How can we explain the standing waves on a string?

When a wave reaches the end of a string, it is reflected and inverted, so in a fixed string in which we've caused vibrations, such as a guitar string, we have two sinusoidal waves travelling in opposite directions. In certain places, where the two waves are exactly out of phase, we observe destructive interference (crest meets trough, and the two waves cancel each other out) and the point remains static. These points are called nodes. Midway between them, we can observe the opposite: constructive interference (where the two waves coincide and produce an even bigger displacement); these points of greatest amplitude are called antinodes. The fixed ends of the string are always nodes, and the number of nodes and antinodes depends on how long the string is relative to the wavelength. For example, in a string which is one-half wavelength long, we have two nodes (at both ends of the string) and one antinode; if the string is one wavelength long, we have three nodes and two antinodes, and so on.