Final answer:
Standing waves occur at certain frequencies that correspond to the natural resonant frequencies of a confined medium, such as a vibrating string. Only waveforms that fit an integer number of half-wavelengths within the medium can form, leading to specific frequencies where the incident and reflected waves constructively and destructively interfere to create the standing wave pattern.
Step-by-step explanation:
A standing wave only occurs at certain frequencies because it is dependent on the size of the region where the wave is confined. When the ends of a vibrating medium, like a string, are fixed, only waves that fit integer numbers of half-wavelengths between these fixed points can establish a standing wave pattern. This is owing to the boundary conditions of the medium, leading to quantization of the possible waveforms; each mode corresponds to a standing wave with a frequency that matches the natural resonant frequencies of the system, which depends on the length of the string and the speed of wave propagation.
For a standing wave to form, the incident wave and the reflected wave must interfere constructively at specific points, called antinodes, and destructively at certain other fixed points, called nodes. The latter are points of no motion and occur at the fixed ends and at positions along the medium determined by the specific resonant frequency. Therefore, because the wave must meet these specific conditions, standing waves only occur at distinct frequencies known as the natural or resonant frequencies of the medium.