The Doppler Effect — Science Sage

Definition and Cause

The Doppler effect is the apparent change in frequency (and wavelength) of a wave as perceived by an observer when there is relative motion between the source and the observer.

The effect occurs because the relative motion changes the number of wavefronts that reach the observer per second:

Source approaching observer: wavefronts are compressed (shorter wavelength) → observer hears higher frequency (higher pitch)
Source receding from observer: wavefronts are stretched (longer wavelength) → observer hears lower frequency (lower pitch)

The frequency of the source does NOT change — only the perceived (observed) frequency changes due to relative motion.

The Doppler Formula

The observed (listener) frequency f_L is given by:

f_L = ((v ± v_L) / (v ∓ v_S)) × f_S

Symbol	Meaning	Unit
f_L	Frequency heard by listener (observer)	Hz
f_S	Frequency emitted by source	Hz
v	Speed of sound in medium	m·s⁻¹
v_L	Speed of listener	m·s⁻¹
v_S	Speed of source	m·s⁻¹

Sign Convention for the Formula

Numerator (v ± v_L):
• Use +v_L if the listener is moving toward the source
• Use −v_L if the listener is moving away from the source

Denominator (v ∓ v_S):
• Use −v_S if the source is moving toward the listener
• Use +v_S if the source is moving away from the listener

Memory tip: Motion that brings source and listener closer → f_L increases. Motion that moves them apart → f_L decreases.

Standard value: speed of sound in air = 340 m·s⁻¹ (at approximately 20°C).

Moving Source Only (Listener Stationary)

If the listener is stationary (v_L = 0):

Source approaching: f_L = v / (v − v_S) × f_S
Source receding: f_L = v / (v + v_S) × f_S

Moving Listener Only (Source Stationary)

If the source is stationary (v_S = 0):

Listener approaching: f_L = (v + v_L) / v × f_S
Listener receding: f_L = (v − v_L) / v × f_S

Real-World Applications

Emergency sirens: ambulance or police siren sounds higher pitched as it approaches, lower as it recedes
Echolocation (bats): bats emit ultrasound pulses; reflected Doppler shift gives speed and direction of prey
Radar speed guns: emit radio waves at known frequency; reflected wave has Doppler-shifted frequency; shift gives vehicle speed
Medical Doppler ultrasound: measures blood flow speed in arteries and veins from frequency shift of reflected ultrasound
Redshift / blueshift (light): light from stars/galaxies moving away is red-shifted (lower f); light from approaching sources is blue-shifted (higher f)

Worked Example 1 — Moving Source

Problem: An ambulance emits a siren at 800 Hz and moves toward a stationary observer at 30 m·s⁻¹. Speed of sound = 340 m·s⁻¹. Find the frequency heard by the observer.

Given: f_S = 800 Hz, v_S = 30 m·s⁻¹ (toward), v_L = 0, v = 340 m·s⁻¹
f_L = v / (v − v_S) × f_S = 340 / (340 − 30) × 800 = 340/310 × 800 = 876.9 Hz

Worked Example 2 — Both Moving

Problem: A train (f_S = 500 Hz) moves away from an observer at 20 m·s⁻¹. The observer runs toward the train at 5 m·s⁻¹. v = 340 m·s⁻¹.

Source receding: denominator uses +v_S
Listener moving toward source: numerator uses +v_L
f_L = (340 + 5) / (340 + 20) × 500 = 345/360 × 500 = 479.2 Hz

IEB Extension — Redshift and Hubble's Law

For light, the Doppler effect causes redshift (recession) or blueshift (approach). Edwin Hubble discovered that distant galaxies are redshifted, and the recession speed is proportional to distance:

v = H₀ × d (Hubble's Law)

H₀ ≈ 70 km·s⁻¹·Mpc⁻¹ (Hubble constant). This provides evidence for the expanding universe. The fractional shift in wavelength: Δλ/λ = v/c for v ≪ c. IEB questions may ask you to calculate recession speed from observed and emitted wavelengths, then infer distance using Hubble's Law.

Doppler Wavefront Simulator

Source Controls

Source speed 80 m·s⁻¹

Source frequency 400 Hz

Calculated Frequencies

f source

400Hz

f (ahead)

—Hz

f (behind)

—Hz

Question 1 of 8

A source emits sound at 600 Hz and moves toward a stationary listener at 34 m·s⁻¹. Speed of sound = 340 m·s⁻¹. The frequency heard is:

Question 2 of 8

A stationary source emits at 500 Hz. A listener moves away from the source at 34 m·s⁻¹. Speed of sound = 340 m·s⁻¹. The frequency heard is:

Question 3 of 8

Which of the following correctly explains why a passing ambulance sounds higher pitched as it approaches than when it recedes?

