A person hears the whistle of an approaching train

train

A person standing near a railway track hears the whistle of an approaching train as having a frequency of 1600 Hz. The train’s velocity is 40 m/s.
What is the true frequency of the sound emitted by the whistle?

Solution.

The Doppler effect is the change in frequency of a wave for an observer. The classical Doppler effect formula is given by

f = f0(c+vr)/(c+vs) ,

where f is observed frequency, f0 is emitted (true) frequency, c is the velocity of waves in the medium, vr is the velocity of the receiver relative to the medium and vs is the velocity of the source relative to the medium.

We know that the person standing near a railway track so vr = 0.
Also, the velocity of the source is negative because the source is approaching the receiver.

Hence the formula becomes

f = f0*c/(c-vs)

or we can rewrite it as

f0 = f(1-vs/c)

We take the velocity of sound waves in air as 340 m/s and substitute the values into formula.

f0 = 1600*(1-40/340) ≈ 1411.76 Hz

 

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