How do a string and an air column produce higher harmonics?

In the context of musical sound, both stretched strings and air columns produce higher harmonics through the formation of standing waves when the vibrating medium is divided into smaller, equal segments called loops.12

Stretched Strings

A string fixed at both ends must have nodes (points of no motion) at those boundaries. The fundamental frequency occurs when the string vibrates as a whole. Higher harmonics are produced when the string vibrates in multiple equal segments, where each segment is one-half wavelength long. As the number of segments ($n$) increases, the wavelength decreases, resulting in frequencies that are integral multiples of the fundamental frequency ($f_n=n\times f_1$).2

Air Columns

Air columns function analogously to strings, with standing waves forming based on boundary conditions.3
* Open Tubes: These have antinodes at both ends, allowing air motion. Like a string, higher harmonics are created by dividing the column into additional loops, resulting in frequencies that are integral multiples of the fundamental.4
* Closed Tubes: One end is closed, creating a node, while the open end remains an antinode. This configuration alters the available harmonic series compared to open tubes.3