Doppler Effect?

Question

You are standing on a train station platform as a train goes by close to you. As the train approaches, you hear the whistle sound at a frequency of f1 = 92 Hz. As the train recedes, you hear the whistle sound at a frequency of f2 = 79 Hz. Take the speed of sound in air to be v = 340 m/s.

Find the numeric value, in hertz, for the frequency of the train whistle fs that you would hear if the train were not moving.

9044 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-05-19T22:38:41

$sf(85color(white)(x)Hz)$

As an estimate I would expect the frequency to be about halfway between the two values i.e $~=$ 85.5 Hz.

We can show this as follows:

When the train is approaching we get:

$sf(f_(obs)=[(v)/(v-v_(source))]f_(source))$

Where v is the speed of sound.

We can find the speed of the train $sf(v_(source)$ , from which we can get $sf(f_(source))$ .

When the train is receding we get:

$sf(f_(obs)=[(v)/(v+v_(source))]f_(source))$

Putting in the numbers we get:

$sf(92=[(340)/(340-v_(source)]]f_(source)" "color(red)((1)))$

$sf(79=[(340)/(340+v_(source))]f_(source)" "color(red)((2)))$

Now divide $sf(color(red)((1))$ by $sf(color(red)((2))rArr)$

$sf(92/79=([(340)/(340-v_(source))])/([(340)/(340+v_(source))])xxcancel(f_(source))/(cancel(f_(source)))$

$sf(92/79=[(cancel(340))/(340-v_(source))]xx[(340+v_(source))/(cancel(340))])$

$sf(1.1645=(340+v_(source))/(340-v_(source)))$

$sf(1.1645(340-v_(source))=(340+v_(source))$

$sf(395.95-1.1645v_(source)=(340+v_(source))$

$sf(2.1645v_(source)=395.95-340=55.949)$

$:.$ $sf(v_(source)=55.949/2.1645=25.85color(white)(x)"m/s")$

This is the speed of the train.

Now we can substitute this value back into $sf(color(red)((1))rArr)$

$sf(92=[(340)/(340-25.85)]f_(source))$

$sf(92=[1.0823]f_(source))$

$sf(f_(source)=92/1.0823=85color(white)(x)Hz)$

From the prediction this seems reasonable.