Question 4 of 8

A bat emits ultrasound at 80 000 Hz while flying toward a wall at 10 m·s⁻¹. Speed of sound = 340 m·s⁻¹. The frequency of the echo reflected from the wall (wall acts as a stationary reflector) back to the bat is approximately:

Question 5 of 8

In which situation would the Doppler formula give f_L = f_S?

Question 6 of 8

The Doppler effect for light from a distant galaxy moving away from Earth results in:

Question 7 of 8

A car sounds its horn at 500 Hz while moving toward a cyclist at 20 m·s⁻¹. The cyclist is riding toward the car at 5 m·s⁻¹. Using v(sound) = 340 m·s⁻¹, what frequency does the cyclist hear?

Question 8 of 8

Two distant stars, A and B, both emit light at the same rest wavelength. Star A's spectral lines are observed shifted toward the BLUE end of the spectrum; Star B's are shifted toward the RED end. What can you conclude?

IEB Extended Questions

IEB Question 1

A galaxy emits light at a wavelength of 656 nm (hydrogen-alpha line). An astronomer on Earth observes the same line at 689 nm. Using Δλ/λ ≈ v/c, the recession speed of the galaxy is approximately (c = 3×10⁸ m·s⁻¹):

Show all formulae, substitutions, and units. Use v = 340 m·s⁻¹ for sound in air unless otherwise stated. Identify which quantity is the source and which is the listener in each problem.

Question 1 — Moving Source (6 marks)

A train blows its horn at 480 Hz as it approaches a station at 25 m·s⁻¹. A passenger stands on the platform. (a) State the Doppler formula and identify all variables. (b) Calculate the frequency heard by the passenger as the train approaches. (c) Calculate the frequency heard as the train recedes. (d) Explain why the perceived frequency changes even though the source frequency remains constant. (e) Calculate the change in perceived frequency between approach and recession.

Question 2 — Moving Listener (5 marks)

A loudspeaker emits a steady tone of 700 Hz. A student on a bicycle rides toward the speaker at 8 m·s⁻¹, then turns and rides away at 8 m·s⁻¹. (a) Calculate the frequency heard while approaching. (b) Calculate the frequency heard while receding. (c) Explain how this scenario differs physically from a moving source with a stationary listener. (d) Would the frequency change be larger if the student moved at 16 m·s⁻¹? Justify quantitatively.

Question 3 — Radar and Speed Measurement (6 marks)

A traffic speed gun emits radio waves at 24 GHz (2.4 × 10¹⁰ Hz). A car moves toward the gun. The reflected signal is detected at 24.002 GHz. (a) Is this an example of the Doppler effect? Explain. (b) Calculate the change in frequency (Δf). (c) Using Δf/f = 2v/c (for a reflector), where c = 3×10⁸ m·s⁻¹, calculate the speed of the car. (d) Convert this speed to km·h⁻¹. (e) The speed limit is 120 km·h⁻¹. Should the driver be fined?

Question 4 — Medical Doppler (5 marks)

A medical ultrasound probe emits sound at 5 × 10⁶ Hz directed toward blood flowing in an artery. The speed of sound in tissue is 1540 m·s⁻¹. Blood flows toward the probe at 0.5 m·s⁻¹. (a) Using the Doppler formula, calculate the frequency of the reflected signal detected by the probe. (b) Calculate the frequency shift (Δf). (c) Explain why the Doppler effect is useful in diagnosing blocked arteries. (d) What would a zero Doppler shift indicate about blood flow at that location?

Question 5 — Cross-Checking Redshift with Multiple Spectral Lines (7 marks)

A galaxy's spectrum is analysed using three different spectral lines. For each line, the laboratory (rest) wavelength and the observed wavelength from the galaxy are measured:

Spectral line	Rest wavelength λ₀ (nm)	Observed wavelength λ (nm)
Hydrogen-alpha	656.3	660.7
Hydrogen-beta	486.1	489.3
Sodium D	589.0	592.9

(a) For EACH line, calculate Δλ = λ − λ₀, then use v = (Δλ/λ₀) × c to calculate the recession speed of the galaxy implied by that line (c = 3 × 10&sup8; m·s−1).
(b) Calculate the average of your three speed values.
(c) The three lines give slightly different individual values for v. Explain why using three lines (and averaging) gives a more reliable estimate of the galaxy's speed than using just one line.
(d) Is this galaxy moving toward or away from Earth? Explain how you can tell directly from the wavelength data, without doing any calculation.