2.2 Speed of Sound in Various Media

Factors Influencing Sound Speed

Medium Properties and Temperature Effects

The speed of sound through a material depends on two core properties: the medium's elasticity (how well it springs back when compressed) and its density (how much mass is packed into a given volume). These two factors show up in every sound speed equation you'll encounter.

Temperature plays a major role in gases and liquids. Higher temperature means particles have more kinetic energy, which increases the medium's elastic response to compression. In solids, temperature matters much less because the rigid lattice structure dominates the elastic behavior.

• Density inversely influences sound speed when other factors are held constant. A denser medium means more inertia for particles to overcome, which slows propagation.
• Humidity changes the average molecular mass of air. Water vapor ($M \approx 18 \text{ g/mol}$) is lighter than dry air ($M \approx 29 \text{ g/mol}$), so humid air actually carries sound slightly faster.
• Pressure has minimal effect on sound speed in gases under normal conditions. This surprises many students: increasing pressure raises both density and elastic modulus proportionally, so they cancel out. Pressure only becomes significant at extremes, like the deep ocean or dense planetary atmospheres.

Material-Specific Influences

In solids, sound speed is set primarily by the elastic modulus and density. Metals typically have high elastic moduli and relatively low compressibility, which is why sound travels through steel far faster than through rubber or wood.

Composition changes also matter:

• Salinity in water: Ocean water carries sound faster than freshwater because dissolved salts increase the bulk modulus more than they increase density.
• Alloying in metals: Steel and pure iron have different sound speeds because alloying elements alter the crystal structure and elastic properties.

Calculating Sound Speed

General Equations

The fundamental equation for sound speed in any medium is:

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

• $v$: speed of sound
• $B$: bulk modulus (a measure of the medium's resistance to uniform compression)
• $\rho$: density of the medium

For an ideal gas, this can be rewritten using thermodynamic quantities:

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

• $\gamma$: adiabatic index (ratio of specific heats, $C_p / C_v$)
• $R$: universal gas constant ($8.314 \text{ J/(mol·K)}$)
• $T$: absolute temperature in Kelvin
• $M$: molar mass of the gas

Notice that pressure doesn't appear in this equation. Temperature and molecular mass are what determine sound speed in an ideal gas.

Medium-Specific Calculations

Different media use different elastic moduli depending on the type of wave:

• Liquids: $v = \sqrt{\frac{K}{\rho}}$, where $K$ is the bulk modulus of the liquid.
• Solids (longitudinal waves): $v = \sqrt{\frac{Y}{\rho}}$, where $Y$ is Young's modulus. These are compression waves traveling along the length of the material.
• Solids (transverse/shear waves): $v = \sqrt{\frac{G}{\rho}}$, where $G$ is the shear modulus. Transverse waves can only propagate in solids (and very viscous liquids) because fluids don't resist shear deformation.

For practical applications like predicting sound speed in air at different altitudes, temperature-dependent versions of these equations are used. A common approximation for air near room temperature is $v \approx 331.3 + 0.606\,T_C$ m/s, where $T_C$ is the temperature in degrees Celsius.

Sound Speed: Materials vs Conditions

Comparative Sound Speeds

The general ranking is: solids > liquids > gases. This follows from the fact that solids have the strongest intermolecular bonds and highest elastic moduli, even though they're also denser. The elasticity factor wins out over the density factor.

Some reference values to know:

Sound in water is roughly 4.3 times faster than in air at room temperature. Steel carries sound about 15 times faster than air. These large differences have real consequences for engineering and acoustics.

Environmental Factors

• In gases, sound speed increases with temperature but is largely independent of pressure under normal conditions.
• In water, salinity and temperature both raise sound speed. Ocean water (salinity ~35 g/kg) carries sound noticeably faster than freshwater.
• In solids, composition and internal stress affect speed. For example, pre-stressed concrete has a different sound speed than unstressed concrete, which is actually used as a non-destructive testing method.
• At extreme conditions, pressure becomes important. In the deep ocean, the high pressure raises the bulk modulus enough to significantly increase sound speed, even as temperature drops.

Medium Properties and Sound Propagation

Wave Behavior in Changing Media

When the speed of sound varies continuously through a medium, waves bend. This is refraction, and it follows the same principles as light refraction.

• Temperature gradients in the atmosphere cause sound to curve. On a hot day, air near the ground is warmer (faster sound speed), so waves bend upward, creating sound shadows where intensity drops. On cool nights with warm air above, waves bend downward, and sound can carry surprisingly far.
• Sound channels form where speed reaches a minimum between two faster regions. Sound waves get trapped and guided along these channels. The SOFAR channel in the ocean (at roughly 700-1200 m depth) allows low-frequency sounds to travel thousands of kilometers.
• Acoustic impedance mismatches at boundaries between media cause partial reflection and partial transmission. Impedance is the product $Z = \rho v$. The greater the mismatch between two media, the more sound reflects at the interface. This is why so little sound passes from air into water (and vice versa).

Complex Propagation Effects

• Scattering occurs when sound encounters inhomogeneities like suspended particles, bubbles in liquids, or grain boundaries in solids. Bubbles in water are particularly effective scatterers.
• Boundaries guide wave propagation. Walls in buildings create reverberations through repeated reflections. The ocean floor and surface act as boundaries that form underwater sound channels.
• Frequency-dependent absorption means high frequencies attenuate faster than low frequencies over distance. This is why thunder sounds like a low rumble from far away but has a sharp crack up close: the high-frequency components have been absorbed.
• External stresses alter a medium's elastic properties and therefore its sound speed. Temperature inversions in the atmosphere, mechanical stress in solids, and pressure changes in the ocean all modify propagation paths in ways that matter for real-world acoustics.