Vibrations of air column in a pipe closed at one end

Consider a long cylindrical tube, closed at one end, with a vibrating tuning fork held near its open end sending sound waves in. The reflection here works exactly like a transverse wave reflecting at the ends of a string, except the resulting phase behaviour is fixed by the different boundary conditions of an air column: the closed end, where particles have essentially no freedom to move, must be a NODE, while the open end, where particles are comparatively free, must be an ANTINODE (formed slightly beyond the physical opening, per the end correction of 6.7.1).

The simplest (FUNDAMENTAL) mode has just a single node at the closed end rising to a single antinode at the open end -- exactly one quarter of a wavelength fits into the air column, so $L=\lambda/4$, i.e. $\lambda=4L$. Using $v=n\lambda$ (Eq. 6.15), the fundamental frequency is $n=\dfrac{v}{\lambda}=\dfrac{v}{4L}=\dfrac{v}{4(l+e)}$ (Eq. 6.16) -- also called the FIRST HARMONIC, the lowest possible frequency of vibration for this pipe.

The next possible mode inserts one additional node and antinode: the air column now contains a node at the closed end, an antinode at the open end, and one more node-antinode pair in between -- THREE quarter-wavelengths fit into the same length L, so $L=3\lambda_1/4$ (Eq. 6.17), giving $\lambda_1=4L/3$ and, from $v=n_1\lambda_1$, a frequency $n_1=\dfrac{3v}{4L}=3n$ (Eq. 6.18) -- exactly three times the fundamental, so this is the THIRD HARMONIC, and (being the next higher frequency actually present) the FIRST OVERTONE. …