4.9 The speed of sound — same number, four meanings

Four routes, four derivations, one wave equation:

But $c$ itself has four different forms — the same number, but with different physical content depending on which route you came in on:

For air at 20°C they all evaluate to $343$ m/s. They are all the same constant — the same physical $c$ — just labelled differently.

What each route teaches about $c$

• Route 1 teaches that $c$ is a thermodynamic quantity. Different fluids at different states have different speeds of sound, computable from the equation of state alone. Water has $c \approx 1480$ m/s. Helium at room temperature has $c \approx 970$ m/s (lower $m$, higher $\gamma$). The temperature dependence in air, $c \propto \sqrt{T}$, follows from this form: hotter air is faster sound.

• Route 2 teaches that $c$ is a structural quantity. Any medium with stiffness $K$ and density $\mu$ has the same speed-of-sound expression. Sound in a solid (longitudinal P-wave) uses the same form with $K$ = bulk modulus and $\mu$ = mass density: $c_\text{water} = \sqrt{2.2 \text{ GPa} / 1000\,\text{kg/m}^3} = 1480$ m/s. Sound in a steel rod, with $K \sim 200$ GPa and $\rho \sim 8000$ kg/m³, gives $c \sim 5000$ m/s. The form is universal; the inputs differ.

• Route 3 teaches that $c$ is bounded by molecular kinematics. Sound cannot propagate faster than the molecules carrying it. The exact relationship $c = \sqrt{\gamma/3}\, v_\text{rms}$ tells us how much slower — by a factor near $0.7$ for diatomic gases. The form $c = \sqrt{\gamma k_B T / m}$ is the most predictive: increase $T$ → faster sound; increase molecular mass $m$ → slower sound (helium voices vs. sulfur hexafluoride voices).

• Route 4 teaches that $c$ is a field-theory speed. It is the propagation speed of the linear excitations of a Lorentz-invariant field, in formal analogy with the speed of light in electrodynamics. Acoustic plane waves obey $\omega = c |\mathbf{k}|$ — linear dispersion — and the wave equation is invariant under the acoustic-Lorentz boosts. This perspective will resurface in chapter 9 when we put the medium in motion.

Numerical sanity

Plug in:

• $\gamma = 1.4$ for diatomic air.
• $R = 8.314$ J/mol/K.
• $T_0 = 293$ K (20°C).
• $M = 29 \times 10^{-3}$ kg/mol (air’s average molar mass).

A useful rule of thumb: $c \approx 331 + 0.6\, T_\text{C}$ m/s, where $T_\text{C}$ is air temperature in Celsius. At 0°C, $c \approx 331$. At 30°C, $c \approx 349$. The variation is small but enough to matter when comparing two sources at different temperatures, or measuring a tube length to better than 1%.

What is not in this number

The speed of sound we derived is linear, non-dispersive, adiabatic, and isotropic. It is what air does for small-amplitude sounds at audible frequencies, at one temperature, in one direction. It will need corrections for:

• Very loud sounds (nonlinear — chapter 10).
• Very high frequencies (dispersion — molecular relaxation, lesson 10.2).
• Sound in a moving medium (chapter 9).
• Sound in inhomogeneous media (refraction — lesson 7.2).

For the rest of this book, until we explicitly relax these assumptions, $c$ is one number for one medium.

What we have at the end of chapter 4

The acoustic wave equation, derived four times, with the speed of sound as a well-understood parameter. From here on, the book studies its consequences: how the wave carries energy (chapter 5), how it is emitted by sources (chapter 6), how it interacts with boundaries (chapter 7), what its frequency content looks like (chapter 8), what changes when the medium moves (chapter 9), and where the linear theory finally breaks (chapter 10).

We have built the foundation. The chapters that follow are the building.