10.3 The frequency dependence of attenuation

Combining the dissipation mechanisms from lessons 10.1 and 10.2 gives the full absorption coefficient $\alpha(f)$ as a function of frequency. The result varies dramatically across the audio and ultrasonic ranges, with profound consequences for how sound propagates over long distances.

The pieces

For air at typical conditions, the total absorption coefficient is

where:

• $\alpha_\text{classical}(f) \propto f^2$ throughout, from viscosity and thermal conduction.
• \alpha_\text{N}_2(f) has a relaxation peak around 100 Hz–1 kHz (humidity-dependent), then plateaus at higher frequencies.
• \alpha_\text{O}_2(f) has a relaxation peak around 4 kHz–40 kHz (humidity-dependent), then plateaus at higher frequencies.

The combined curve is roughly:

• 20–100 Hz: very low absorption, dominated by classical $\propto f^2$.
• 100 Hz – 1 kHz: N₂ relaxation peak; absorption rises faster than $f^2$.
• 1–4 kHz: plateau between relaxation peaks.
• 4–40 kHz: O₂ relaxation peak; absorption rises again faster than $f^2$.
• Above 40 kHz: classical $\propto f^2$ dominates, increasing rapidly.

In log–log coordinates the absorption curve has two bumps (one per relaxation), embedded in a slowly-rising baseline.

Humidity dependence

The relaxation rates of N₂ and O₂ depend on collisions with water molecules — H₂O is a particularly effective “catalyst” for energy transfer. This makes atmospheric absorption strongly humidity-dependent:

• Dry air (0% RH): very long relaxation times; absorption peaks shift to low frequencies and become broader.
• Moist air (60% RH): much shorter relaxation times; peaks shift up and become sharper.

The practical upshot: outdoor sound propagation changes by significant amounts between dry and moist conditions. Music played outdoors on a humid day sounds brighter than the same music on a dry day, because high-frequency absorption is paradoxically less at 60% RH than at very low humidity (in some frequency ranges).

ISO 9613-1 provides the exact formulas. They are widely implemented in noise-propagation software.

In water

For water, the dominant relaxation mechanisms are different: B(OH)₃ (boric acid) relaxation around 1 kHz and MgSO₄ relaxation around 100 kHz, both important in seawater. Pure freshwater has neither, and classical absorption ($\propto f^2$) dominates over almost the entire ultrasonic range.

Seawater absorption is much greater than freshwater absorption — important for sonar engineering and for the long-range sound channel (SOFAR) properties.

In tissue

Medical ultrasound encounters tissue with $\alpha \propto f^n$ where $n \approx 1$–$1.5$ (intermediate between $f^1$ for pure relaxation and $f^2$ for pure classical). The mechanism is complex: combinations of viscous, thermal, and relaxation losses in heterogeneous biological materials. Practical values for tissue absorption are about 0.5 dB/cm at 1 MHz, increasing roughly linearly with frequency — so 5 MHz ultrasound penetrates 4 cm, 10 MHz penetrates 2 cm. This is the central design tradeoff in ultrasound imaging (frequency vs penetration).

Looking at the linear regime as a whole

We have, in lessons 10.1–10.3, described the full linear-acoustic loss budget: classical mechanisms plus molecular relaxation, all with their own frequency dependences. The linear-acoustic medium is well-characterised: a wave equation with corrections that depend on frequency and on the medium’s microscopic properties.

But the linear theory assumes small amplitudes. If we drive a sound source hard enough — and the threshold for “hard enough” is set by the wave speed and the propagation distance — nonlinear effects become important. The leading nonlinear effect is wave steepening: the crest of a strong sound wave travels faster than the trough, so the wave gradually distorts into a sawtooth, eventually forming a shock.

Lesson 10.4 begins the nonlinear story.