Problem — Dimensional analysis of the speed on a string

Problem

The speed of a transverse wave on a string depends on the tension $T$ (force) and the linear density $\mu$ (mass per length). Use dimensional analysis to derive the functional dependence of $v$ on $T$ and $\mu$.

Solution

Setup. We look for $v = k\,T^{a}\mu^{b}$ with $k$ a pure number. Dimensions:

$[v] = \mathrm{m\,s^{-1}}, \qquad [T] = \mathrm{N} = \mathrm{kg\,m\,s^{-2}}, \qquad [\mu] = \mathrm{kg\,m^{-1}}$

Dimensional balance.

$\mathrm{m\,s^{-1}} = (\mathrm{kg\,m\,s^{-2}})^{a}\,(\mathrm{kg\,m^{-1}})^{b} = \mathrm{kg}^{a+b}\,\mathrm{m}^{a-b}\,\mathrm{s}^{-2a}$

System of exponents (kg, m, s):

$a+b = 0, \qquad a-b = 1, \qquad -2a = -1$

From which $a = \tfrac{1}{2}$ and $b = -\tfrac{1}{2}$. Hence:

$\ev{v = k\sqrt{\frac{T}{\mu}}}$

Dimensional analysis fixes the form $v \propto \sqrt{T/\mu}$; the full calculation gives $k = 1$, i.e. $v = \sqrt{T/\mu}$.

Collegamenti

Argomenti: Waves Concetti: Harmonic Wave Competenze: Dimensional Analysis