As shown by Rayleigh, a considerable number of acoustic phenomena are known which involve the viscosity of the medium and require the solution of the hydrodynamic equations to a higher degree of approximation than is customary in elementary treatments of the theory of sound. Among these are the fluid streams that occur near intense sources of sound (e.g.: the "quartz wind").

The general equations of these second-order acoustic phenomena are developed in a systematic manner. When viscous forces are neglected, the effects are of three kinds: (1) those that can be ascribed to the inertia of acoustic energy, (2) those arising from radiation pressure, and (3) those caused by the variable compressibility of the medium. All of them result in the production of overtones of the fundamental vibration. In certain cases, this distortion can become very large, being unlimited except by the viscous forces. However, even when the average value of the gradient of the radiation pressure does not vanish, it does not, on the average, cause an acceleration of the fluid. Such gradients are balanced by the elastic rather than by the viscous forces.

When the latter are introduced into the calculation, a fourth effect appears: the irrotational motion in the sound wave generates vorticity as a second-order effect. This vortex motion will ultimately approach a steady state, being generated and resisted by forces that are independent of the time. Both generating and resisting forces are viscous, and consequently the steady motion is independent of the magnitude of the coefficient of viscosity. However, the resisting forces depend only on the shear viscosity of the medium, while the generating forces depend also on the bulk viscosity. It is suggested that the ratio of the bulk and shear coefficients of viscosity can be determined by studying these phenomena.

Calculations of the velocity of the stream generated by a beam of sound show that it is proportional (1) to $b=\left(\frac{4}{3}\right)+\left(\frac{{\nu }^{\prime }}{\nu }\right)$, where ${\nu }^{\prime }$ and $\nu$ are the bulk and shear viscosities, (2) to the power being radiated in the beam, (3) inversely to the square of the wave-length, and (4) inversely to ${\rho }^2{c}^3$, where $\rho$ is the density and c the sound velocity of the medium. The maximum value of the steady-streaming velocity depends on the resistance offered by the walls of the vessel or room in which the experiment is performed. The time required to set up the steady state is, of course, inversely proportional to this resistance, and the flow is apt to become turbulent when the resistance is low.

• Received 28 August 1947

DOI:https://doi.org/10.1103/PhysRev.73.68