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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)
![\[ f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right) \]](https://on22.com/wp-content/ql-cache/quicklatex.com-216a8132ab2a39391cdf12b0cac4181e_l3.png "Rendered by QuickLaTeX.com")
Where:
![f']( "Rendered by QuickLaTeX.com")
: Observed frequency (Hz)![f]( "Rendered by QuickLaTeX.com")
: Source frequency (Hz)![v]( "Rendered by QuickLaTeX.com")
: Wave speed (340 m/s for sound in air)![v_o]( "Rendered by QuickLaTeX.com")
: Observer’s velocity (+ if moving toward source)![v_s]( "Rendered by QuickLaTeX.com")
: Source’s velocity (+ if moving toward observer)
: Observed frequency (Hz)
: Source frequency (Hz)
: Wave speed (340 m/s for sound in air)
: Observer’s velocity (+ if moving toward source)
: Source’s velocity (+ if moving toward observer)Light Waves (Relativistic Doppler Effect)
![\[ \frac{\Delta \lambda}{\lambda} = \frac{v}{c} \]](https://on22.com/wp-content/ql-cache/quicklatex.com-4d6360493308eb90adb8b2e2be27387b_l3.png "Rendered by QuickLaTeX.com")
Where:
![\lambda]( "Rendered by QuickLaTeX.com")
: Original wavelength![\Delta \lambda]( "Rendered by QuickLaTeX.com")
: Observed wavelength shift![c = 3 \times 10^8 \, \text{m/s}]( "Rendered by QuickLaTeX.com")
: Speed of light
: Original wavelength
: Observed wavelength shift
: Speed of Applications
Everyday Phenomena
- Emergency vehicle sirens: Pitch drops as vehicle passes
- Weather radar: Measures precipitation velocity
Astronomy
- Redshift (
![z = \frac{\Delta \lambda}{\lambda}]( "Rendered by QuickLaTeX.com")
): Indicates cosmic expansion - Blueshift: Reveals approaching celestial objects
): Indicates cosmic expansionMedical 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).
- Identify formula: Source moving toward observer
![\[ f' = f \left( \frac{v}{v - v_s} \right) \]](https://on22.com/wp-content/ql-cache/quicklatex.com-a88f2d7fd29992494fdeb1d9cc86247b_l3.png "Rendered by QuickLaTeX.com")
- Calculate:
![\[ f' = 500 \left( \frac{340}{340 - 30} \right) \approx 548.4 \, \text{Hz} \]](https://on22.com/wp-content/ql-cache/quicklatex.com-051ce324f36b3a22d8470f730d5ce16e_l3.png "Rendered by QuickLaTeX.com")
Common Mistakes
Practice Problems
- Calculate the observed frequency when a car (20 m/s) emitting 400 Hz approaches a stationary observer.
- Derive the redshift formula
![z = v/c]( "Rendered by QuickLaTeX.com")
for ![v \ll c]( "Rendered by QuickLaTeX.com")
. - Explain how Doppler ultrasound measures valve stenosis.
for 
.
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