10.2 Molecular relaxation and atmospheric absorption

The largest loss mechanism for sound in air at audio frequencies is not viscosity or thermal conduction. It is molecular relaxation — the slow conversion of acoustic energy into the rotational and vibrational degrees of freedom of nitrogen and oxygen molecules. The mechanism is subtle, but its effect is large and frequency-selective.

Internal degrees of freedom

A diatomic molecule like N₂ has three translational degrees of freedom (motion in $x$, $y$, $z$), two rotational degrees of freedom (rotation about the two axes perpendicular to the bond), and vibrational degrees of freedom (stretching the bond like a spring). At room temperature, only the translational and rotational modes are thermally populated — the vibrational modes have energy gaps much greater than $k_B T$ and are essentially in the ground state.

When a sound wave compresses the air, it deposits energy almost entirely in the translational mode (by speeding up the molecules’ centre-of-mass motion). To reach thermal equilibrium, this energy must redistribute among the rotational and vibrational modes via molecular collisions. That takes time.

Relaxation time

The rate of energy transfer from translation to internal modes is characterised by a relaxation time $\tau$. For N₂ rotation, $\tau \sim 1$ ns at room temperature — fast compared to acoustic periods, so rotational relaxation is essentially instantaneous and doesn’t cause much loss.

For N₂ vibration, $\tau$ depends strongly on humidity and temperature. In dry air at 20°C, $\tau \sim 10^{-2}$ s. In moist air, water molecules catalyse vibrational excitation and $\tau$ drops to $\sim 10^{-4}$ s. For O₂ vibration, $\tau \sim 10^{-3}$ s in dry air.

When the acoustic period is comparable to $\tau$, the relaxation lags behind the compression, energy is irreversibly converted to heat, and the wave is attenuated.

Frequency dependence

The absorption due to a relaxation process with time constant $\tau$ peaks at a characteristic frequency $f_r = 1/(2\pi \tau)$. The absorption coefficient as a function of frequency is

At low frequencies ($\omega \tau \ll 1$), $\alpha \propto \omega^2 \tau$ — quadratic, like classical absorption, but with a $\tau$ prefactor.

At high frequencies ($\omega \tau \gg 1$), $\alpha \propto 1/\tau$ — independent of $\omega$, asymptoting to a constant.

In between, the absorption peaks. Each relaxation process contributes its own peak at its own $f_r$.

Atmospheric absorption — the full story

Real atmospheric absorption is the sum of:

• Classical (viscous + thermal): $\alpha \propto f^2$, always present.
• N₂ vibrational relaxation: peak around 100 Hz in dry air, shifting to higher frequencies (and getting larger) in humid air.
• O₂ vibrational relaxation: peak around 4 kHz in dry air, shifting similarly.

The total absorption is strongly humidity- and temperature-dependent, and at audio frequencies it is dominated by relaxation processes that depend non-trivially on which gas, which mode, and what the humidity is.

For practical purposes, ISO 9613-1 provides a standard formula for atmospheric absorption that accounts for all of these mechanisms. Some representative values for dry air at 20°C:

• 100 Hz: $\sim 0.0001$ dB/m
• 1 kHz: $\sim 0.004$ dB/m
• 4 kHz: $\sim 0.03$ dB/m
• 10 kHz: $\sim 0.15$ dB/m
• 20 kHz: $\sim 0.5$ dB/m
• 100 kHz: $\sim 30$ dB/m

The 20 kHz value of 0.5 dB/m means high-frequency content loses 50 dB over 100 metres. This is why bats can’t echolocate distant prey: their ultrasound is absorbed in flight.

Practical consequences

• Outdoor music. Bass sounds carry farther than treble. A distant rock concert sounds like a steady low rumble — the treble has been absorbed away.
• Anechoic chambers vs reverberation rooms. The high-frequency absorption coefficient of typical acoustic foams ($\sim 0.9$ at 1 kHz, falling at lower frequencies) is set partly by the same kind of dissipative mechanisms.
• Ultrasound imaging. Medical ultrasound at MHz frequencies penetrates only a few cm in tissue before being absorbed. Higher frequencies give better spatial resolution but shorter penetration.
• The cocktail party effect. Speech at conversational levels is intelligible at about 5 m in air, partly limited by atmospheric absorption of the 2–4 kHz formants.

How relaxation enters the wave equation

To incorporate relaxation in the wave equation, one introduces a frequency-dependent speed of sound and a frequency-dependent damping. Equivalently, the equation of state $p = c^2 \rho'$ from chapter 4 is replaced by a non-local-in-time relation, where the pressure at time $t$ depends on the density history. Below the relaxation peak frequency the medium responds nearly instantaneously; above, the response lags significantly.

This is the most complete linear theory of acoustic absorption. Nonlinear effects — wave steepening and shock formation — come from a quite different mechanism and are the topic of the remaining lessons in this chapter.