Speed of Sound

Speed of sound propagation in gases, liquids, and solids: dependence on temperature, medium, and its elastic properties.

Conceptual context

Sound requires a material medium to propagate: it does not exist in a vacuum. But once in a medium, at what speed does it travel? The answer is not trivial. Newton attempted to calculate it theoretically in the seventeenth century and obtained a value 15 % below the experimental one. Laplace resolved the problem a century later by recognizing that acoustic compressions are adiabatic, not isothermal. This historical episode encapsulates the essential physics of the leaf: the speed of sound depends on the elastic stiffness of the medium and its inertia, and for gases, the thermodynamic nature of the compression process determines which elastic modulus (adiabatic or isothermal) is the correct one.

Understanding $v$ is a necessary condition for calculating wavelengths, propagation times, Doppler effects, tube and cavity resonances, and for designing sonar systems, medical ultrasound diagnostic techniques, and industrial ultrasound quality control.

🟢 Essential level

When a tuning fork is struck, the pressure perturbation it generates moves away from it in all directions. The speed at which that perturbation moves away is $v$: the speed of sound in the medium. It is not the speed at which air molecules oscillate back and forth; it is the speed at which the wavefront travels, like the crest of a wave on the sea.

$v$ is a property of the medium, not the source. A cannon and a flute produce very different sounds, but both travel at the same $v$ in the same medium and at the same temperature. What distinguishes the two sources is amplitude and frequency, not propagation speed.

In air at room temperature (20 °C), $v$ is approximately 343 m/s. This is equivalent to covering 1 km in just under 3 seconds, which explains the delay between lightning (travels at the speed of light, practically instantaneous) and thunder (travels at $v$). Counting the seconds between lightning and thunder and dividing by 3 gives approximately the distance to the storm in kilometers.

$v$ depends on the medium and temperature. In water, $v$ ≈ 1480 m/s (nearly 4.4 times faster than in air); in steel, $v$ ≈ 5100 m/s (15 times faster than in air). In air, $v$ increases with temperature at a rate of about 0.6 m/s per degree Celsius.

[!NOTE] The idea that sound travels faster in denser media is a very common error. Steel is thousands of times denser than air and yet transmits sound much faster, because it is also enormously stiffer. It is the stiffness of the medium, not its density, that accelerates sound.

🔵 Formal level

Basic formulas

The physics of acoustic propagation shows that $v$ results from the balance between two properties of the medium: its stiffness (measured by the bulk modulus $B$) and its inertia (measured by the density $\rho$). The Newton-Laplace formula expresses this balance:

Speed of sound in an elastic medium (Newton-Laplace)

This formula is valid for any continuous medium. $B$ quantifies how much pressure must be applied to compress the medium by 1 % in volume; $\rho$ is the mass per unit volume. A stiffer medium (higher $B$) propagates sound faster; a denser medium (higher $\rho$) propagates it more slowly, because the greater inertia makes it harder to accelerate the particles.

Real student questions

If I shout louder, does the sound arrive sooner? No. The amplitude and intensity increase, but $v$ remains fixed by the medium as long as propagation stays linear.

Why does sound travel faster in water than in air if water is denser? Because the bulk stiffness of water grows much more than its inertia. Density alone does not decide the speed.

Final synthesis

The speed of sound $v$ in a medium is the square root of the ratio between the medium's elastic stiffness ($B$) and its inertia ($\rho$). In ideal gases, $v$ depends only on $T$, $\gamma$, and $M$, and is independent of pressure. Laplace's correction (factor $\gamma$ in the modulus) resolves the historical 15 % discrepancy predicted by Newton. In air, $v$ increases 0.6 m/s per degree Celsius. These principles determine the design of musical instruments, sonar systems, ultrasound equipment, and underwater acoustic communication protocols.