How does interference relate to standing waves?

Difficult question!
A way to produce standing waves is to make two waves interfere with each other.

Consider a long rope and you producing a wave on it (shaking the rope) propagating to the right with amplitude #A# (1). Consider now that your wave meets another one quite similar but going in the opposite direction (2)...the two interfere.

The superposition of the two waves, after a while, produces a standing wave, i.e., a wave that oscillates vertically but is fixed horizontally...it doesn’t' travel anymore (3)!!!

Each point on the rope is now forced to have a vertical oscillation that cannot have all the amplitudes!!! For example, a node (where you have perfect destructive interference) is a point P of your rope that will never oscillate again (while before the formation of the standing wave could move up and down according to the propagation of the wave to the right); an anti-node will now have double the amplitude, #2A#, it had before the formation of the standing wave (construction interference)!!!

Mathematically:

Hope it helps!