10.1 Viscosity and thermal absorption

The linearised wave equation said nothing about losses. A plane wave in our chapter-4 medium propagates forever without decay. Real fluids dissipate, slowly. The dominant mechanisms are viscosity (a velocity-gradient resistive force in the fluid) and thermal conduction (heat flowing from compressed regions to rarefied regions, smoothing the temperature variation that the wave creates).

The damped wave equation

Adding viscous and thermal dissipation to the linearised wave equation gives — at lowest order — a modified equation of the form

\frac{\partial^2 p'}{\partial t^2} \;=\; c^2 \nabla^2 p' \;+\; \delta\, \frac{\partial}{\partial t}\, \nabla^2 p',

where $\delta$ is a small dissipation coefficient with units of m²/s. The extra term is a kind of “viscous wave equation” — its effect is to damp high-frequency components more than low ones.

Plane-wave attenuation

For a plane wave $p' = P_0\, e^{i(\omega t - k x)}$ with frequency $\omega$ , the dispersion relation becomes (to lowest order in $\delta$ )

\omega^2 \;=\; c^2 k^2 + i \delta \omega k^2,

giving a complex wavenumber $k = k_r + i k_i$ where $k_r \approx \omega/c$ and $k_i \approx \delta \omega^2 / 2 c^3$ .

The imaginary part of $k$ produces exponential decay:

p'(x, t) \;\sim\; e^{-k_i x}\, \cos(\omega t - k_r x).

The absorption coefficient is $\alpha \equiv k_i$ , with units of nepers per metre (or, more usually, dB per metre after multiplying by $20/\ln 10 \approx 8.686$ ).

The classical absorption formula

For air the classical (viscous + thermal) absorption coefficient is

\alpha_\text{classical} \;=\; \frac{2 \pi^2 f^2}{\rho_0 c^3}\!\left[\frac{4}{3}\eta + (\gamma - 1)\frac{\kappa}{c_p}\right],

where $\eta$ is the shear viscosity and $\kappa$ is the thermal conductivity. The two terms in brackets are the viscous and thermal contributions; for air they are of similar magnitude.

The key feature: $\alpha_\text{classical} \propto f^2$ . Absorption rises as the square of frequency. A 1 kHz wave travels much farther than a 10 kHz wave; a 100 Hz wave is essentially undamped by classical absorption.

Quantitatively for dry air at 20°C:

100 Hz: $\alpha_\text{classical} \approx 8 \times 10^{-6}$ dB/m. Negligible over kilometres.
1 kHz: $\alpha_\text{classical} \approx 8 \times 10^{-4}$ dB/m. Still small over hundreds of metres.
10 kHz: $\alpha_\text{classical} \approx 0.08$ dB/m. About 8 dB lost over 100 m.
100 kHz: $\alpha_\text{classical} \approx 8$ dB/m. Strong attenuation; relevant for ultrasonic propagation.

Why this is “classical”

The formula above accounts only for the two simplest, “classical” dissipation mechanisms — viscous shear and thermal conduction in an ideal gas. Real air at audio frequencies has a third important loss mechanism: molecular relaxation of N₂ and O₂. Below about 10 kHz, relaxation losses dominate over the classical losses by an order of magnitude or more. The full atmospheric absorption is the topic of the next lesson.

In other fluids

For water, classical absorption gives much smaller $\alpha$ — by a factor of $\sim 10^3$ at the same frequency — making seawater an excellent acoustic medium for long-range communication (SOFAR channel propagation distances of thousands of km, exploited by whales and submarines).

For solids, viscous-like dissipation is replaced by various internal friction mechanisms (dislocation motion, grain boundary sliding, etc.), with their own frequency dependences. For ultrasonic NDT (non-destructive testing) in metals, attenuation at MHz frequencies is typically a few dB/m and rises with frequency.

Why thermal conduction shows up at all

A surprising feature: thermal conduction contributes to acoustic absorption. The mechanism: a compression zone is briefly hotter than the surrounding medium (adiabatic compression heats the gas). Heat flows out of the compression toward the cooler rarefactions. This heat flow carries energy out of the wave and into a thermal “tail” of the medium that doesn’t propagate.

The same effect makes the gas-compression process slightly less than fully adiabatic — the entropy is not exactly conserved. The deviation is tiny at audio frequencies but accumulates as a measurable loss over distance.

Next lesson: how molecular relaxation in N₂ and O₂ adds another, larger loss mechanism, with its own characteristic frequency dependence.