Physics: How fast would the car need to be going to have the change in frequency be one octave (frequency halved) difference?

Question

At very fast car races the sound of the engine as a car approaches is much different than when it passes you.

How fast would the car need to be going to have the change in frequency be one octave (frequency halved) difference? (assume velocity of sound in air is 343 m/s)

Answer 1

2 formulas for doppler shift frequency (well 8 actually) 

 

1 for source moving toward stationary listener

Ft = [ v / (v - Vs) ] *f  

 

1 for source moving away from listener

Fa = [ v / (v + Vs) ] *f

 

v is sped of sound -- 343 m/s

2Fa = Ft  (frequency halved)

 

solve each for Vs (speed of car, source) , set 2 equations equal (Vs=Vs)  f cancels 

subsititute Ft = 2Fa 

solve for Fa, then obtain Ft (1/2 fa).

 

answer is 1/3 of Vsound