Question

The speed of sound v in a medium depends on its wavelength λ, the young modulus E, and the density ρ, of the medium. Use the method of dimensional analysis to derive a formula for the speed of sound in a medium

Answer 1

Answer on Question #81251 Physics / Mechanics

The speed of sound $v$ in a medium depends on its wavelength $\lambda$, the young modulus $E$, and the density $\rho$, of the medium. Use the method of dimensional analysis to derive a formula for the speed of sound in a medium.

Solution:

Let

$$v = \lambda^{\alpha} E^{\beta} \rho^{\gamma}$$

Since

$$[\lambda] = [\mathrm{m}] = \mathrm{L}$$ $$[E] = [\mathrm{Pa}] = \mathrm{ML}^{-1} \mathrm{T}^{-2}$$ $$[\rho] = [\mathrm{kg/m^3}] = \mathrm{ML}^{-3}$$ $$[v] = [\mathrm{m/s}] = \mathrm{LT}^{-1}$$

We get

$$\mathrm{LT}^{-1} = \mathrm{L}^{\alpha} (\mathrm{ML}^{-1} \mathrm{T}^{-2})^{\beta} (\mathrm{ML}^{-3})^{\gamma}$$

Or

$$\left\{ \begin{array}{c} \alpha - \beta - 3\gamma = 1 \\ \beta + \gamma = 0 \\ -2\beta = -1 \end{array} \right.$$

The solution of this system of linear equations

$$\left\{ \begin{array}{l} \alpha = 0 \\ \beta = 1/2 \\ \gamma = -1/2 \end{array} \right.$$

Therefore

$$v = \lambda^{0} E^{1/2} \rho^{-1/2} = \sqrt{\frac{E}{\rho}}$$

Answer: $v = \sqrt{\frac{E}{\rho}}$

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