Doppler Shift: Deriving a Formula for Both Moving Objects

[StephenPrivitera]

My book derives two formulas for Doppler shift. One for when the source moves and one for when the observer moves.
What about when both are moving?
I tried deriving it myself... but I couldn't :(
If you want, you could just give me a hint on how to derive it.

 

[JasonZ]

The general formula for the doppler effect should be:

$$f'=f_o(\frac{v \pm v_d}{v \pm v_s})$$

In this, $$v$$ should be the velocity the wave is traveling at (like 343 m/s for sound in air). So if you have both the detector and source moving, just make sure you use the correct sign in front of each and you should be set. Does this help?

-Jason

 

[StephenPrivitera]

Yes that helps a lot. Do you use - on the botton when they are moving apart and + on the top when they are moving apart?
Also, can you clearly define f and f'?
f is the frequency as observed by the source?
f' is the frequency as observed by the detector?
Also, these speed v_d and v_s are relative to the medium in which the wave travels correct?

 

[JasonZ]

Let me try and answer these one at a time:

If they are moving away from each other, than yes you would use + on top, and - on bottom I believe.

You have the f's correct as well. $$f'$$ will be your new frequency (as observed by the detector), and $$f_o$$ will be the initial frequency (as sent out by the source.

As for the velocities, once again you are correct. They are relative to the medium.

-Jason