Q.The fundamental frequency of an air column in a pipe closed at one end is in unison with the third overtone of an open pipe. Calculate the ratio of lengths of their air columns.

A pipe closed at one end must always have a NODE at the closed end (where particle motion is essentially impossible) and an ANTINODE at the open end (where particles are comparatively free, formed slightly beyond the physical opening per the end correction). The fundamental mode fits exactly one quarter-wavelength into the corrected air-column length L, so $L=\lambda/4$, giving fundamental frequency $n=v/4L$. Because both boundary conditions (node at one end, antinode at the other) must be preserved simultaneously in every higher mode, each successive allowed mode must add exactly TWO more quarter-wavelength segments (never just one), so the general p-th overtone has frequency $n_p=(2p+1)n$. …