Speed of sound in three media

Problem

Rank from slowest to fastest: sound in air, in water, in steel. Argue based on the stiffness and density of the medium.

Solution

Principle. The speed of sound in a medium increases with stiffness (elastic modulus $E$) and decreases with density $\rho$:

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

The dominant factor is stiffness: going from gas to liquid to solid, the elastic modulus increases by orders of magnitude, far more than the density does.

Order.

• Air (gas, not very stiff): $v \approx 343\;\mathrm{m/s}$.
• Water (liquid, nearly incompressible): $v \approx 1480\;\mathrm{m/s}$.
• Steel (solid, very stiff): $v \approx 5000\;\mathrm{m/s}$.

$\ev{\text{air} < \text{water} < \text{steel}}$

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Topics: Onde Concepts: Onda armonica Skills: Ragionamento di ranking