Langevin radiation force as a consequence of linear wave fields

The question of whether acoustic radiation force is a consequence of linear or nonlinear wave fields has been debated for over a century. It is understood that the Rayleigh radiation pressure depends on the quadratic term in the equation of state, whereas the Langevin radiation pressure does not [R. T. Beyer, J. Acoust. Soc. Am. 63, 1025-1030 (1978)]. Nevertheless, traditional derivations of the Langevin radiation force begin with the nonlinear momentum equation. Here, the Langevin radiation force is derived from the linearized continuity, state, and momentum equations, proving that "for the calculation of the average force correct up to terms of the second order in the velocity, it is sufficient to find the solution of the linear scattering problem" [L. P. Gor'kov, Sov. Phys. Dokl. 6, 773-775 (1962)]. The linear origins of the Langevin radiation force are clarified by drawing parallels with radiation forces exerted by electromagnetic waves, surface gravity waves, and waves on a string.