Two waves of the same frequency and amplitude travelling in opposite directions can superpose to produce a standing wave. The pattern oscillates in place rather than travelling as a whole.

For example,

$$A\cos(kx-\omega t)+A\cos(kx+\omega t) =2A\cos(kx)\cos(\omega t).$$

The spatial factor and temporal factor are separated.

Nodes and antinodes

A node is a position where the displacement is always zero. An antinode is a position where the oscillation amplitude is maximal.

Adjacent nodes are separated by

$$\frac{\lambda}{2}.$$

Boundary conditions

Only certain wavelengths fit a system with fixed boundary conditions. For a string of length $L$ fixed at both ends,

$$L=n\frac{\lambda_n}{2},\qquad n=1,2,3,\ldots$$

so

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

These allowed frequencies are the normal modes.

Standing waves arise from interference, but their discrete mode structure comes from the boundaries of the physical system.