Speed of a Transverse Wave on a Stretched String

For a transverse wave travelling along a taut string, the elastic (restoring) property is provided by the tension $T$ in the string -- it is the tension that pulls a displaced element of the string back towards the straight equilibrium line -- and the inertial property is provided by $\mu$, the mass per unit length (linear mass density) of the string, since a heavier string per unit length has more inertia resisting the restoring pull. The speed of a transverse wave on a stretched string is $v = \sqrt{\frac{T}{\mu}}$ with $T$ in newtons and $\mu$ in $\text{kg m}^{-1}$, giving $v$ in $\text{m s}^{-1}$. This formula shows that a string under greater tension carries transverse waves faster, while a heavier (thicker, denser) string of the same tension carries them more slowly -- exactly the adjustments a musician makes, consciously or not, when tuning a stringed instrument: tightening a string (raising $T$) raises the wave speed and hence (for a fixed string length, as the next section on stationary waves will show) raises the pitch. Crucially, $v=\sqrt{T/\mu}$ depends only on the string's own tension and mass per unit length -- not on the …