Physics HL · Chapter 16: The Doppler Effect
16.3 The Doppler Effect for Sound: Moving Source
Derive and apply moving-source formulas for observed frequency and wavelength, while preserving medium-determined wave speed reasoning.
Estimated time: 40 minutes
Deriving Frequency for a Moving Source
A practical derivation starts from one second of emission. The source emits f wavefronts in one second. If the source moves toward a stationary observer with speed u_s, those f fronts are packed into distance v-u_s instead of v. Shorter spacing means higher observed frequency.
Denominator decreases for approach, so frequency rises; denominator increases for recession, so frequency falls.
Wavelength Shift and Wave Speed Invariance
For sound, the medium determines wave speed. In still air, both source and stationary observer use the same v for wave speed. What changes is wavelength ahead of and behind the source because emission points move between successive wavefront launches.
Approach compresses spacing; recession stretches it.
This is the cleanest way to avoid a common error: do not replace wave speed by v plus-or-minus u_s in observer equations for this case. The source moves, but the medium remains still air, so wave propagation speed relative to air is still v.
Sign Selection Workflow Under Time Pressure
When choosing signs, first predict direction of change qualitatively. If source approaches, observed frequency must be larger than emitted. Pick the algebraic form that guarantees this before substituting numbers. This quick sanity check prevents denominator-sign mistakes.
After frequency is found, recover wavelength from v = f'lambda' if asked. Keeping the speed relation explicit in your written steps earns method marks and demonstrates physical consistency.
Simulation: Moving-Source Sound Formula Lab
Adjust source velocity and frequency to compare observed pitch and wavefront spacing, then verify sign choices against equation predictions.
Doppler Effect Lab
Observed frequency
1.36e+3 Hz
Wavefront spacing near observer
0.250 m
Source-frame wavelength
0.272 m
Wave speed relative to observer
340.00 m/s
Wavefront geometry (positive velocity points to the right)
Moving source, stationary observer
f' = f v/(v - u_s) for an approaching source and f' = f v/(v + u_s) for a receding source. The wave speed in still air stays v.
Stationary source, moving observer
f' = f (v + u_o)/v for an observer moving toward the source and f' = f (v - u_o)/v for moving away. The observer's motion changes encounter rate.
Wavelength interpretation
Source motion changes wavefront spacing in the medium. Observer motion does not change spacing in the medium, but it changes how quickly fronts are met.
Check your intuition
Increasing loudness as a source approaches is a distance/intensity effect. Frequency shift is separate and can remain constant while speed is constant.
Test Yourself
A siren emits 1250 Hz and approaches a stationary observer at 28.0 m/s. Take v = 340 m/s. Enter observed frequency in Hz.
Hint: Use f' = f v/(v-u_s).