Laws of Transverse Vibrations in Stretched Strings

Three separate experimental laws describe how the fundamental frequency of a stretched string responds to changing exactly one of its three controlling physical variables -- vibrating length, tension, and mass per unit length -- while the other two are held fixed; all three laws are, in fact, nothing more than the single sonometer formula $f=(1/2l)\sqrt{T/\mu}$ examined one variable at a time. The law of length states that, for a given wire under fixed tension $T$ and fixed mass per unit length $\mu$, frequency varies inversely with the vibrating length, $f\propto1/l$, so that $l\times f=C$ for some constant $C$ -- doubling the vibrating length of a string, all else unchanged, halves its fundamental frequency. The law of tension states that, for a given vibrating length $l$ and mass per unit length $\mu$, frequency varies directly with the square root of the tension, $f\propto\sqrt{T}$, so $f=A\sqrt{T}$ for some constant $A$ -- tightening a string (increasing its tension) raises its pitch. The law of mass states that, for a given vibrating length $l$ and tension $T$, frequency varies inversely with the square root of the mass per unit length, $f\propto1/\sqrt{\mu}$, so $f=B/\sqrt{\mu}$ for some constant $B$ -- a heavier (th …