1. **Problem Statement:** A standing wave is formed in a pipe that is closed at one end and open at the other. The pipe length is $L$ and the speed of sound in the pipe is $V$. We want to find the correct expression for the frequencies of the harmonics in this pipe, where $n$ is a positive integer. 2. **Understanding the Setup:** In a pipe closed at one end and open at the other, the standing wave pattern has a node (point of zero displacement) at the closed end and an antinode (point of maximum displacement) at the open end. 3. **Formula for Frequencies in a Closed-Open Pipe:** The harmonics in such a pipe occur at odd multiples of the fundamental frequency. The fundamental frequency $f_1$ is given by: $$f_1 = \frac{V}{4L}$$ The frequencies of the harmonics are: $$f_n = (2n - 1) f_1 = \frac{(2n - 1)V}{4L}$$ where $n = 1, 2, 3, \ldots$. 4. **Explanation of Why Other Options Are Incorrect:** - Option A: $\frac{(2n - 1)V}{2L}$ is incorrect because it doubles the denominator, which does not match the quarter-wavelength condition. - Option C and D: These correspond to harmonics in pipes open at both ends or closed at both ends, where frequencies are integer multiples of $\frac{V}{2L}$ or $\frac{V}{4L}$, but not with the odd multiple pattern. 5. **Summary:** The correct expression for the frequencies of the harmonics in a pipe closed at one end and open at the other is: $$\boxed{f_n = \frac{(2n - 1)V}{4L}}$$ which corresponds to option B. 6. **Why Your Approach Might Be Wrong:** If you used a formula with $2L$ in the denominator or did not consider the odd harmonics pattern, the result would be incorrect. Remember, the closed end forces a node, and the open end an antinode, leading to only odd harmonics being present.