C5-1. The Doppler Effect

1. Fundamentals of the Doppler Shift

When a source of sound, such as the whistle of a train or the siren of an ambulance, moves away from an observer, it appears to decrease in frequency, i.e., it sounds lower in pitch. The source of the sound however, remains at a constant frequency. This frequency change due to the relative motion between a source of sound or light and an observer is known as the Doppler effect (or Doppler shift).

  • When the observer and the source of sound (e.g. ambulance siren) are both stationary, the waves appear to remain at the same frequency for both the observer and the source.
  • When the observer and the source of sound are moving relative to each other, the waves appear to have a different frequency for both the observer and the source.
  • The frequency is increased when the source is moving towards the observer.
  • The frequency is decreased when the source is moving away from the observer.

Electromagnetic Waves (Light):

  • The Doppler effect is the apparent change in frequency and wavelength of a wave due to the relative motion between the wave source and the observer. The same phenomena occurs for electromagnetic waves, such as light.
  • Waves moving away from the observer are red-shifted. Their wavelengths shift to the red end of the electromagnetic spectrum. This is equivalent to sound waves appearing at a lower frequency to the observer.
  • Waves moving towards the observer are blue-shifted. Their wavelengths shift to the blue end of the electromagnetic spectrum. This is equivalent to sound waves appearing at a higher frequency to the observer.

2. Visualising the Doppler Effect with Wavefronts

In the following diagrams, wavefronts are even in a stationary object but are squashed in the direction of the moving wave source.

  • When the source starts to move towards the observer, the wavelength of the waves is shortened. The sound, therefore, appears at a higher frequency to the observer.
  • If the observer was standing behind the moving source, they would hear the sound at a lower frequency due to the wavelength of the waves broadening.
  • A moving object will cause the wavelength $\lambda$ (and frequency) of the waves to change. Let $\Delta\lambda$ be the change in wavelength.
  • The wavelength of the waves in front of the source decreases $(\lambda - \Delta\lambda)$ and the frequency increases.
  • The wavelength behind the source increases $(\lambda + \Delta\lambda)$ and the frequency decreases.
Source (v = 0) Observer A (Behind/Receding) Standard λ Standard f Observer B (In front/Approaching) Standard λ Standard f Observed Frequency & Wavelength are standard Source v Observer A (Source moving away) RED SHIFT Stretched λ (↑) Lower f (↓) Observer B (Source approaching) BLUE SHIFT Compressed λ (↓) Higher f (↑)
EXAMINER TIP

The relationship between frequency and wavelength is determined by the wave equation, which is given in your data booklet. The speed $v$ of the wave does not change.

3. Examples

Example 1

Problem: A cyclist rides a bike ringing their bell past a stationary observer. Which of the following correctly describes the Doppler shift caused by the sound of the bell as they move away?
A. Shorter Wavelength, Higher Frequency, Lower Pitch
B. Longer Wavelength, Lower Frequency, Higher Pitch
C. Shorter Wavelength, Lower Frequency, Higher Pitch
D. Longer Wavelength, Lower Frequency, Lower Pitch

Solution:

If the cyclist is riding past the observer and moving away, the wavelength of sound waves are going to become longer (stretched). This rules out options A and C.
A longer wavelength means a lower frequency (from $v = f\lambda$).
Lower frequency creates a lower sound pitch. Therefore, the answer is row D.

Example 2: Pitch Dynamics of a Passing Train

Problem: A high-speed train travels through a station without stopping, sounding a horn at a constant $600\text{ Hz}$. Describe qualitatively how the pitch of the horn heard by a stationary observer on the platform changes as the train approaches, passes directly in front of them, and then recedes into the distance.

Solution:

1. Approaching: As the train heads towards the observer, the wavefronts are compressed. The observer hears a steady but distinctly higher pitch than $600\text{ Hz}$.
2. Passing: At the exact instantaneous moment the train is directly alongside the observer, there is no relative radial velocity. The observer hears the true pitch of $600\text{ Hz}$.
3. Receding: As the train moves away, the wavefronts expand behind it. The pitch drops abruptly as it passes and remains at a steady lower pitch than $600\text{ Hz}$ as it travels away.

Example 3: Wavefront Interpretation

Problem: Two observers, A and B, are standing on a long straight road. An ambulance with its siren on is driving between them. Observer A notes that the siren sounds higher in pitch than its standard frequency, while Observer B notes that the siren sounds lower in pitch. Using the concept of wavefronts, determine the direction the ambulance is moving.

Solution:

The frequency is increased (higher pitch) only when the source is moving towards an observer, as the wavefronts are physically squashed in the direction of motion, decreasing the wavelength in front of the source ($\lambda - \Delta\lambda$). Because Observer A hears the higher pitch, the ambulance must be moving towards Observer A.
Conversely, the frequency decreases (lower pitch) when the source moves away, as wavefronts broaden behind the source ($\lambda + \Delta\lambda$). Since B hears a lower pitch, the ambulance is moving away from Observer B. Therefore, the ambulance travels in the direction from B towards A.