Question

0. The hobo is on his way back through town on a faster train. Turns out he’s also a competent trombone player.
   1. What frequency would he actually be playing if the train is approaching you at 25 m/s and you hear a perfect 220 Hz note (an A)?
1. 2. As the train approaches, you happen to be watching a friend at band practice. When she stands still, you hear a 2 Hz beat frequency as she, too, is holding out the same note (though not quite in tune). What are the two possible frequencies that you are hearing from her (in combination with the hobo)?
2. 3. When she slightly extends her slide, the beat pattern effectively disappears (i.e. she’s perfectly playing 220 Hz). Which frequency did you hear her playing in part b? Explain.

Answer 1

Frequency heard by observer ( f_o  ) will be different than the original one ( f ) due to Doppler's Effect.

We know, f_o = vf / ( v - v_s ),

where, v = speed of sound in air = 332 m / s,

f_o = 220 Hz,

v_s = speed of sound source towards observer = 25 m / s.

Hence, f = f_o ( v - v_s ) / v

or, f = 220 * ( 332 - 25 ) / 332

or, f = 203.43 Hz

Hence, the hobo is actually playing a sound of 203.43 Hz.

Let, two possible frequencies be f₁ Hz and f₂ Hz.

Hence, beat frequency = 2 Hz gives,

either f₁ - f_o = 2 f₁ = f_o + 2 = 220 + 2 f₁ = 222;

or, f_o - f₂ = 2 f₂ = fo - 2 = 220 - 2 f₂ = 218.

Hence, two possible frequencies are : 218 Hz and 222 Hz.

Extending the slide will reduce frequency, and here, the frequency is reduced to 220 Hz.

So, previously the frequency was higher than 220 Hz.

Hence, in part b, the frequency heard from her was 222 Hz.