M ore on resonant frequency. When you find resonance, you are making standing waves. Here is a video that shows what standing waves are for a vibrating string:
The important thing to see here is that a single shake of the rope makes a single wave that travels down the rope, bounces, and comes back. If many of these shakes happen, there will be lots of little waves, moving back and forth along the rope simultaneously – a real mess.
That is, it’s a real mess until you time the shakes just right. As soon as that happens, the waves start lining up with each other. What you are doing is matching the resonant frequency of the rope, and you’ll get what is called a standing wave. If you shake the rope twice as fast, you’ll see two smaller standing waves (the octave pitch), shake it three times as fast, and you’ll get the next harmonic, and so forth.
If the rope is shorter, then the waves have less distance to travel, and so they make the trip quicker. You have to shake the rope more rapidly to get those standing waves. This is why shorter strings vibrate at higher frequencies, and result in higher pitches. Waves travel more slowly through thicker ropes, so thicker strings give lower frequencies, and so forth.