Radiation force

Introduction

It might be said that radiation pressure is a phenomenon that the observer thinks he understands—for short intervals, and only every now and then.

R. T. Beyer [1]

Radiation force refers to the time-averaged force exerted by waves on objects they encounter. "Time-averaged" refers to integration of a periodic function over a cycle and division by the period. "Waves" can refer to waves on a string, surface waves in water, acoustic waves, and electromagnetic waves, for example. The mechanism underlying radiation force is the conservation of momentum at quadratic order.

Both traveling and standing waves are capable of exerting radiation forces, but these forces are different in their dependence on both position and frequency. The differences can be understood qualitatively by noting that traveling waves carry time-averaged intensity, while standing waves do not; meanwhile, standing waves create time-averaged gradients of energy, while traveling waves do not. As such, radiation forces due to traveling and standing waves are studied separately.

Below are some real-world examples of radiation force:

In 1619, Johannes Kepler posited the existence of radiation force to explain why the tails of comets point away from the sun. Comet Hale-Bopp (1997) and Comet Neowise (2020) are two of the "great comets" of our era that memorably demonstrate this effect. See The Shifting Tails of Comet NEOWISE for an image of Comet NEOWISE's changing appearance over the course of a week in July 2020, and note how the tail points away from the Sun, rather than opposite to the direction of its apparent or proper motion. Also seen in these images is the (fainter) ion tail, which generally does not coincide with the dust tail. The ion tail is subject not only to the radiation force of the Sun but also to the Lorentz force associated with the Sun's magnetic field. For a stunning example of how misaligned the dust and ion tails can be, see Hale-Bopp's Fickle Ion Tail.
Laser cooling refers to the cooling of systems using the radiation force of light.
Optical tweezers are beams of light used to trap and manipulate particles.
Acoustic fountain: an ultrasonic transducer in water exerts a time-averaged force on the surface of the water, causing a jet to form. Because the transducer is often made of a piezoelectric crystal, these fountains are sometimes referred to as "crystal fountains."

For not-so-real (but very illustrative) examples of radiation force, see:

A note about the earlier literature

I had originally attempted to reproduce Rayleigh's study of radiation forces in "simpler" oscillatory systems, namely the pendulum and string constrained by a ring, which he discussed in the papers

Philos. Mag. 3, 338-346 (1902)
Philos. Mag. 10, 364-374 (1905).

But the setup of these problems (especially the role that the ring plays, and the forces it experiences) makes them not so simple. For example, in the 1902 paper, I do not understand why the ring experiences an upward force \(W(1-\cos\theta)\), where \(W\) is the weight of the bob, and where \(\theta\) is the angle measured from the downward direction.

Since Rayleigh's examples seem rather contrived, and since I struggled to relate those examples to the study of acoustic and electromagnetic radiation forces, I abandoned my attempts. Instead, I consider the radiation force exerted by waves on a string. The energy and momentum conservation theorems for waves on a string prove to be analogous to acoustical versions. A convenient parallel can also be made between the radiation stresses of waves on a string and acoustic waves.

These notes may contain typos—please contact me if you find any.

Acknowledgements

These notes were inspired by a lunch discussion with Profs. G. W. Swift, R. Waxler, L. Zhang, A. A. Atchley, J. Mobley, J. D. Maynard, and M. F. Hamilton at the 2024 Physical Acoustics Summer School (PASS). I am grateful to Prof. L. A. Ostrovsky for our email discussions about acoustic radiation force. Thanks to Prof. P. S. Wilson for the "acoustic fountain" demo. Also, thanks to Dr. R. P. Williams for helping me think through Rayleigh's pendulum [Philos. Mag. 3, 338-346 (1902)], and to Mr. J. S. Hallveld for his contributions to the discussion on whether acoustic radiation force should be regarded as a linear or nonlinear effect.

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