Vibrations In Air Columns

In a pipe closed at one end, the closed end must be a displacement node while the open end is an antinode, so the fundamental mode fits only a quarter wavelength of the resonant standing wave into the pipe length, giving $\lambda =4L$. This makes the fundamental frequency $f_1=v/(4L)$. Substituting $v=340\text{ m/s}$ and $L=0.25\text{ m}$ gives $f_1=\frac{340}{4\times 0.25}=\frac{340}{1.0}=340\text{ Hz}$. The value $680\text{ Hz}$ uses $\lambda =2L$, which is the rule for an open pipe, not a closed one. The value $170\text{ Hz}$ wrongly uses $\lambda =8L$, doubling the pipe relation. The value $425\text{ Hz}$ has no valid harmonic correspondence for a closed pipe. This applies the NCERT analysis of air columns, where a closed pipe sounds only odd harmonics built on $f_1=v/4L$. As a plausibility check, a closed pipe of given length produces a lower fundamental than an open pipe of the same length, and the next allowed mode here is the third harmonic at 1020 Hz, confirming the odd-harmonic series consistent with 340 Hz.