Q.Show that all harmonics are present in case of an air column vibrating in a pipe open at both ends.

A pipe open at BOTH ends must have an ANTINODE at each open end (both ends being comparatively free, each formed slightly beyond its own physical opening per the end correction). The fundamental mode fits exactly half a wavelength into the corrected air-column length L (an antinode at each end, one node in between), so $L=\lambda/2$, giving fundamental frequency $n=v/2L$. Because BOTH boundary conditions are of the SAME kind (antinode at each end, unlike the closed-pipe case), each successive allowed mode can add just ONE more half-wavelength segment at a time, so the general p-th overtone has frequency $n_p=(p+1)n$. …