Waves and the Doppler Effect: Understanding Frequency Shifts in A-Level Science

What Is the Doppler Effect?

The Doppler Effect describes the perceived change in frequency when there is relative motion between a wave source and observer.

Key Equations

Sound Waves (Classical Doppler Effect)

\[
f’ = f \left( \frac{v \pm v_o}{v \mp v_s} \right)
\]
Where:

• \(f’\): Observed frequency (Hz)
• \(f\): Source frequency (Hz)
• \(v\): Wave speed (340 m/s for sound in air)
• \(v_o\): Observer’s velocity (+ if moving toward source)
• \(v_s\): Source’s velocity (+ if moving toward observer)

Light Waves (Relativistic Doppler Effect)

\[
\frac{\Delta \lambda}{\lambda} = \frac{v}{c}
\]
Where:

• \(\lambda\): Original wavelength
• \(\Delta \lambda\): Observed wavelength shift
• \(c = 3 \times 10^8 \, \text{m/s}\): Speed of light

Applications

Everyday Phenomena

• Emergency vehicle sirens: Pitch drops as vehicle passes
• Weather radar: Measures precipitation velocity

Astronomy

• Redshift (\(z = \frac{\Delta \lambda}{\lambda}\)): Indicates cosmic expansion
• Blueshift: Reveals approaching celestial objects

Medical Technology

• Doppler echocardiography: Measures blood flow up to 5 m/s
• Fetal heart rate monitoring

Example Problem

Scenario: Train (30 m/s) approaches observer, emitting 500 Hz sound (v = 340 m/s).

1. 1. Identify formula: Source moving toward observer

\[
f’ = f \left( \frac{v}{v – v_s} \right)
\]

1. 1. Calculate:

\[
f’ = 500 \left( \frac{340}{340 – 30} \right) \approx 548.4 \, \text{Hz}
\]

Common Mistakes

1. Sign errors in velocity terms
2. Using sound speed for light waves
3. Neglecting relativistic effects when \(v > 0.1c\)

Practice Problems

1. Calculate the observed frequency when a car (20 m/s) emitting 400 Hz approaches a stationary observer.
2. Derive the redshift formula \(z = v/c\) for \(v \ll c\).
3. Explain how Doppler ultrasound measures valve stenosis.