Why do fixed strings have harmonic modes?

One plucked string can have a low overall note and a brighter, more complex sound than a pure electronic tone. A string can vibrate in several organized patterns. Its fixed ends help decide which patterns can fit, making the attachment points part of the sound-making mechanism.

A plucked string can vibrate in organized loops because traveling disturbances reflect from its fixed ends and overlap. At certain frequencies, their combined motion forms a standing pattern with still positions called nodes. Both attached endpoints must stay nodes, limiting which patterns fit the string.

The simplest pattern, called the fundamental mode, has one loop between the ends. Mode 2 has two loops separated by one interior node. Mode 3 has three loops separated by two interior nodes. The endpoint nodes are additional to these interior nodes.

The fundamental fits one half-wavelength between the endpoints. Two loops fit two half-wavelengths into the same length, and three fit three. Wavelength is the spacing of a repeated wave pattern; fitting more loops makes it shorter. For this unchanged uniform string, wave speed stays the same, so shorter wavelengths require faster repetition. A fundamental of 100 hertz therefore gives modes 2 and 3 at 200 and 300 hertz.

The solid and dashed curves show opposite extreme positions of each standing mode, not separate strings or a moving object’s path. Equal drawn heights compare shapes without asserting equal loudness. A real pluck can combine modes, with different amounts of each contributing to the sound. The same string length therefore supports a family of related vibrations: the fixed ends constrain the shapes, and those shapes organize the frequencies.