Mechanical Waves

1. Introduction

A mechanical wave is a wave that requires a material medium (solid, liquid, or gas) for its propagation, because it travels by the successive vibration of the particles of the medium about their mean positions. Sound is the most important mechanical wave. This chapter deals with the speed of mechanical waves, in particular the velocity of sound in solids, liquids, and gases, the correction Laplace made to Newton's formula, and the factors that affect the speed of sound.

The speed of a wave is the distance travelled by the disturbance per unit time and, like every wave, is linked to frequency $f$ and wavelength $\lambda$ by:

$$v = f\lambda$$

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2. Velocity of a Wave in an Elastic Medium

A mechanical wave travels through a medium because of the medium's elasticity (which provides the restoring force) and its inertia (density). In general the speed of a mechanical wave is:

$$ \begin{aligned} v &= \sqrt{\frac{\text{elasticity of the medium}}{\text{density of the medium}}} \ &= \sqrt{\frac{E}{\rho}} \end{aligned} $$

The appropriate elastic modulus $E$ depends on the state of the medium.

2.1 Velocity of sound in a solid

In a solid rod, sound (a longitudinal wave) is governed by Young's modulus $Y$:

$$v_{\text{solid}} = \sqrt{\frac{Y}{\rho}}$$

where $\rho$ is the density of the solid. Solids have large $Y$, so sound travels fastest in solids.

2.2 Velocity of sound in a liquid

In a liquid, the relevant modulus is the bulk modulus $B$:

$$v_{\text{liquid}} = \sqrt{\frac{B}{\rho}}$$

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3. Velocity of Sound in a Gas

3.1 Newton's formula

Newton assumed that when sound propagates through a gas, the compressions and rarefactions occur isothermally (at constant temperature), because the heat developed in a compression has time to conduct away. For an isothermal change, Boyle's law gives:

$$PV = \text{constant}$$

Differentiating both sides:

$$P,dV + V,dP = 0 \implies P = -\frac{dP}{dV/V}$$

The quantity $-\dfrac{dP}{dV/V}$ is precisely the isothermal bulk modulus, so for an isothermal process $B_{\text{iso}} = P$. Substituting $B = P$ into $v = \sqrt{B/\rho}$ gives Newton's formula:

$$\boxed{v = \sqrt{\frac{P}{\rho}}}$$

Value at NTP: with $P = 1.01\times10^{5}\ \text{N m}^{-2}$ and $\rho_{\text{air}} = 1.293\ \text{kg m}^{-3}$:

$$ \begin{aligned} v &= \sqrt{\frac{1.01\times10^{5}}{1.293}} \ &= \sqrt{7.81\times10^{4}} \approx 280\ \text{m s}^{-1} \end{aligned} $$

The experimental value of the speed of sound in air at NTP is about $332\ \text{m s}^{-1}$. Newton's value ($280\ \text{m s}^{-1}$) is about 16% too low, so his isothermal assumption must be wrong.