5.6 Momentum carried by sound; radiation pressure

A sound wave carries momentum as well as energy. The momentum density is small (it’s a second-order quantity in the perturbation), but it produces real, measurable effects — most notably radiation pressure on obstacles in the path of the wave.

Momentum density

The linear-acoustic momentum density is $\rho_0 \mathbf{v}'$ — mass density times perturbation velocity. But that quantity time-averages to zero over a period: $\mathbf{v}'$ is sinusoidal with zero mean. The momentum we measure is second-order: the time-averaged $\rho' \mathbf{v}'$ , which is non-zero.

For a plane wave with $p' = P_0 \cos(\omega t - k x)$ and $v' = (P_0/\rho_0 c) \cos(\omega t - k x)$ :

\rho' v' \;=\; \frac{p' v'}{c^2} \;=\; \frac{P_0^2}{\rho_0 c^3}\, \cos^2(\omega t - k x),

and the time average

\langle \rho' v' \rangle \;=\; \frac{P_0^2}{2 \rho_0 c^3} \;=\; \frac{\langle I \rangle}{c^2}.

The mean momentum density carried by a plane wave is its mean intensity divided by $c^2$ . Compare with the photon case: $p_\text{photon} = E_\text{photon}/c$ , which gives the same relation in the appropriate limit. Acoustic momentum is a second-order analogue.

Radiation pressure on an obstacle

If a plane wave is absorbed by an obstacle, the obstacle gains momentum at the rate $\langle I \rangle / c$ per unit area facing the wave. That gain rate is a force per unit area — i.e. a pressure:

P_\text{rad}^\text{absorb} \;=\; \langle I \rangle / c.

If the obstacle reflects the wave perfectly back, the momentum change is doubled (the photon-like analogue), so $P_\text{rad}^\text{reflect} = 2 \langle I \rangle / c$ .

For a 1 W/m² wave in air ( $\sim$ 120 dB, painfully loud), the radiation pressure on a fully-reflecting surface is

P_\text{rad} \;\approx\; \frac{2 \cdot 1}{343} \;\approx\; 6 \times 10^{-3}\, \text{Pa},

— about $6$ µPa, six millionths of atmospheric pressure. Small but measurable. Acoustic radiation pressure was first measured by Rayleigh in the late 19th century using sensitive torsion balances.

What this is not

It is not the everyday “I can feel the bass” sensation, which is mostly vibration of your body driven by the time-varying $p'$ field, not the time-averaged radiation pressure. The time-varying $p'$ is first-order in perturbation and large enough to feel; the radiation pressure is second-order and microscopic.

It also is not the same as “speakers pushing air across the room.” Speakers move air locally, but the flow of air dies away within a wavelength or so of the source. The radiation that travels far is a wave, not a wind.

Where radiation pressure matters

Several specialised contexts use radiation pressure as a primary tool:

Acoustic levitation. Standing-wave ultrasonic fields can suspend small objects at nodes or antinodes against gravity. The radiation pressure from the surrounding field provides the lift force.
Acoustic tweezers. Focused ultrasound beams can trap and move cells or microparticles. Used in biological assays.
Cavitation seeding. In a high-intensity field, the local time-averaged pressure (which includes both ambient and radiation contributions) can drop below the vapour pressure of the liquid, nucleating a bubble. This is the first event in cavitation — and the bridge to the planned Cavitation book.
Acoustic streaming. When sound propagates through a viscous medium, the absorbed momentum drives a steady flow. The flow is much slower than the wave but transports mass — useful in microfluidics, audible as the small puff of air from a powerful loudspeaker.

Looking ahead

We started chapter 5 with the plane wave and ended with the field’s mechanical action on matter. The picture is now: plane waves carry energy, intensity, momentum; their interaction with material is mediated by impedance; their amplitude is encoded by decibels. We have the field, we know what it carries, and we know what it does.

Chapter 6 turns to the question of how a wave is produced — how a vibrating surface radiates sound into the surrounding medium. We have been studying the wave in flight; we are about to look at how it leaves the source.