Laws of a Vibrating String

The fundamental-frequency formula derived above, $n=\dfrac{1}{2l}\sqrt{\dfrac{T}{m}}$ (Eq. 6.32), can be broken apart into three separate proportionality statements, each obtained by holding TWO of the three quantities (l, T, m) fixed and varying the third -- known as the three LAWS OF A VIBRATING STRING.

(1) LAW OF LENGTH: if the tension T and linear density m are both kept constant, the fundamental frequency is INVERSELY proportional to the vibrating length: $n\propto\dfrac{1}{l}$ (Eq. 6.33) -- a shorter vibrating length gives a higher note, exactly as a guitarist raises pitch by pressing a string shorter against the fretboard.

(2) LAW OF TENSION: if the length l and linear density m are both kept constant, the fundamental frequency is DIRECTLY proportional to the SQUARE ROOT of the tension: $n\propto\sqrt{T}$ (Eq. 6.34) -- tightening a string (e.g. turning a guitar's tuning peg) raises its pitch.

(3) LAW OF LINEAR DENSITY: if the tension T and length l are both kept constant, the fundamental frequency is INVERSELY proportional to the SQUARE ROOT of the linear density: $n\propto\dfrac{1}{\sqrt{m}}$ (Eq. 6.35) -- a heavier (thicker, or denser-material) string of the same length and under the same tension sounds a lower note, which is exactly why the thickest strings on a guitar or violin are used for the lowest notes. …