Deriving the Doppler Effect: Where Did I Go Wrong?

Question

It has been a while since I did any physics, but I woke up today with the strange craving to derive the Doppler Effect.

To my dismay I couldn't produce a sensible formula, though I believe my reasoning is sound.

Let us suppose we have a source moving with velocity $v_s$ and it is at the origin initially. Its position is given by:

$$p_s(t)=v_st$$

Similarly, suppose there is an observer at position $X$ initially moving towards the source at velocity $v_o$. Then its position is:

$$p_o(t)=X-v_o t$$

The source emits pulses at every time $T$ and I believe these should travel at speed $v$. As a matter of fact, I remember seeing it derived in a fluid mechanics lecture at college that it was something like $v=\sqrt{\gamma R T_m}$ and doesn't depend on the source's speed.

Let me call $\tau_N$ the time it takes for the pulse emmited at position $v_sTN$ to reach the observer initially at position $X-v_oTN$. We have

$$v_s NT+v\tau_N=X-v_oNT-v_o \tau_N$$ $$\therefore\tau_N(v+v_o)=X-NT(v_o+v_s)$$

Clearly, the observer perceives pulses every:

$$\Delta \tau=\tau_N-\tau_{N+1}=T \frac{v_o+v_s}{v+v_o}$$

Which yields the appalling formula:

$$f_o=f_s\frac{v_o+v}{v_s+v_o}$$

Where did this rusty mathematician go wrong?

Answer 1

$\tau_N$ is the time it takes for the Nth pulse to reach the observer.

The time when the observer perceives the pulse is actually:

$$NT+\tau_N$$

Hence, the actual time diference between pulses perceived by the observer is:

$$T_o=T+(\tau_{n+1}-\tau_n)=T-T\frac{v_o+v_s}{v_o+v}=T\frac{v-v_s}{v_o+v}$$

Which yields the correct formula: $$f=f_o\frac{v+v_o}{v-v_s}$$