The Second Order

In 1847, George Gabriel Stokes showed that water particles in surface waves do not trace closed orbits. Linear wave theory predicts circles — each particle rises as the crest passes, moves forward, descends as the trough arrives, and returns to its starting position. The orbit closes. No net transport occurs.

Stokes kept the second-order terms. When the wave amplitude is not infinitesimally small, the forward velocity at the crest is slightly greater than the backward velocity at the trough, because the particle is slightly higher during the forward phase and slightly lower during the backward phase. The orbital speed varies with depth, and the particle spends unequal times at different heights. The orbit does not close. Each cycle leaves the particle slightly advanced in the direction of wave propagation.

The displacement per cycle is proportional to the square of the wave amplitude divided by the wavelength, multiplied by an exponential decay with depth. For typical ocean swell — two meters high, ten-second period — the drift at the surface is roughly six centimeters per second. Slow enough to be invisible in any single wave. Fast enough to move floating debris into convergence lines, to spread oil spills faster than wind stress predicts, to generate Langmuir circulations when the drift interacts with wind-driven surface currents.

The transport is real. It moves mass. But it vanishes from the equations the moment you linearize them — the moment you assume the wave amplitude is small enough that its square can be neglected. The drift exists in the square of the amplitude, and linear theory sees only the amplitude.

In 1862, James Clerk Maxwell calculated that electromagnetic waves carry momentum. The prediction followed from his equations: the Poynting vector, which describes the energy flux in the wave, divided by the square of the speed of light, gives a momentum density. A wave that oscillates electric and magnetic fields back and forth — an oscillation with zero time-average field — nonetheless exerts a steady pressure on any surface it strikes.

Adolfo Bartoli derived the same conclusion in 1876 from thermodynamics alone, without reference to Maxwell's equations. If radiation could not exert pressure, Bartoli argued, it would be possible to construct a perpetual motion machine using mirrors and radiating bodies. The pressure must exist because its absence would violate the second law.

The measurement proved difficult. In 1901, Pyotr Lebedev in Moscow and Ernest Nichols and Gordon Hull at Dartmouth independently succeeded. Both used torsion balances with reflective and absorbing vanes in evacuated chambers. The principal obstacle was the radiometric effect: residual gas molecules bouncing off the warm illuminated side of a vane produce a force thousands of times larger than radiation pressure. Only at pressures below a hundredth of a millimeter of mercury did the radiation pressure emerge from the thermal noise. Lebedev's result agreed with Maxwell's prediction to within twenty percent.

The force is small. Solar radiation pressure at Earth's orbit is four and a half micropascals. But it acts continuously, and it acts on everything the light touches. Cometary tails point away from the sun because radiation pressure pushes dust particles outward. Small asteroids spin up over millions of years as asymmetric thermal re-emission applies a torque — the Yarkovsky-O'Keefe-Radzievskii-Paddack effect. Solar sails — IKAROS in 2010, LightSail 2 in 2019 — produce measurable thrust from sunlight alone. The pressure was always there, in every beam of light that ever struck a surface. It was invisible because it lives in the square of the field amplitude, and the oscillating fields — the ones you see and feel — are first order.

In 1884, Lord Rayleigh demonstrated that sound waves in a viscous medium generate steady flows. A tuning fork vibrating near a surface produces an oscillating air velocity that, time-averaged, should yield zero net motion. But viscosity breaks the symmetry. In the thin boundary layer near the surface, the oscillating flow experiences friction. The friction is nonlinear — it depends on the square of the velocity. Because the velocity profile is asymmetric within the boundary layer, the time-averaged product of velocity fluctuations produces a net force: the Reynolds stress. The stress drives a steady circulation.

Rayleigh distinguished two patterns. Inner streaming — later called Schlichting streaming — consists of small vortices within the boundary layer itself, rotating in alternating directions along the surface. Outer streaming fills the bulk of the enclosure with larger circulations driven by the inner vortices. James Lighthill unified the theory in 1978, showing that both arise from the same mechanism: the time-averaged divergence of the oscillatory momentum flux.

A different form — Eckart streaming — appears in unbounded media where the sound wave itself is gradually absorbed. The absorption transfers momentum from the wave to the medium. Since the wave travels in one direction, the transferred momentum drives a flow in that direction. The flow velocity is proportional to the acoustic intensity and the absorption coefficient. Louder sound, more absorption, faster flow.

Acoustic streaming drives ultrasonic cleaning: cavitation dislodges particles from surfaces, and the streaming flow carries them away. Surface acoustic wave devices in microfluidics use streaming to mix fluids in channels too small for turbulence. Acoustic levitation suspends small objects at pressure nodes, and streaming stabilizes them against perturbations. In every application, the useful work — the transport, the mixing, the suspension — comes from the second-order terms of an oscillation whose first-order average is zero.

A water wave oscillates surface particles. A light wave oscillates electric and magnetic fields. A sound wave oscillates air pressure. At first order, each oscillation averages to zero — no net displacement, no net force, no net flow. The motion is reversible and carries nothing.

At second order, each oscillation transports. Water waves drift particles downwind. Light waves push surfaces sunward. Sound waves drive flows along their path. The transport arises because the square of the oscillation does not average to zero — the forward phase is not the exact negative of the backward phase once amplitude, position, or friction breaks the symmetry.

Linear theory discards these terms. The discarding is not wrong — it is a deliberate simplification that captures the dominant behavior. But the dominant behavior is oscillation, and oscillation moves nothing. Everything that waves carry — momentum, mass, energy in a preferred direction — lives in the terms that linear theory drops.