Closed Organ Pipe

In a pipe closed at one end and open at the other, the closed end must be a displacement node while the open end must be a displacement antinode. Since the spacing between a node and its nearest antinode is $\lambda/4$, the pipe's length must fit an odd number of quarter-wavelengths (an even number would place a node, not an antinode, at the open end): $L=\frac{(2n-1)\lambda_n}{4}, \qquad n=1,2,3,\dots \;\Rightarrow\; \lambda_n=\frac{4L}{2n-1}, \qquad f_n=\frac{(2n-1)v}{4L}=(2n-1)\,f_1, \qquad f_1=\frac{v}{4L}$ Consequently a closed organ pipe supports only the ODD harmonics of its fundamental: $f_1$ (the fundamental, first harmonic, one quarter-wavelength loop), $3f_1$ (the third harmonic, called the first overtone since it is the next mode actually present), $5f_1$ (the fifth harmonic, second overtone), and so on -- every even harmonic ($2f_1,4f_1,\dots$) is completely absent from the pipe's note, which gives a closed pipe's characteristic timbre, distinctly different from an open pipe's fuller, all-harmonics tone even when both sound at the same fundamental pitch. A further direct consequence, comparing the two boxed fundamental formulas, is that a closed pipe and an open pipe of the identical length $L$, filled with the same gas (same $v$), do not sound …