How are the harmonics of an air column related to its fundamental frequency?

In an air column, the fundamental frequency represents the lowest frequency of vibration for a given tube length, often referred to as the first harmonic. The higher frequencies, known as harmonics or overtones, are integer multiples of this fundamental frequency.123

The relationship depends on the boundary conditions of the tube:

• Open Tubes: A tube open at both ends can produce all integer multiples of the fundamental frequency ($f_n=n·f_1$, where $n=1,2,3,\dots$).12
• Closed Tubes: A tube closed at one end creates a node at the closed end and an antinode at the open end. Because of these boundary conditions, the air column can only support odd-numbered harmonics ($f_n=n·f_1$, where $n=1,3,5,\dots$).34

As the air column is divided into smaller segments—a process known as overblowing—the frequency increases in direct proportion to these divisions. These harmonics, along with the fundamental, form the harmonic series, which defines the unique timbre of a sound.3