I have a question about the interpretation of the Doppler effect, when you look at the results as a change in octaves. Nothing actually changes when you look at the result in octaves instead of frequency shift, obviously, but it suddenly seems a whole lot less intuitive. This makes me wonder whether I understand things correctly.
If we assume a stationary listener and medium and moving sound source ($-c < v_s < c$, where $c \approx 343 \frac{m}{s}$, speed of sound), the observed frequency $f_L$ is:
$$f_L = \left(\frac{c}{c - v_s} \right) f_0$$
where $f_0$ is the emitted frequency. For an object moving away from you (negative $v_s$), the maximum observed pitch is $\frac{1}{2} f_0$, which is exactly one octave down. Intuitively, I find that this makes sense: all frequencies are pitched down, but not by a crazy amount.
When the source moves away ($v > 0$), however, one octave upis observed at half the speed of sound: $v = \frac{1}{2} c \Rightarrow f_L = 2 f_0$. And at $v = \frac{3}{4}c$ , it's two octaves up, and you can go all the way up to infinity. So in this case, all frequencies are suddenly scaled to a larger and larger range of frequencies, which seems odd.
Is this actually a correct way of looking at it? And what happens for the limit of $v_s \rightarrow c$ (just before the sonic boom)?