Waves and the Doppler Effect: Understanding Frequency Shifts in A-Level Science
What Is the Doppler Effect?
The Doppler Effect describes the change in frequency or wavelength of a wave as the source and observer move relative to each other.
Key Equations for the Doppler Effect
For Sound Waves
f′=fv±vov∓vsf’ = f \frac{v \pm v_o}{v \mp v_s}f′=fv∓vsv±vo
Where:
- f′f’f′: Observed frequency (HzHzHz).
- fff: Source frequency (HzHzHz).
- vvv: Speed of sound (m/sm/sm/s).
- vov_ovo: Observer velocity (m/sm/sm/s).
- vsv_svs: Source velocity (m/sm/sm/s).
For Light Waves (Relativistic Doppler Effect)
Δλλ=vc\frac{\Delta \lambda}{\lambda} = \frac{v}{c}λΔλ=cv
Where:
- Δλ\Delta \lambdaΔλ: Change in wavelength (mmm).
- λ\lambdaλ: Original wavelength (mmm).
- vvv: Relative velocity (m/sm/sm/s).
- ccc: Speed of light (3.0×108 m/s3.0 \times 10^8 \, \text{m/s}3.0×108m/s).
Applications of the Doppler Effect
Sound Waves
- Ambulance sirens shift pitch as they pass due to the Doppler Effect.
Astronomy
- Redshift: Light shifts to longer wavelengths as objects move away.
- Blueshift: Light shifts to shorter wavelengths as objects approach.
Medical Imaging
- Doppler ultrasound measures blood flow velocities.
Example Problem
A train emitting a 500 Hz500 \, \text{Hz}500Hz sound travels at 30 m/s30 \, \text{m/s}30m/s toward a stationary observer. The speed of sound is 340 m/s340 \, \text{m/s}340m/s. Find the observed frequency.
- Formula:
f′=fvv−vsf’ = f \frac{v}{v – v_s}f′=fv−vsv
- Substitute Values:
f′=500⋅340340−30=500⋅340310≈548.39 Hzf’ = 500 \cdot \frac{340}{340 – 30} = 500 \cdot \frac{340}{310} \approx 548.39 \, \text{Hz}f′=500⋅340−30340=500⋅310340≈548.39Hz
Common Mistakes in Doppler Effect Problems
- Mixing up the sign convention for source and observer velocities.
- Using incorrect values for wave speed.
- Forgetting relativistic effects for high-speed scenarios.
Practice Questions
- A car moves at 20 m/s20 \, \text{m/s}20m/s toward a stationary observer and emits a sound at 400 Hz400 \, \text{Hz}400Hz. Find the observed frequency (v=340 m/sv = 340 \, \text{m/s}v=340m/s).
- Explain how the Doppler Effect is used in astronomy.
- Describe one application of the Doppler Effect in medical imaging.
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