For a moving source and stationary listener, the source changes the spacing of wavefronts: approach shortens wavelength and raises apparent frequency, while recession lengthens wavelength and lowers it.
Learning Outcomes
A source moving toward a listener shortens wavelength ahead of it.
Approaching source: λ′ = v − vSf and f′ = (vv − vS)f.
A source moving away lengthens wavelength behind it.
Receding source: λ′ = v + vSf and f′ = (vv + vS)f.
The sound speed relative to stationary air remains v.
Approach raises pitch; recession lowers pitch.
Intermediate Level
How Source Motion Changes Wavelength
In one period T = 1f, a source moves distance vST. Each new crest is emitted from a shifted position, changing crest spacing while the crests themselves still travel through air at speed v.
Key equation
Δ x=vST=vSf
Source Moving toward a Stationary Listener
Key equation
λ'=λ-vSf=v-vSf
Key equation
f'=vv-vSf
Because vv − vS is greater than one, f′ > f.
Source Moving away from a Stationary Listener
Key equation
λ'=λ+vSf=v+vSf
Key equation
f'=vv+vSf
Because vv + vS is less than one, f′ < f.
Source motion
Wavelength
Denominator
Frequency
toward listener
shorter
v − vS
higher
away from listener
longer
v + vS
lower
Textbook Example 9.2
A 354 Hz source recedes from a stationary observer at 22 m s⁻¹. If v = 332 m s⁻¹, find the apparent frequency.
1
Working
Receding source: f′ = (vv + vS)f
2
Step 2
f′ = (332332 + 22)(354)
Final answer:f′ = 332 Hz
Textbook Assignment 9.2 Solved
A bus horn emits 500 Hz while the bus moves at 20 m s⁻¹. Find the frequencies heard by a stationary roadside observer as the bus approaches and recedes. Use v = 340 m s⁻¹.
1
Working
Approaching: f′ = (340340 − 20)(500) = 531.25 Hz
2
Step 2
Receding: f′ = (340340 + 20)(500) = 472.22 Hz
Final answer: About 531 Hz approaching and 472 Hz receding
A. It emits successive crests from different positions.
Q. Write the approaching-source formula.
A.f′ = (vv − vS)f.
Q. Write the receding-source formula.
A.f′ = (vv + vS)f.
Q. What happens to wavelength ahead of the source?
A. It becomes shorter, so the frequency heard ahead is higher.
Sticky Notes
A source moving toward a listener shortens wavelength ahead of it.
Approaching source: λ′ = v − vSf and f′ = (vv − vS)f.
A source moving away lengthens wavelength behind it.
Receding source: λ′ = v + vSf and f′ = (vv + vS)f.
The sound speed relative to stationary air remains v.
Approach raises pitch; recession lowers pitch.
60-Second Revision
A moving source shifts crest spacing. Toward: λ′ = v − vSf and f′ = vf/(v − vS). Away: λ′ = v + vSf and f′ = vf/(v + vS).
Summary
For a moving source and stationary listener, the source changes the spacing of wavefronts: approach shortens wavelength and raises apparent frequency, while recession lengthens wavelength and lowers it.
Ask-the-Tutor Modes
Explain the Doppler effect simply.
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Exam preparation
Exam Focus
Important points
A source moving toward a listener shortens wavelength ahead of it.
Approaching source: λ′ = v − vSf and f′ = (vv − vS)f.
A source moving away lengthens wavelength behind it.
Receding source: λ′ = v + vSf and f′ = (vv + vS)f.
The sound speed relative to stationary air remains v.
