Question

Why can standing wave patterns only occur at specific wavelengths and frequencies?

Answer 1

Because you can only get a stable pattern if there is a whole number of half wavelengths along the length of the oscillator.

Explanation:

The wavespeed in any given medium (includes tension for a string) is fixed, so if you have a particular number of half wavelengths along the length the frequency is also fixed. Thus we see /hear harmonics at particular frequencies where all the particles in between two nodes are in phase (i.e. all reach their amplitude simultaneously.)

There are known equations relating these variables here and also good explanations of the field.