Two speakers playing the same note in step make a pattern of loud and quiet spots, as in Superposition. What matters at a spot is how much farther one speaker is than the other. 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
Path difference $\Delta = |L_2 - L_1|$ and $\lambda = v/f$. $$\text{loud: } \Delta = m\lambda, \qquad \text{quiet: } \Delta = (m + \tfrac{1}{2})\lambda.$$ The lowest frequency that makes a spot quiet has $\Delta = \lambda/2$, so $f = v/(2\Delta)$.
Worked example: Loud or quiet?
Speakers $3\ \text{m}$ apart; a listener $4\ \text{m}$ straight out from one of them. Sound travels at $340\ \text{m/s}$. Find (1) the path difference and (2) the lowest frequency that is quiet there.
- (1) The far speaker is $\sqrt{3^2 + 4^2} = 5\ \text{m}$ away, so $\Delta = 1\ \text{m}$.
- (2) $\lambda = 2\Delta = 2\ \text{m}$, so $f = 340/2 = 170\ \text{Hz}$.
Answer: $1\ \text{m}$, $170\ \text{Hz}$
Common mistake
Comparing a single distance with the wavelength. A spot $5\ \text{m}$ from one speaker tells you nothing alone; only the difference between the two paths decides how the waves meet.