Why does the length of an air column affect the pitch of a wind instrument?

The length of an air column is a primary factor in determining the pitch of a wind instrument because it dictates the characteristic frequencies at which the air inside the tube can vibrate.1

When air is set into vibration within a tube, the sound waves reflect off the ends, creating a standing wave. The physical length of the tube ($l$) constrains these waves, as they must satisfy specific boundary conditions to persist. The fundamental frequency ($\nu$) of an open tube is defined by the formula:21

$$\nu =\frac{v}{2l}$$

In this equation, $v$ represents the speed of sound in the tube. Because the frequency is inversely proportional to the length of the air column, a longer tube results in a lower frequency (lower pitch), while a shorter tube results in a higher frequency (higher pitch). By using mechanisms like valves, slides, or finger holes to alter the effective length of the column, musicians change these boundary conditions, thereby selecting different resonant frequencies to produce various musical notes.12