10.4 Energy, intensity, and radiation pressure

A wave transports energy and momentum without transporting matter. This lesson accounts for both: the energy density a wave carries and its equipartition between kinetic and potential forms, the intensity that measures its flux, and the pressure it exerts when that momentum is delivered to a surface.

Energy density and equipartition

A plane sound wave carries energy in two forms — the kinetic energy of the moving fluid and the potential energy of its compression:

\mathcal{E} \;=\; \underbrace{\tfrac12\rho_0|\mathbf{v}'|^2}_\text{kinetic} \;+\; \underbrace{\frac{p'^2}{2\rho_0 c^2}}_\text{potential}.

where

$\mathcal{E}$: energy density J/m³
$\mathbf{v}'$: particle velocity perturbation m/s
$p'$: pressure perturbation Pa
$\rho_0$: equilibrium density kg/m³
$c$: sound speed m/s

For a travelling plane wave the two parts are in phase — both peak at the crest and both vanish at the node — because pressure and velocity are locked in phase by the impedance relation of the previous lesson. This is unlike a standing wave or a mass on a spring, where kinetic and potential energy trade off a quarter-cycle apart.

In a plane sound wave, kinetic energy density (red) and potential energy density (green dashed) oscillate *in phase* — they are simultaneously maximum at the wave crests and zero at the nodes. On time-average they are equal (acoustic equipartition), each contributing ½ to the total energy density. The total (black) is twice the time-average of either part — a clean instance of the chapter's energy bookkeeping.

On time-average the two parts are equal — acoustic equipartition — so the mean energy density is either one doubled. This equality is a general feature of linear travelling waves: the energy divides evenly between the field’s “velocity-like” and “displacement-like” halves.

Intensity

The intensity is the time-averaged energy flux — power per unit area carried normal to the wavefront:

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

Intensity scales with the square of the pressure amplitude: doubling $P_0$ quadruples the intensity. It is the physical quantity the decibel scale references, and the $P_0^2$ dependence is why a $6\,\text{dB}$ change corresponds to a doubling of amplitude but a fourfold change in power.

Radiation pressure

A wave carries momentum as well as energy, at a rate of $I/c$ per unit area. When that momentum is delivered to a surface, the surface feels a steady radiation pressure:

P_\text{rad} \;=\; \frac{I}{c}\;\;(\text{absorbed}), \qquad \frac{2I}{c}\;\;(\text{reflected}).

The reflected case gives twice the force because the wave’s momentum is not merely stopped but reversed. Though small at ordinary intensities, radiation pressure becomes significant in intense beams: it is what lets a focused ultrasonic field levitate and manipulate small particles — acoustic levitation and acoustic tweezers — and it is the direct mechanical analogue of the light pressure that drives solar sails and optical tweezers. Energy sets what a wave delivers per second; momentum sets the force it exerts while doing so.