Doppler: how to derive formule "source moving" from formule "observer moving"

Question

A question like this has already been asked, with a satisfying explication (the relative motion of the source and the observer with respect to the medium (air) is different), but without a derivation on the ground of the problem.

When a source is moving and the observer (receiver) at rest, the frequency $f_\text{r}$ measured is $$f_\text{r}=f_\text{o}\left(\frac{v_\text{o}}{v_\text{o}+v_\text{s}}\right) (1) $$with $v_\text{s}$ the velocity of the source.

When the observer is moving and the source at rest, frequency $f_\text{r}$ measured is $$f_\text{r}=f_\text{o}\left(\frac{v_\text{o}-v_\text{r}}{v_\text{o}}\right) (2)$$ with $v_\text{r}$ the velocity of the receiver (observer).

I have no idea, but is it (easy) possible to derive formula (2) from formula (1) (or inverse) taking into account that the motion of the source and the receiver is relative?

Answer 1

To answer your question, no it is not possible to derive the equations you call (1) and (2) from each other because $v_s$ and $v_r$ are independent of each other, and independent of the constants $f_0$ and $v_0$.

However, we can very easily either derive the equations separately and then combine them by multiplication, or simply derive them together.   Let's do the latter:

Using your notation, let
      $v_0$ = the speed of sound, positive in the direction from source to observer.
      $v_r$ = the velocity of the observer, positive in the direction away from the source
      $v_s$ = the velocity of the source, positive in the direction toward the observer
       (Everything is positve from left to right, if the source is left of the observer)
      $f_0$ = the source frequency
      $f_r$ = the observed frequency
      $\lambda_0$ = the source wavelength

We'll use the relation $$f_0=\frac{v_0}{\lambda_0},$$ but for an observer moving toward or away from the source, $v_0$ is replaced by $v'$:$$ v' = v-v_r$$
and for a moving source chasing or running away from its own waves, $\lambda_0$ is replaced by $\lambda'$ $$\lambda' = \frac{v-v_s}{f}$$ so $$f_r=\frac{v'}{\lambda'}=f_0(\frac{v-v_r}{v-v_s})$$ Note that for either a stationary observer or a stationary source, the above expression reduces to your equations (1) and (2), respectively (except that you didn't specify positive and negative directions).   This is what dmckee meant in his comment above by saying that your equations were special cases.