3.1 Factors affecting the speed of sound

Sound speed in materials depends on physical properties like temperature, density, elasticity, humidity, and molecular structure. Understanding what controls sound speed lets you predict how sound behaves in different environments, which is foundational to everything else in acoustics.

Physical Properties Affecting Sound Speed

The speed of sound comes down to how quickly molecules can pass energy to their neighbors. Anything that changes how fast molecules move or how strongly they're coupled together will affect sound speed.

• Temperature increases molecular motion, which speeds up wave propagation in gases.
• Elasticity (stiffness) directly increases sound speed. Stiffer materials transmit waves faster because molecules spring back more forcefully.
• Density on its own works against sound speed. More mass per unit volume means more inertia for the wave to overcome.
• Humidity changes the composition and effective molecular weight of air, slightly altering sound speed.
• Pressure has minimal effect on sound speed in ideal gases, because increasing pressure raises both density and restoring force proportionally.
• Molecular structure matters in solids and liquids, where crystal lattice arrangement and bond strength determine how efficiently energy transfers between molecules.

The general relationship that ties these together is:

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

where $v$ is sound speed, $E$ is the relevant elastic modulus (a measure of stiffness), and $\rho$ is density. This single equation is the backbone of nearly every sound speed calculation. The specific form of $E$ changes depending on the medium, but the logic stays the same: more stiffness means faster sound, more density means slower sound.

Temperature Effects on Sound

Temperature is the dominant factor controlling sound speed in gases. Higher temperature means molecules move faster, so they collide and transfer energy more quickly.

In air, sound speed increases by roughly 0.6 m/s for every 1 °C rise in temperature. At 20 °C, sound travels at about 343 m/s; at 0 °C, it drops to about 331 m/s.

The formal relationship for an ideal gas is:

$v = \sqrt{\frac{\gamma R T}{M}}$

• $v$: sound speed
• $\gamma$: ratio of specific heats (1.4 for air)
• $R$: universal gas constant (8.314 J/(mol·K))
• $T$: absolute temperature in Kelvin
• $M$: molar mass of the gas (0.029 kg/mol for air)

Notice that $v$ depends on $\sqrt{T}$, so the relationship is technically a square root, not perfectly linear. But over the temperature ranges you'll normally encounter (say, -20 °C to 40 °C), it's close enough to linear that the "0.6 m/s per °C" approximation works well.

Density and Elasticity in Sound Propagation

Density and elasticity often pull in the same direction in real materials, which can be counterintuitive. You might expect sound to travel slowly through dense materials, but steel (dense, very stiff) carries sound at about 5,960 m/s, while rubber (less dense, very flexible) carries it at only around 1,600 m/s. The stiffness of steel far outweighs its density.

Some patterns to remember:

• High elasticity, moderate density → fast sound speed. Steel (~5,960 m/s) and diamond (~12,000 m/s) are classic examples.
• High density, low elasticity → slow sound speed. Lead is dense but soft, so sound only travels at about 1,200 m/s.
• In solids and liquids, density and elasticity tend to increase together, so you can't just look at one property in isolation. The ratio $E/\rho$ is what matters.

Calculating Sound Speed in Materials

Different media require different elastic moduli in the general formula $v = \sqrt{E/\rho}$:

• Ideal gases: $v = \sqrt{\frac{\gamma R T}{M}}$
• Liquids: $v = \sqrt{\frac{K}{\rho}}$ where $K$ is the bulk modulus (resistance to uniform compression)
• Solids, longitudinal waves: $v = \sqrt{\frac{Y}{\rho}}$ where $Y$ is Young's modulus (resistance to stretching/compression along one axis)
• Solids, transverse (shear) waves: $v = \sqrt{\frac{G}{\rho}}$ where $G$ is the shear modulus (resistance to shape deformation)

Transverse wave speed is always slower than longitudinal wave speed in the same solid, because $G$ is always less than $Y$ for a given material.

When solving problems, keep these practical steps in mind:

1. Identify the medium (gas, liquid, or solid) and choose the correct formula.
2. Use the appropriate elastic modulus for the wave type (bulk, Young's, or shear).
3. Make sure temperature is in Kelvin for gas calculations.
4. For air, account for humidity if precision matters, since water vapor ($M$ ≈ 0.018 kg/mol) is lighter than dry air ($M$ ≈ 0.029 kg/mol), which slightly increases sound speed in humid conditions.