C.5 The Doppler Effect: Catching the Wave Shift

Hello future physicists! Welcome to one of the most relatable and mind-bending topics in Wave Behaviour: the Doppler Effect. Don't worry if the name sounds complicated—you experience this phenomenon every day!

In this chapter, you will learn why the pitch of a siren changes as it passes you, and, perhaps more dramatically, how this same principle allows astrophysicists to determine that the entire universe is expanding. This topic links mechanics, waves, and even cosmology, making it crucial for your IB studies.

Understanding Relative Motion and Frequency

What is the Doppler Effect?

The Doppler Effect describes the change in the observed frequency or wavelength of a wave due to the relative motion between the wave source and the observer.

It is crucial to understand that the source itself is *not* changing the frequency it produces. If a siren emits a constant 500 Hz sound, it always emits 500 Hz. The change occurs only because of the motion, which squishes or stretches the wavefronts before they reach the observer.

Key Term: Observed Frequency (\(f'\)) is the frequency the listener actually hears or measures.

Imagine you are standing by the roadside as an ambulance approaches, passes, and moves away:

  • Approaching: The sound waves are compressed in front of the vehicle. You observe a higher frequency (higher pitch).
  • Receding (Moving Away): The sound waves are stretched behind the vehicle. You observe a lower frequency (lower pitch).


Visualizing Wavefronts

Think of a boat in a pond.

  • If the boat is stationary, the ripples (wavefronts) are uniform circles centered on the boat.
  • If the boat is moving, the ripples pile up in the direction of motion (compression) and spread out behind the boat (stretching).

The Doppler Effect applies to all types of waves, including sound (mechanical) and light (electromagnetic).

Quick Review: The Doppler Effect is caused by relative speed, not by the source changing its true frequency.

The Doppler Effect for Sound Waves (Non-Relativistic)

When dealing with sound, the analysis is non-relativistic because the speeds of the source (\(u_s\)) and the observer (\(u_o\)) are usually much, much smaller than the speed of sound (\(v\)) in the medium.

We must consider four key variables:

  • \(f\): Frequency emitted by the source (true frequency).
  • \(f'\): Frequency observed by the listener.
  • \(v\): Speed of the wave in the medium (speed of sound).
  • \(u_s\): Speed of the source.
  • \(u_o\): Speed of the observer.
The General Equation for Sound

The IB data booklet provides a comprehensive equation that covers all scenarios for sound waves where the source and observer are moving along the same line:

$$f' = f \left( \frac{v \pm u_o}{v \mp u_s} \right)$$

Don't worry about memorizing the derivation, but you must know how to apply the sign conventions!

Sign Convention: The Crucial Part!

The sign conventions determine whether the observed frequency \(f'\) increases or decreases.

Trick to Remember: We want the numerator (Observer) to make \(f'\) bigger when moving TOWARDS and the denominator (Source) to make \(f'\) bigger when moving TOWARDS.

1. For the Observer (\(u_o\)) - Numerator:

  • If the observer moves TOWARDS the source: Use the + sign. (Numerator increases, \(f'\) increases)
  • If the observer moves AWAY from the source: Use the – sign. (Numerator decreases, \(f'\) decreases)

2. For the Source (\(u_s\)) - Denominator:

  • If the source moves TOWARDS the observer: Use the – sign. (Denominator decreases, \(f'\) increases)
  • If the source moves AWAY from the observer: Use the + sign. (Denominator increases, \(f'\) decreases)

Common Mistake Alert! Students often mix up the signs, especially for the source. Remember, when the source moves TOWARDS you, it compresses the waves, so the observed wavelength (\(\lambda'\)) must decrease, meaning \(f'\) must increase. To make the fraction larger, you need a smaller denominator, hence the MINUS sign in the source term!

Example Scenario: A police car (\(u_s = 30 \text{ m/s}\)) is chasing a fleeing pedestrian (\(u_o = 5 \text{ m/s}\)). The speed of sound is \(v = 340 \text{ m/s}\).

  • Source (car) is moving TOWARDS the observer (pedestrian): Use \(-u_s\).
  • Observer (pedestrian) is moving AWAY from the source (car): Use \(-u_o\).

$$f' = f \left( \frac{v - u_o}{v - u_s} \right)$$

Key Takeaway for Sound: Always determine if the motion is TOWARDS or AWAY. Towards means a higher observed frequency (\(f'\)); Away means a lower observed frequency (\(f'\)). Adjust the signs in the formula accordingly!

