A wall turns one Doppler problem into two: first the wall hears a moving source, then it sends that sound back as a still source. Builds on The Doppler Effect. Each problem here is one short scene and a chain of questions. Answer them in order: each later question uses what you found before, so a slip early on carries through. Keep full precision between steps and round only at the end.
Key idea
$$f' = f\,\frac{v \pm v_o}{v \mp v_s}$$ (upper signs when moving toward each other). A source moving toward a wall at $u$: the wall gets $f\,\dfrac{v}{v - u}$. A listener moving toward the wall at $u$ hears the echo raised again by $\dfrac{v + u}{v}$. Beats: the difference of two frequencies.
Worked example: The driver's echo
A $500\ \text{Hz}$ siren drives at $20\ \text{m/s}$ toward a cliff ($v = 340\ \text{m/s}$). Find (1) the frequency at the cliff and (2) the echo the driver hears.
- (1) $500 \cdot \dfrac{340}{320} = 531.2\ \text{Hz}$.
- (2) $531.2 \cdot \dfrac{360}{340} = 562.5\ \text{Hz}$, so the driver hears $62.5\ \text{Hz}$ of beats against the siren.
Answer: $531.2\ \text{Hz}$, $562.5\ \text{Hz}$
Common mistake
Putting the source's speed in the numerator. A moving source changes the spacing of the waves, $v \mp v_s$ in the denominator; a moving listener changes how fast they meet the waves, $v \pm v_o$ on top. Check: approaching must raise the pitch.