Nonlinear Acoustic Theory Research Articles

Sound Field Visualization of Specified Frequency by Combining Scanning System with Lock-in Amplifier

Abstract Sound field visualization of specified frequency is an essential method for acoustic studies, especially in nonlinear acoustics. In this paper, we designed a sound field visualization system of specified frequency (SVSF). By combining the traditional sound field scanning system with the lock-in amplifier (LIA), the SVSF can visualize the amplitude distribution of multiple arbitrary specified frequencies with only one-time sound field scanning. Further, depending on the SVSF, we analysed the nonlinear acoustic effects around the focus of a spherical coronal ultrasonic transducer array. The excitation signal is a sinusoidal signal at 40 kHz. The results show that the acoustic waves at the focus contain fundamental frequencies at 40 kHz and harmonic frequencies at 80 kHz, 120 kHz, and 160 kHz. Comparing the amplitude distribution of those frequencies, we found that higher harmonics own smaller focus areas as well as lower amplitude, which conforms to the nonlinear acoustic theory. The system is convenient for those similar studies involving the multi-frequency amplitude field comparison.

Nonlinear acoustic theory on flowing liquid containing multiple microbubbles coated by a compressible visco-elastic shell: Low and high frequency cases

Microbubbles coated by visco-elastic shells are important for ultrasound diagnosis using contrast agents, and the dynamics of single coated bubbles has been investigated in the literature. However, although a high number of contrast agents are used in practical situations, there has long been an absence of a nonlinear acoustic theory for multiple coated bubbles, except for our recent work by Kikuchi and Kanagawa [“Weakly nonlinear theory on ultrasound propagation in liquids containing many microbubbles encapsulated by visco-elastic shell,” Jpn. J. Appl. Phys. 60, SDDD14 (2021)], under several assumptions to be excluded. Aiming for generalization, in this study, we theoretically investigate weakly nonlinear propagation of ultrasound in liquid containing multiple bubbles coated by a visco-elastic shell with compressibility. Leveraging the method of multiple scales, both the Korteweg–de Vries–Burgers (KdVB) equation for a low-frequency long wave and nonlinear Schrödinger (NLS) equation for a high-frequency short wave are derived from the volumetric averaged equations for bubbly liquids based on a two-fluid model and the up-to-date model for single coated bubbles with shell compressibility. Neglected factors in our previous paper, i.e., compressibility of the shell and liquid, drag force acting on bubbles, bubble translation, and thermal conduction, are incorporated in the present KdVB and NLS equations; the proposed model will be regarded as a generic physico-mathematical model. The results show that shell compressibility attenuated ultrasound strongly and decreased nonlinearity of ultrasound. Finally, we compared the magnitudes of six dissipation factors (shell compressibility, shell viscosity, liquid compressibility, liquid viscosity, thermal effect, and drag force) for five typical ultrasound contrast agents, and a similar tendency between KdVB and NLS equations was revealed.

Theory of nonlinear acoustic forces acting on inhomogeneous fluids

Recently, the phenomena of streaming suppression and relocation of inhomogeneous miscible fluids under acoustic fields were explained using the hypothesis on mean Eulerian pressure. In this work, we derive the expression for the acoustic body force without relying on any prior assumptions regarding the second-order Eulerian pressure. We present a theory of nonlinear acoustics for inhomogeneous fluids from first principles, which explains streaming suppression and acoustic relocation in both miscible and immiscible inhomogeneous fluids inside a microchannel. This theory predicts the relocation of higher impedance fluids to pressure nodes of the standing wave, which agrees with recent experiments.

Weakly nonlinear acoustic theory on pressure propagation in bubbly flows with different cases of nonuniform distribution of initial flow velocities of gas and liquid phases

Three cases of weakly nonlinear propagation of plane progressive pressure waves in flowing water containing many uniformly separated spherical microbubbles are theoretically investigated, with a particular focus on different cases of initial flow patterns. The gas and liquid phases initially have a high velocity with uniform distribution, and a low velocity with nonuniform distribution for each phase. Basic conservation laws based on a two-fluid model are employed to govern different velocity distributions for gas and liquid phases. From the method of multiple scales with perturbation expansions, we derive three cases of nonlinear wave equation describing the long-range propagation of waves: (i) the KdVB (Korteweg–de Vries–Burgers) equation for a long wave, (ii) the NLS (nonlinear Schrödinger)-I equation for a short wave in slow nonuniform flow, (iii) the NLS-II equation for a short wave in fast nonuniform flow. As a result, all the initial velocities contribute to an advection effect of waves, initial nonuniform flow distribution induces a variable coefficient of advection term into the KdVB, NLS-I, and NLS-II equations, and an important role of relative velocity between the gas and liquid phases is clarified.

Parametric Acoustic Array and Its Application in Underwater Acoustic Engineering.

