6.6 Diffusion timescales and acoustic attenuation

A diffusivity, having units of $\text{m}^2/\text{s}$ , immediately sets the timescale on which a transport process acts over a given distance. This final lesson turns the three diffusivities into estimates of how fast, and then applies them to a culminating example: the absorption of a sound wave by the viscosity and heat conduction of the medium it travels through.

The diffusion time

Because $\langle x^2\rangle \sim Dt$ , the time for a diffusive process to act across a length $L$ is

\tau \;\sim\; \frac{L^2}{D},

with $D$ standing for whichever diffusivity is relevant — $\nu$ for momentum, $\alpha$ for heat, $D$ for matter. The scaling is quadratic in length, and that single fact dominates the physics of transport: halving the distance quarters the time. Diffusion is extraordinarily effective over micrometres and hopeless over metres.

D1.00e-5 m²/s

L1.00e-4 m

τ = L²/D1.00e-3 s

log₁₀ D = -5.00 log₁₀ L = -4.00

The diffusion timescale scales quadratically with length: doubling L quadruples τ. This is the key fact that makes microscopic processes fast and macroscopic ones slow. Sugar takes seconds to diffuse across a 100 μm cell; minutes across a 1 mm tube; *years* across a meter of unstirred water. The L² scaling is why mixing must usually be done by *convection*, not diffusion alone.

A nutrient diffuses across a $10\,\mu\text{m}$ cell in a fraction of a second but would take years to cross a metre of the same medium; heat penetrates a thin metal sheet instantly on the scale that it would take hours to soak through a thick wall. The quadratic law is why living cells must be small, why thin boundary layers form quickly while thick ones lag, and why stirring — which replaces slow diffusion over a large $L$ with fast diffusion over the small $L$ between folds — accelerates mixing so dramatically.

Fast or slow? The adiabatic criterion

The same estimate decides whether a rapid compression is isothermal or adiabatic. Compress a parcel of gas of size $L$ over a time $t_\text{comp}$ . Heat can equalise the temperature only if it has time to diffuse across the parcel, that is, if $t_\text{comp} \gtrsim L^2/\alpha$ . When the compression is faster than the thermal-diffusion time, heat cannot escape the parcel and the process is adiabatic; when it is slower, the parcel stays at the temperature of its surroundings and the process is isothermal. The dimensionless ratio $L^2/(\alpha\,t_\text{comp})$ — or equivalently $\omega L^2/\alpha$ for an oscillation of angular frequency $\omega$ — is what decides. A compression that is adiabatic when fast becomes isothermal when slow, with the crossover at the diffusion time.

Acoustic attenuation

A sound wave is an oscillating compression, and the transport coefficients of this chapter make it gradually decay. Two mechanisms drain its energy: viscosity, which dissipates the shearing motion between regions moving at different velocities, and heat conduction, which leaks heat from the compressed (warm) crests to the rarefied (cool) troughs. The classical Kirchhoff–Stokes absorption coefficient combines both:

\alpha_\text{abs}(f) \;=\; \frac{(2\pi f)^2}{2\rho c^3}\left[\tfrac{4}{3}\mu \;+\; (\gamma - 1)\frac{k}{c_p}\right].

where

$\alpha_\text{abs}$: amplitude absorption coefficient 1/m
$f$: sound frequency Hz
$\rho$: fluid density kg/m³
$c$: speed of sound m/s
$\mu$: dynamic viscosity Pa·s
$k$: thermal conductivity W/(m·K)
$c_p$: specific heat at constant pressure J/(kg·K)
$\gamma$: ratio of specific heats —

Both terms scale as $f^2$ : the absorption grows with the square of frequency, because higher frequencies pack steeper gradients into each wavelength and diffusion acts more fiercely on steep gradients.

Medium:

Classical absorption scales as f² — doubling the frequency *quadruples* the attenuation per metre. Air absorbs sound much more strongly than water; ultrasonic frequencies (MHz) are essentially absorbed within metres in air but propagate hundreds of metres in water. Below 100 kHz absorption in both is utterly negligible on room scales, which is why ordinary acoustics happily treats sound as undamped.

The viscous term comes directly from the momentum diffusion of the first lesson; the thermal term from the heat conduction of the second; the $(\gamma - 1)$ factor measures how much warmer a compression gets, which is the lever heat conduction pulls on. The combined coefficient is utterly negligible for low-frequency sound in air but becomes dominant for megahertz ultrasound in water — which is why ultrasonic imaging trades penetration depth for resolution as frequency rises. The same single equation, built from two of the three transport coefficients introduced in this chapter, governs the entire trade-off.