Doppler Effect With Source And Observer Moving

Question

Two trains approach each other on parallel tracks, one sounding a 500 Hz whistle while moving at 20 m per second toward the other, which moves at 10 m per second; sound travels at 340 m per second. What frequency does a passenger on the second train hear?

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When both source and observer move along the line joining them, the general Doppler formula is $f' = f\left(\frac{v + v_o}{v - v_s}ight)$, where $v_o$ is the observer's speed toward the source and $v_s$ is the source's speed toward the observer. The numerator grows because the observer rushes into the wavefronts, and the denominator shrinks because the source chases its own sound. Here $f = 500	ext{ Hz}$, $v = 340	ext{ m/s}$, $v_o = 10	ext{ m/s}$, and $v_s = 20	ext{ m/s}$, giving $f' = 500\left(\frac{340 + 10}{340 - 20}ight) = 500 	imes \frac{350}{320} = 500 	imes 1.094 \approx 547	ext{ Hz}$. The value $515	ext{ Hz}$ accounts for only the observer's motion and omits the source term. The value $530	ext{ Hz}$ uses only the source motion. The value $485	ext{ Hz}$ wrongly treats the trains as receding, lowering the pitch. This is the NCERT combined Doppler relation. As a plausibility check, both motions act to raise the pitch since the trains close in, so the result must exceed 500 Hz and be larger than either single-effect value, which the 547 Hz answer correctly reflects.

Doppler Effect With Source And Observer Moving

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