Question
A train approaches a stationary observer at while sounding a horn of frequency . The speed of sound in air is . What frequency does the observer hear when the train is approaching, and what does the observer hear after the train has passed and is receding at the same speed?
Solution — Step by Step
For a moving source and stationary observer:
The minus sign applies when the source approaches; plus when it recedes.
The pitch jumps from about 548 Hz down to about 460 Hz as the train passes — a drop of nearly 90 Hz. This is the characteristic “neeeeoooom” of a fast vehicle going past.
Final: approaching Hz, receding Hz.
Why This Works
When the source moves toward you, each successive wavefront is emitted from a closer position, so the wavefronts crowd together — shorter wavelength, higher frequency. When the source recedes, wavefronts spread apart and frequency drops.
Note that the shift is not symmetric — the drop on receding is slightly larger than the rise on approaching, because the formula has the source velocity in the denominator. For small velocities (), the shifts become approximately symmetric.
Alternative Method
For small Mach numbers, use the linearised approximation:
This gives a shift of Hz, predicting roughly 544 and 456 Hz. Close to the exact answers but not identical.
Common Mistake
Students mix up source-moving and observer-moving formulas. For a moving source, goes in the denominator. For a moving observer, goes in the numerator. Memorise these positions or rederive — never guess.