As a sound transmitting device based on the nonlinear acoustic theory, parametric acoustic array (PAA) is able to generate high directivity and low frequency broadband signals with a small aperture transducer. Due to its predominant technical advantages, PAA has been widely used in a variety of application scenarios of underwater acoustic engineering, such as sub-bottom profile measurement, underwater acoustic communication, and detection of buried targets. In this review paper, we examine some of the important advances in the PAA since it was first proposed by Westervelt in 1963. These advances include theoretical modelling for the PAA, signal processing methods, design considerations and implementation issues, and applications of the PAA in underwater acoustic engineering. Moreover, we highlight some technical challenges which impede further development of the PAA, and correspondingly give a glimpse on its possible extension in the future. This article provides a comprehensive overview of some important works of the PAA and serves as a quick tutorial reference to readers who are interested to further explore and extend this technology, and bring this technology to other application areas.

Basic Study on Nonlinear Acoustic Theory for Water Flow Containing Many Microbubbles with Acceleration and Deceleration

Basic Study on Nonlinear Acoustic Theory for Water Flow Containing Many Microbubbles with Acceleration and Deceleration

Non-linear acoustic analysis of the pressure rise of the compression wave generated by a train entering a tunnel

Hara's formula is used to predict the pressure rise of a compression wave generated by a high-speed train entering a tunnel. The acoustic theory is used to predict its waveform. Hara's formula was derived from the shock theory. Meanwhile, the acoustic theory produces the same non-dimensional formula as Hara's formula up to order OM4, where M is the train Mach number, when the non-linear acoustic effect in the vortex sound theory is considered. However, the relationship between the shock and acoustic theories remains unclear because the closed formula produced by the vortex sound theory apparently differs from that of Hara's formula. This study shows that Hara's formula can be deductively obtained from the non-linear acoustic theory. A general equation used to predict the compression waveform with a relative error of Op/ρ0c02, where p/ρ0c02 is the non-dimensional acoustic pressure to the atmospheric density ρ0 and the sound speed c0, is shown herein using the non-linear acoustic theory.

Ultrasound processing of Chlorella vulgaris and a novel functional classification of power ultrasound test systems

This study evaluated energy transfer and efficiency in power ultrasound systems applied to processing of Chlorella vulgaris. Power transfer to a series of system combinations was characterized via calorimetry, and energy per cell ruptured was calculated from observed cell disruption. The calorimetric power transferred to solution ranged from 0.028 to 0.348 W/ml. Differentiation of reactor combinations via mixing delay and analysis in the context of non-linear acoustic theory led to a novel approach to reactor classification, allowing inference of cavitation based on streaming and heating behavior. Calorimetric efficiencies of the systems were relatively consistent, and the power transfer of a smooth probe tip was not distinguishable from that of an eroded tip face. The power ultrasound treatment of C. vulgaris required an average ultrasound energy input per cell ruptured ranging from 18 to 76 μJ. This value is 6 orders of magnitude higher than the energy requirement reported for the disruption of a single algae cell with an atomic force microscope. However, the specific energy requirement, estimated as 430 MJ per kg of dry algae cell disrupted, was within the range of values reported in the literature. This difference may be due to power transfer efficiencies inherent in existing algal pretreatment methods.

Active Nonlinear Acoustic Sensing of an Object with Sum or Difference Frequency Fields

A number of nonlinear acoustic sensing methods exist or are being developed for diverse areas ranging from oceanic sensing of ecosystems, gas bubbles, and submerged objects to medical sensing of the human body. Our approach is to use primary frequency incident waves to generate second order nonlinear sum or difference frequency fields that carry information about an object to be sensed. Here we show that in general nonlinear sensing of an object, many complicated and potentially unexpected mechanisms can lead to sum or difference frequency fields. Some may contain desired information about the object, others may not, even when the intention is simply to probe an object by linear scattering of sum and difference frequency incident waves generated by a parametric array. Practical examples illustrating this in ocean, medical, air and solid earth sensing are given. To demonstrate this, a general and complete second-order theory of nonlinear acoustics in the presence of an object is derived and shown to be consistent with experimental measurements. The total second-order field occurs at sum or difference frequencies of the primary fields and naturally breaks into (A) nonlinear waves generated by wave-wave interactions, and (B) second order waves from scattering of incident wave-wave fields, boundary advection, and wave-force-induced centroidal motion. Wave-wave interactions are analytically shown to always dominate the total second-order field at sufficiently large range and carry only primary frequency response information about the object. As range decreases, the dominant mechanism is shown to vary with object size, object composition, and frequencies making it possible for sum or difference frequency response information about the object to be measured from second-order fields in many practical scenarios. It is also shown by analytic proof that there is no scattering of sound by sound outside the region of compact support intersection of finite-duration plane waves at sum or difference frequencies, to second-order. Analytic expressions for second-order fields due to combinations of planar and far-field wave-wave interactions are also derived as are conditions for when wave-wave interactions will dominate the second order field.

