+v_0)}{(v-v_s)}$

Where:

• ƒ' = observed frequency
• ƒ = source frequency
• v = speed of the wave (e.g., speed of sound in air)
• $v_o$ = speed of the observer (positive if moving toward the source, negative if moving away)
• $v_s$ = speed of the source (positive if moving toward the observer, negative if moving away)

Doppler Effect for Light

The human eye perceives a change in the frequency or wavelength of light when a light-emitting source, such as the sun, stars, or moon, moves closer or farther away. The Doppler effect for light is the name given to this shift in wavelength or frequency.

When the distance between the observer and the light source decreases, the apparent frequency of light increases, and vice versa.

• Red shift: When the source is moving away from the observer, the observer perceives a decrease in light frequency or an increase in light wavelength. Redshift is the term used to describe the displacement of the spectral lines towards the red end.
• Blue shift: In this phenomenon, the viewer perceives an increase in light frequency or a decrease in light wavelength as the source gets closer. Blueshift is the term used to describe the displacement of the spectral lines towards the blue end.

The relativistic Doppler Effect formula for light is given by:

$f^{\prime }=f\times \sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$

Where:

• c = speed of light in a vacuum
• v = relative speed between the source and observer

Difference Between Doppler Effect for Sound and Light

Doppler Effect Equation

The Doppler effect equation for sound waves is:

$f^{\prime }=f\times \frac{v+v_0}{v+v_s}$

For electromagnetic waves:

${\lambda }^{\prime }=\lambda \times \sqrt{\frac{1+v/c}{1-v/c}}$

Where:

• λ' = observed wavelength
• λ = source wavelength
• c = speed of light

Doppler Effect Example

A classic Doppler Effect example is the change in pitch of a siren on an ambulance as it approaches and then moves away from an observer. As the ambulance approaches, the siren sounds higher in pitch. Once it passes and moves away, the pitch becomes lower.

Calculation Example:
If an ambulance siren emits a frequency of 700 Hz and it approaches an observer at a speed of 30 m/s (speed of sound = 343 m/s), calculate the frequency heard by the observer.

$f^{\prime }=f\times \frac{v+v_0}{v-v_s}$

$f^{\prime }=700\times \frac{343}{343-30}$

$f^{\prime }\approx 767Hz$

Doppler Effect Uses

The Doppler Effect has several practical applications, including:

Practice Problems Doppler Effect

Q1. A train approaches a platform at a speed of 25 m/s. The train's horn emits a frequency of 400 Hz. Calculate the frequency heard by a stationary observer. (Speed of sound = 343 m/s)

Q2. A light source moves away from an observer at 0.1c. If the source emits light at a frequency of 5 × 10^14 Hz, calculate the observed frequency.

Q3. An ambulance moves away from an observer at a speed of 20 m/s. The siren emits a sound at a frequency of 750 Hz. Calculate the observed frequency.

Solved Problems for the Doppler Effect

Solution to Q1 

Given:

• Source frequency (f) = 400 Hz
• Speed of source ($v_s$) = 25 m/s
• Speed of sound (v) = 343 m/s
• Speed of observer ($v_o$) = 0 m/s (stationary)

Using the Doppler Effect formula:

$f^{\prime }=f\times \frac{v}{v-v_s}$

$f^{\prime }=400\times \frac{343}{343-25}$

$f^{\prime }\approx 400\times \frac{343}{318}$

$f^{\prime }\approx 431.76Hz$

The observed frequency is approximately 431.76 Hz.

Solution to Q2

Given:

• Source frequency (f) = $5X10^{14}Hz$
• Speed of source (v) = 0.1c
• Speed of light (c) = c

Using the relativistic Doppler Effect formula:

$f^{\prime }=f\times \sqrt{\frac{1-v/c}{1+v/c}}$

$f^{\prime }=(5\times 10^{14})\times \sqrt{\frac{1-0.1}{1+0.1}}$

$f^{\prime }\approx (5\times 10^{14})\times \sqrt{\frac{0.9}{1.1}}$

$f^{\prime }\approx (5\times 10^{14})\times 0.9055$

$f^{\prime }\approx 4.53\times 10^{14}Hz$

The observed frequency is approximately $4.5310^{14}Hz$.

Solution to Q3

Given: 

• Source frequency (f) = 750 Hz
• Speed of source ($v_s$) = 20 m/s (moving away)
• Speed of sound (v) = 343 m/s
• Speed of observer ($v_o$) = 0 m/s (stationary)

Using the Doppler Effect formula:

$f^{\prime }=f\times \frac{v}{v+v_s}$

$f^{\prime }=750\times \frac{343}{343+20}$

$f^{\prime }\approx 750\times \frac{343}{363}$

$f^{\prime }\approx 707.44Hz$

The observed frequency is approximately 707.44 Hz.