The Doppler Effect for Light (Electromagnetic Waves)

The Doppler Effect for light (and all electromagnetic radiation) is fundamentally different from sound because light does not require a medium and its speed (\(c\)) is the same for all observers (a principle of special relativity).

Since light speeds are so high, we generally consider the relative speed between the source and observer, rather than treating them separately with respect to a stationary medium.

Redshift and Blueshift

Instead of a change in pitch (sound frequency), the Doppler shift for light results in a change in observed color (light frequency/wavelength).

1. Blueshift (\(f'\) is higher, \(\lambda'\) is shorter)

  • Occurs when the source of light is moving TOWARDS the observer.
  • The observed wavelength is shortened, shifting the light spectrum towards the blue/violet end of the visible spectrum.

2. Redshift (\(f'\) is lower, \(\lambda'\) is longer)

  • Occurs when the source of light is moving AWAY from the observer.
  • The observed wavelength is lengthened, shifting the light spectrum towards the red end of the visible spectrum.

Did you know? Analyzing the redshift of distant galaxies is the primary method astronomers use to confirm that the universe is expanding. Almost all distant galaxies show significant redshift, meaning they are moving away from us!

The Approximate Formula (Non-Relativistic Speeds)

When the relative speed of the source and observer (\(v\)) is small compared to the speed of light (\(c\)), we use a simplified, non-relativistic approximation:

$$\frac{\Delta f}{f} \approx \frac{v}{c} \quad \text{or} \quad \frac{\Delta \lambda}{\lambda} \approx \frac{v}{c}$$

Where:

  • \(\Delta f\) (or \(\Delta \lambda\)) is the change in frequency (or wavelength).
  • \(f\) (or \(\lambda\)) is the emitted frequency (or wavelength).
  • \(v\) is the relative speed between the source and observer.
  • \(c\) is the speed of light (\(3.00 \times 10^8 \text{ m/s}\)).

This formula relates the fractional change in frequency (or wavelength) directly to the ratio of the relative speed to the speed of light.

Note: In advanced (HL) and specialized scenarios, the full relativistic Doppler equation is needed, but for most standard IB problems involving small speeds relative to \(c\), the approximate formula is sufficient and commonly used.

Remember: If the relative motion is AWAY, \(v\) is positive (for the redshift calculation) and \(\Delta \lambda\) is positive (wavelength increases).

Key Takeaway for Light: Motion away causes Redshift (longer wavelength/lower frequency); Motion towards causes Blueshift (shorter wavelength/higher frequency). The magnitude of the shift is proportional to the relative speed \(v\).

Real-World Applications of the Doppler Effect

The Doppler Effect is not just an academic curiosity; it is a vital tool used across many scientific and engineering fields:

1. Police Speed Traps (Radar)

Police use radar guns that emit electromagnetic waves towards a vehicle. The waves reflect off the moving car and return to the gun. Because the car is moving (either towards or away), the frequency of the reflected wave is Doppler-shifted. By measuring the change in frequency (\(\Delta f\)), the relative speed \(v\) of the car can be calculated.

2. Medical Ultrasound

In medicine, the Doppler Effect is used to measure blood flow velocity.

A transducer sends high-frequency sound waves into the body. These waves reflect off moving red blood cells. The frequency of the reflected waves shifts according to the speed and direction of the blood cells. This allows doctors to detect blockages or abnormal flow rates.

3. Astronomy and Cosmology

As previously mentioned, Doppler shifts of light are essential for studying celestial objects:

  • Radial Velocity: Measuring the speed at which a star or galaxy is moving towards or away from Earth.
  • Galaxy Expansion: Observation of widespread redshift confirms that the universe is expanding (Hubble's Law).
  • Exoplanet Detection: Tiny Doppler shifts in a star's light can be caused by the gravitational tug of orbiting planets, revealing their presence.

4. Weather Forecasting (Doppler Radar)

Weather radar uses the Doppler shift principle to determine the speed and direction of precipitation (rain, snow, hail) and wind within storms, helping forecasters predict severe weather events like tornadoes.

Don't worry if the formulas seem tricky at first. Practice applying the sign conventions for sound and understanding the blueshift/redshift concepts for light. These are fundamental skills for success in this chapter!


Final Key Takeaway: The Doppler Effect is the measurement of motion using wave frequency change, allowing us to measure speeds from cars on the highway to galaxies billions of light-years away.