Question

A train is moving away from an observer at 31.0 m/s blows a 305 Hz whistle. What frequency is detected by the stationary observer? Assume standard pressure and 20°C for the speed of sound.

Answer 1

The apparent frequency v' that would be heard by stationary listener when source of sound having velocity V away from it is given by

v'= (V/V+u​​​​​​s )*v

where v is the actual frequency of sound emitted by source(train) and u​​​​​​s is the velocity of sound

Given, V = 31 m/s

v = 305 Hz

Also, at standard pressure and temp = 20 degree celcius = 20+273= 293 K , velocity u​​​​​​s of the sound is

u​​​​​​s = 343 m/s

Using these values of u​​​​​​s, V, v in eqn 1, we get

v' = (31/31+343)*305

= (31/374)*305

= 0.08288*305

v' = 25.280 Hz

It is obvious that v' <v, i.e frequency of sound heard by the listener is much lesser than that of actual frequency emitted by the train. This is exactly what we would expect as the train is moving away from the stationary listener.