9.4 The Doppler Effect

The Doppler Effect describes the apparent change in the frequency or pitch of a wave when there is relative motion between the wave source and an observer. This common phenomenon explains why the siren of an ambulance sounds higher in pitch as it approaches and lower as it moves away.

General Formula

The frequency perceived by the observer ($f^{\prime }$) is given by a general formula that accounts for the motion of both the source and the observer.

$f^{\prime }=f\left(\frac{v\pm v_o}{v\mp v_s}\right)$

Where:

• $f^{\prime }$ = Observed frequency
• $f$ = Actual frequency of the source
• $v$ = Speed of the wave in the medium
• $v_o$ = Speed of the observer
• $v_s$ = Speed of the source

Sign Convention Rule:

• Numerator ($v\pm v_o$): Use + for "towards" (observer moving towards the source).
• Denominator ($v\mp v_s$): Use - for "towards" (source moving towards the observer).

Case 1: Observer Moves Towards Stationary Source

When the observer moves towards a stationary source, they encounter wave crests more frequently, leading to a higher perceived frequency.

• Velocity of sound = $V$
• Velocity of observer = $b$

The new frequency perceived by the observer is: $f_A=\frac{V+b}{\lambda }$ Since $\lambda =V/f$, we can rewrite this as: $f_A=\left(\frac{V+b}{V}\right)\times f$ The result is $f_A>f$, meaning the frequency increases.

Case 2: Observer Moves Away from Stationary Source

When the observer moves away from the source, they encounter wave crests less frequently, resulting in a lower perceived frequency.

The new frequency is: $f_B=\frac{V-b}{\lambda }$ Using $\lambda =V/f$: $f_B=\left(\frac{V-b}{V}\right)\times f$ The result is $f_B<f$, meaning the frequency decreases.

Case 3: Source Moves Towards Stationary Observer

When the source moves towards a stationary observer, the waves in front of it are compressed, leading to a shorter wavelength and a higher frequency.

• Source velocity = $a$
• The new, shorter wavelength is: ${\lambda }_C=\frac{V-a}{f}$ The new frequency becomes: $f_C=\frac{V}{{\lambda }_C}=\left(\frac{V}{V-a}\right)\times f$ The result is $f_C>f$.

Case 4: Source Moves Away from Stationary Observer

When the source moves away from a stationary observer, the waves behind it are stretched, leading to a longer wavelength and a lower frequency.

• The new, longer wavelength is: ${\lambda }_D=\frac{V+a}{f}$ The new frequency is: $f_D=\frac{V}{{\lambda }_D}=\left(\frac{V}{V+a}\right)\times f$ The result is $f_D<f$.

Case 5: Source and Observer Both Move Towards Each Other

When both move towards each other, the two frequency-increasing effects combine, resulting in the highest perceived frequency.

• Change in wavelength: ${\lambda }_C=\frac{V-a}{f}$
• Change in velocity of sound relative to the observer: $V+b$ The new observed frequency is: $f^{\ast }=\frac{V+b}{V-a}\times f$ The result is $f^{\ast }>f$.

Case 6: Source and Observer Both Move Away from Each Other

When both move away from each other, the two frequency-decreasing effects combine, resulting in the lowest perceived frequency.

• Change in wavelength: ${\lambda }_D=\frac{V+a}{f}$
• Change in velocity of sound relative to the observer: $V-b$ The new observed frequency is: $f^{\ast \ast }=\frac{V-b}{V+a}\times f$ The result is $f^{\ast \ast }<f$.

Summary of Cases

Applications of the Doppler Effect

The Doppler Effect has wide-ranging practical applications across science and technology:

1. Radar (Radio Detection and Ranging)

Police radar guns and weather radar systems emit radio waves at a known frequency. When these waves reflect off a moving vehicle or storm system, the returned frequency is shifted. A higher returned frequency indicates the object is approaching; a lower frequency indicates it is receding. The speed is calculated from the magnitude of the frequency shift.

2. SONAR (Sound Navigation and Ranging)

SONAR systems used in submarines and ships emit ultrasonic pulses through water. The Doppler shift of the echo from a moving underwater object (submarine, fish school) allows determination of both the distance and speed of the target.

3. Medical Doppler Ultrasound

Doctors use Doppler ultrasound to measure blood flow velocity in arteries and veins. Ultrasonic waves reflect off moving red blood cells; the frequency shift reveals the speed and direction of blood flow, helping diagnose conditions like blocked arteries.

4. Astronomy — Red Shift and Blue Shift

Red Shift: When a star or galaxy moves away from Earth, the observed light frequency decreases (wavelength increases toward the red end of the spectrum). Edwin Hubble (1929) observed that virtually all distant galaxies are red-shifted, providing the first direct evidence that the universe is expanding.

Blue Shift: When a celestial body moves towards Earth, the observed light frequency increases (wavelength decreases toward the blue end). The Andromeda Galaxy shows a blue shift, indicating it is approaching the Milky Way.