A string fixed at both ends can support many normal modes. Each mode is a standing wave whose wavelength fits the fixed boundaries.

For a string of length $L$, the $n$th mode has shape

$$y_n(x,t)=A_n\sin\left(\frac{n\pi x}{L}\right)\cos(\omega_n t+\phi_n),$$

with

$$f_n=n\frac{v}{2L}.$$

For a stretched string,

$$v=\sqrt{\frac{T}{\mu}},$$

so its mode frequencies depend on length, tension and linear density.

A real pluck is not one mode

The familiar standing-wave shapes describe individual modes, but a plucked string usually begins in a shape that is not any single one of them. Because the wave dynamics are approximately linear, the actual motion can be written as a superposition of modes:

$$y(x,t)=\sum_{n=1}^{\infty} y_n(x,t).$$

The modes are therefore building blocks of the motion rather than competing descriptions of it.

Fundamental and harmonics

The lowest mode is the fundamental. Higher modes have frequencies $2f_1,3f_1,\ldots$ and are the string's harmonics.

How the string is plucked determines how strongly the different modes are excited. Plucking near a position that is a node of a particular mode suppresses that mode, while other modes can remain strong.

The fundamental largely determines the perceived pitch, while the mixture of harmonics helps determine the resulting waveform and timbre.

Two equivalent pictures

The same vibration can be viewed either as a superposition of standing-wave modes or as travelling disturbances repeatedly reflecting from the fixed ends. These are two descriptions of the same wave motion.