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Figure 7. Top: particle separation in an acousto-microfluidic device accomplished with acoustic radiation forces produced by standing waves generated in the fluid by interdigital transducers (IDT; after Jo and Guldiken, 2012). Bottom: acoustic streaming vortices produced by the absorption of a sound beam in an enclosed fluid. See text for further description.
shocks at the focus (Figure 6). In this case, a symmetrical sawtooth waveform does not occur due to the presence of diffraction, which results in different frequencies focusing at slightly different distances, producing the highly asymmetri- cal waveform with a strong short peak and a longer trough of lower amplitude. The shocks are thus superfocused and con- fined to a much narrower focal region. This occurs because the very short rise times of the shocks are associated with very high frequencies generated during nonlinear propagation of the wave toward the focus, and they are less affected by dif- fraction (spreading) than the frequencies radiated directly by the source.
Acoustic cavitation is a common by-product of HIFU, espe- cially histotripsy (Maxwell et al., 2012) and shock-wave lithotripsy (Bailey et al., 2006). Bubbles are created when the negative pressure phase of the acoustic wave drops below the vapor pressure, and in the case of shock-wave lithotripsy, the subsequent bubble collapse is often suf- ficiently violent to become a secondary source of shock waves. Alternatively, various therapeutic applications of acoustically driven bubbles, which are strongly nonlinear
oscillators, employ micron- and submicron-size bubbles injected into the bloodstream (Gray et al., 2019).
Radiation Force and Streaming
Other physical effects of high-intensity sound are related to the fact that a wave carries momentum that can be trans- ferred to the medium. This momentum transfer creates a volume force, called the radiation force, which depends nonlinearly on the amplitude of the wave. In liquids, the resulting force generates hydrodynamic flows (acoustic streaming), and in elastic media such as soft biological tis- sues, it generates shear waves.
The relevant quantity is the time average of the product of the mass density and particle velocity, a quadratic quantity equal to the average momentum per unit volume. When a progressive sound wave encounters an obstruction, whether it be an object that scatters sound in different directions or a planar interface that produces reflected and transmitted waves, the magnitude and direction of the wave momentum change. From Newton’s second law, a force equal to the rate of change of the wave momentum acts on the obstruction. Although standing and multidimensional wave fields add complexity, the physical principle is the same.
For a particle that is small relative to a wavelength, the magnitude of the radiation force is proportional to the differences in compressibility and density of the par- ticle compared with the corresponding properties of the surrounding fluid (Gor’kov, 1962). The force acting on larger objects is similarly related to these parame- ters (Sapozhnikov and Bailey, 2013; Ilinskii et al., 2018). This relationship is exploited to separate small particles in microfluidic devices. An example of one such device described by Jo and Guldiken (2012) is illustrated in Figure 7, top.
Radiation force can separate particles not only in a stand- ing wave but also in a traveling wave. A possibility of biological cell sorting based on this principle is discussed by Matula et al. (2018). Even larger particles like kidney stones can be effectively manipulated (Simon et al., 2017).
In contrast with acoustic radiation force acting on scatter- ers, radiation force that creates acoustic streaming is due to momentum transferred to the bulk of the liquid caused by absorption of the wave; it thus accompanies energy dissi- pation in a sound field. Whereas energy dissipation due to
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