Wave Speed and the Medium

A wave on a string moves because each piece of string pulls on its neighbor. Tension, T, is the force doing that pulling, so more tension means each piece yanks its neighbor into motion faster, and the wave races along quicker. Mass per length, mu, is how much string has to be dragged along for every meter of wave. A heavier string resists being yanked around, so the wave crawls. These two properties combine as v = √(T/mu): speed grows with tension and shrinks with mass per length.

This is why tightening a guitar string raises its pitch before you ever touch a fret. Tightening increases T, which increases v. Since frequency f = v/wavelength, and the wavelength is set by the fixed length of the string, a higher v means a higher frequency, a higher pitch. Nothing about how you pluck it enters that formula. Shake the end harder and you pump more energy and bigger amplitude into the wave, and you can wiggle it faster or slower to change frequency, but the wave still travels down the string at the same speed, because v is a property of the medium, not the disturbance.

This same logic carries over to sound. Sound speed in air depends on properties of the air itself, and heating air makes its molecules move faster and collide more often, so sound travels faster through hot air than cold air. The medium, not the source, always sets the speed. That separation between source properties (amplitude, frequency) and medium properties (speed) is the idea to carry into every wave topic that follows.