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 Oleg A. Sapozhnikov
Vera A. Khokhlova
Physics Faculty M. V. Lomonosov Moscow State University Moscow 119991, Russia
Robin O. Cleveland
Institute of Biomedical Engineering University of Oxford Oxford OX3 7DQ, United Kingdom
Philippe Blanc-Benon
Laboratoire de Mécanique des Fluides et d’Acoustique Centre National de la Recherche Scientifique (CNRS) Ecole Centrale de Lyon 36 Avenue Guy de Collongue 69134 Ecully Cedex, France
Mark F. Hamilton
Walker Department of Mechanical Engineering and Applied Research Laboratories The University of Texas at Austin Austin, Texas 78712, USA
Nonlinear Acoustics Today
Nonlinear acoustics can remove particulates from air, quiet sonic booms, create audio spotlights, and improve medical ultrasound imaging and therapy.
The world around us is inherently nonlinear. Linearity of any process is an approxi- mation for the case of small, possibly infinitesimal, motion. From this point of view, it would seem that it should be linearity rather than nonlinearity that is considered unusual. Linearity is commonly assumed in wave physics, which allows, for example, the principle of superposition, according to which two waves pass freely through each other, not interacting or influencing one another in any way.
Nonlinear waves that we have all seen many times are those on the surface of water. When such a wave approaches the shore, its profile begins to distort; the crest of the wave travels fastest, causing the wave to steepen and eventually overturn, ultimately breaking and creating a cloud of foam (Figure 1, top). Unlike waves on a water sur- face, it is physically impossible for sound waves to become multivalued (e.g., possess three different sound pressures at one point in space) and subsequently break. However, exact formulations of the equations for ideal fluids by mathematicians in the eighteenth and nineteenth centuries (luminaries such as Euler, Riemann, Poisson, Earnshaw, and Airy) predicted multivalued solutions for sound waves that generated considerable controversy. The matter was ultimately resolved by Stokes, who explained that viscos- ity prevents nonlinear acoustic waves from overturning and, instead, discontinuities
appear in their profiles, which are referred to as shocks (see Hamilton and Blackstock, 2008, Chap. 1).
As depicted in Figure 1, the frequency spectrum of a nonlinearly distorted sound wave contains, along with the initial frequency, many frequencies referred to as harmonics. The process of waveform distortion, especially shock formation, and the accompanying generation of higher frequencies are distinguishing features of nonlinear acoustic waves
(Atchley, 2005). Several examples are considered below, first in the frequency domain and then in the time domain.
Nonlinear distortion results in energy being exchanged among different frequencies, not only frequencies that were present at the source but other frequencies too. For example, if a wave is radiated at a single frequency (f0), it interacts with itself to transfer energy from the signal at f0 to frequencies 2f0 (called the second harmonic), 3f0, 4f0, and higher, and these harmonics grow with the propagation distance (Figure 1, bottom).
Diagnostic Ultrasound Imaging
One area where generation of the second harmonic plays an important role is in diag- nostic ultrasound imaging. The basis of imaging in medical diagnostics is the pulse-echo method. An ultrasonic transducer emits short tone bursts into the patient’s body, and due to scattering from tissue inhomogeneities and organ interfaces, echoes propagate back to the probe and are used to build an image in the form of a brightness pattern, a B-mode image (Figure 2, bottom left). Images are traditionally formed assuming
©2019 Acoustical Society of America. All rights reserved. volume 15, issue 3 | Fall 2019 | Acoustics Today | 55

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