# Nonlinear Acoustics

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## Nonlinear acoustics

The study of amplitude-dependent acoustical phenomena. The amplitude dependence is due to the nonlinear response of the medium in which the sound propagates, and not to the nonlinear behavior of the sound source. According to the linear theory of acoustics, increasing the level of a source by 10 dB results in precisely the same sound field as before, just 10 dB more intense. Linear theory also predicts that only frequency components radiated directly by the source can be present in the sound field. These principles do not hold in nonlinear acoustics. *See* Nonlinear physics

The extent to which nonlinear acoustical effects are strong or even significant depends on the competing influences of energy loss, frequency dispersion, geometric spreading, and diffraction. When conditions are such that nonlinear effects are strong, acoustic signals may experience substantial waveform distortion and changes in frequency content as they propagate, and shock waves may be present. Nonlinear acoustical effects occur in gases, liquids, and solids, and they are observed over a broad range of frequencies. Shock waves present in sonic booms and thunder claps are in the audio frequency range. Principles of nonlinear acoustics form the basis for procedures at megahertz frequencies used in medical ultrasound and nondestructive evaluation of materials. Nonlinearity can also induce changes in nonfluctuating properties of the medium. These include acoustic streaming, which is the steady fluid flow produced by the absorption of sound, and radiation pressure, which results in a steady force exerted by sound on its surroundings. *See* Acoustic radiation pressure, Shock wave, Sonic boom

The principal feature that distinguishes nonlinear acoustics from nonlinear optics is that most acoustical media exhibit only weak dispersion, whereas media in which nonlinear optical effects arise exhibit strong dispersion. Dispersion is the dependence of propagation speed on frequency. In optical media, strong nonlinear wave interactions require that phase-matching conditions be satisfied, which can be accomplished only for several frequency components at one time. In contrast, all frequency components in a sound wave propagate at the same speed and are automatically phase-matched, which permits strong nonlinear interactions to occur among all components in the frequency spectrum. *See* Nonlinear optics

Acoustic streaming is a nonlinear effect because the velocity of the flow depends quadratically on the amplitude of the sound, and the flow is not predicted by linear theory. Absorption due to viscosity and heat conduction results in a transfer of momentum from the sound field to the fluid. This momentum transfer manifests itself as steady fluid flow.

Acoustic streaming produced in sound beams is enhanced considerably when shocks develop. Shock formation generates a frequency spectrum rich in higher harmonics. Because thermoviscous absorption increases quadratically with frequency, attenuation of the wave, and therefore the streaming velocity, increases markedly following shock formation. Streaming is also generated in acoustic boundary layers formed by standing waves in contact with surfaces. Measurements of acoustic streaming have been used to determine the bulk viscosity coefficients of fluids. Thermoacoustic engines and refrigerators are adversely affected by heat transport associated with streaming. *See* Thermoacoustics

Phase conjugation refers to wavefront reversal, also called time reversal, at a single frequency. The latter terminologies more clearly describe this procedure. A waveform is captured by a phase conjugation device and reversed in such a way that it propagates back toward the source in the same way that it propagated toward the conjugator. Sound that is radiated from a point source and propagates through an inhomogeneous medium that introduces phase distortion in the wave field is thus retransmitted by the conjugator in such a way as to compensate for the phase distortion and to focus the wave back on the point source.

Phase conjugation is used to compensate for phase distortion in applications involving imaging and retargeting of waves on sources. The most successful techniques for acoustical phase conjugation are based on modulation of acoustical properties of a material that captures the incident sound wave. The modulation is twice the frequency of the incident sound wave, and it is induced by an electric field applied to piezoelectric material, or a magnetic field applied to magnetostrictive material. Often the modulated property of interest is the sound speed in the material. When the incident wave at frequency *f* propagates through a medium in which the sound speed fluctuates at frequency 2*f*, parametric interaction generates a wave at the difference frequency *f* that propagates backward as though reversed in time. *See* Magnetostriction, Optical phase conjugation, Piezoelectricity

Phenomena associated with nonlinear acoustics have proved useful in both diagnostic and therapeutic applications of biomedical ultrasound. A very significant breakthrough in diagnostic imaging, especially for echocardiography and abdominal ultrasound imaging, is based on second-harmonic generation. Medical ultrasound imaging is performed at frequencies of several megahertz. Images constructed from the backscattered second-harmonic component have substantially reduced clutter and haze associated with the propagation of ultrasound through the outer layers of skin, which is the primary cause of phase aberrations. In another technique, microbubbles are injected into the bloodstream to enhance echoes backscattered from blood flow. The microbubbles are fabricated to make them resonant at diagnostic imaging frequencies, and they become strongly nonlinear oscillators when excited by ultrasound. Imaging is based on echoes at harmonics of the transmitted signal. Frequencies backscattered from the microbubbles differ from those in echoes coming from the surrounding tissue, which highlights the locations of the microbubbles and therefore of the blood flow itself.

A notable therapeutic application is lithotripsy, which refers to the noninvasive disintegration of kidney stones and gallstones with focused shock waves. Nonlinear acoustical effects in lithotripsy are associated not only with propagation of the shock wave but also with the generation of cavitation activity near the stones. Radiation of shock waves due to the collapse of cavitation bubbles is believed to be the dominant cause of stone breakup. An emerging therapeutic application, high-intensity focused ultrasound (HIFU), utilizes the heat dissipated by shock waves that develop in beams of focused ultrasound. The heating is so intense and localized that the potential exists for noninvasive cauterization of internal wounds and removal of tumors and scar tissue. *See* Cavitation, Ultrasonics

## Nonlinear Acoustics

the part of acoustics that deals with phenomena that are inadequately described by the usual approximations of linear acoustic theory and for which the nonlinear terms of the hydrodynamics equations and the equation of state must be taken into account. Ordinarily, such phenomena (called the nonlinear effects) become important only for sufficiently large amplitudes of the sound waves. In this sense the subject of nonlinear acoustics is high-intensity sound fields—for example, the propagation of high-power ultrasonic and sonic (shock) waves and the generation of intensive parasitic oscillations during the operation of rocket motors.

The propagation of intensive acoustic waves (also called waves of finite amplitude) has a number of important features. One such feature is the change in wave form during propagation as a result of the differences in the rates of displacement of various points on the wave’s profile: points corresponding to regions of compression “travel” faster than points corresponding to regions of rarefaction. This occurs because the velocity of sound in a region of compression is higher than in a region of rarefaction; in addition, the wave is carried along by the medium which moves in the direction of the wave propagation in a region of compression and against it in a region of rarefaction. For waves of low amplitude, this velocity difference is negligible; therefore, such waves propagate with virtually no change in shape, in conformity with the solutions of linear acoustics, which assumes the velocity of sound to be constant at all points of the wave profile. In the case of high-intensity waves the cumulative effect of the change in the form of a wave that is initially sinusoidal can cause such an increase of the slope in the separate parts of its profile that a discontinuity occurs in each period, and a periodic shock wave having a sawtooth shape is generated.

Unlike low-amplitude waves, intense acoustic waves do not obey the superposition principle. Also included among the nonlinear effects are acoustic radiation pressure and sound streaming, which are important for certain technological processes.

### REFERENCES

Zarembo, L. K., and V. A. Krasil’nikov.*Vvedenie v nelineinuiu akustiku*. Moscow, 1966.

*Fizika i tekhnika moshchnogo ul’trazvuka*[book 2]. Edited by L. D. Rozenberg. Moscow, 1968.