Standing waves in organ pipes — open and closed pipe harmonics

An open pipe has antinodes at both ends (the air is free to vibrate maximally at open ends). The condition for standing waves is:

All harmonics (even and odd) are present.

Fundamental ($n = 1$): $f_1 = \frac{v}{2L} = \frac{340}{2 \times 0.5} = \mathbf{340 \text{ Hz}}$

Second harmonic ($n = 2$): $f_2 = 2f_1 = \mathbf{680 \text{ Hz}}$

Third harmonic ($n = 3$): $f_3 = 3f_1 = \mathbf{1020 \text{ Hz}}$

Pattern: $f_1, 2f_1, 3f_1, 4f_1, \ldots$ — all integer multiples of the fundamental.

A closed pipe has a node at the closed end (air cannot vibrate) and an antinode at the open end. The condition is:

Only odd harmonics are present. This is the crucial difference from an open pipe.

Fundamental ($n = 1$): $f_1 = \frac{v}{4L} = \frac{340}{4 \times 0.5} = \mathbf{170 \text{ Hz}}$

Third harmonic ($n = 3$): $f_3 = 3f_1 = \mathbf{510 \text{ Hz}}$ (this is the 1st overtone)

Fifth harmonic ($n = 5$): $f_5 = 5f_1 = \mathbf{850 \text{ Hz}}$ (this is the 2nd overtone)

Pattern: $f_1, 3f_1, 5f_1, 7f_1, \ldots$ — only odd multiples.