Question

Singing that is off-pitch by more than about 1% sounds bad. How fast would a singer have to be moving relative to the rest of a band to make this much of a change in pitch due to the Doppler effect

Answer 1

-3.396 m/s or 3.465 m/s

Step-by-step explanation:

v = Speed of sound in air = 343 m/s

`v_s` = Relative speed of the singer

f = Observed frequency

f' = Actual frequency

1% change can mean
`f=1.01f'`

From the Doppler effect equation we have

`f=f'(v)/(v+v_s)\\\Rightarrow 1.01f'=f'(v)/(v+v_s)\\\Rightarrow 1.01=(343)/(343+v_s)\\\Rightarrow v_s=(343)/(1.01)-343\\\Rightarrow v_s=-3.396\ m/s`

The velocity is -3.396 m/s

when
`f=0.99f'`

`f=f'(v)/(v+v_s)\\\Rightarrow 0.99f'=f'(v)/(v+v_s)\\\Rightarrow 0.99=(343)/(343+v_s)\\\Rightarrow v_s=(343)/(0.99)-343\\\Rightarrow v_s=3.46464646465\ m/s`

The velocity is 3.465 m/s