Ed Carstensen and the recognition of nonlinear acoustics in biomedical ultrasound

Nonlinear acoustics in biomedical ultrasound was not well recognized, nor utilized in the era leading up to the 1980s. The discipline was experiencing rapid growth and experimentation, but without consideration of nonlinear effects, even though they were significant. Ed Carstensen was one of the first to become suspicious of the “linear assumption,” mostly in his own experiments. In 1979, he contacted me and asked for help in making the community aware of nonlinear acoustics at biomedical frequencies and intensities. The two of us, with the help of David Blackstock and Ed’s graduate students, W.K. Law and N.D. McKay, set out to produce a pair of tutorial manuscripts, to call the community’s attention to acoustic nonlinearity. These were published in the Journal of Ultrasound in Medicine and Biology, Vol. 6, p 345-357, and 359-368 (1980). The first was a description of pertinent nonlinear acoustic theory, as then used, and the second was an experimental demonstration of nonlinear effects, easily recognizable in the laboratory. The present paper recounts the situation at the time (37 years ago) which experienced a turning point, once nonlinear acoustics was recognized. Existing perplexities were resolved and new paths of research and development were identified. Some discussion of here-to-fore unappreciated individuals and history in nonlinear acoustics is also presented. [Work supported by Applied Research Laboratories, The University of Texas at Austin.]

A Novel Noncontact Ultrasonic Levitating Bearing Excited by Piezoelectric Ceramics

A novel ultrasonic levitating bearing excited by three piezoelectric transducers is presented in this work. The transducers are circumferentially equispaced in a housing, with their center lines going through the rotation center of a spindle. This noncontact bearing has the ability to self-align and carry radical and axial loads simultaneously. A finite element model of the bearing is built in ANSYS, and modal analysis and harmonious response analysis are conducted to investigate its characteristics and driving parameters. Based on nonlinear acoustic theory and a thermodynamic theory of ideal gas, the radical and lateral load-carrying models are built to predict the bearing’s carrying capacity. In order to validate the bearing’s levitation force, a test system is established and levitating experiments are conducted. The experimental data match well with the theoretical results. The experiments reveal that the maximum radical and axial levitating loads of the proposed bearing are about 15 N and 6 N, respectively, when the piezoelectric transducers operate at a working frequency of 16.11 kHz and a voltage of 150 Vp-p.

Propagation properties of one-dimensional nonlinear acoustic waves

The propagations of one-dimensional nonlinear acoustical waves are studied numerically and experimentally in this paper. The finite element method (FEM) is used to simulate the propagations of nonlinear acoustic waves. The FEM equation of one-dimensional nonlinear acoustic wave is derived according to the theory of nonlinear acoustics. A three-dimensional matrix appearing in the nonlinear FEM equation represents the nonlinear part of the nonlinear acoustic equation and indicates the complex propagation characteristics of nonlinear acoustic waves. However, there is no corresponding matrix in the linear FEM equation. The matrix correlates with the nonlinear properties of propagating waves such as wave distortion, high order harmonic wave generation and transformation of energy from basic frequency to high order harmonic frequency, etc. Then, an FEM program is coded to compute the propagations of the one-dimensional nonlinear acoustic waves. The results show that the nonlinear acoustic waves are distorted obviously during propagation. After fast Fourier transform processing the original wave signal, the basic frequency signals and high order harmonic signals both appear in the frequency-region signals. To prove the correctness of the FEM results, nonlinear acoustic experiments in water are carried out under different conditions. In the first experiment, the distance between the transmitting and receiving transducers is kept unchanged, but the transmitting transducer is excited with different energies. So with propagation distance fixed, the influences of different exciting energies on the nonlinear properties of acoustic waves are obtained from this experiment. In the second experiment, with the exciting energy fixed, the influences of different propagation distances on the nonlinear properties of acoustic waves are obtained by changing the distance between the transmitting and receiving transducers. Then the numerical results and the experimental results are compared and analyzed carefully. The result shows that the waveforms and the spectra of simulated nonlinear waves are in good agreement with those of experimental signals. These results prove the correctness of the proposed numerical method. It is also noticed that the propagation properties of basic frequency wave and the second order harmonic waves are different. The amplitude of basic frequency wave decreases gradually, but the amplitude of second order harmonic wave first increases and then decreases after propagating some distance. The amplitude of the second harmonic wave changes with propagation distance and energy of the input source amplitude. The relationship between the amplitude of second harmonic wave and the propagation distance is numerically fitted. We find a fitting equation of the relation between high order harmonic acoustic wave and propagation distance, which also brings clear physical meaning for the high order harmonic waves. Finally, the properties of nonlinear acoustic wave propagation in solid are preliminarily discussed. This study provides theoretical and experimental evidence for the nonlinear acoustic wave traveling in liquid.

Contrast enhancement of medical ultrasound imaging with reversal phase-inversion pulse technology

A new ultrasound contrast imaging technique was proposed for eliminating the harmonic components from the emission signal transmitted by the broadband ultrasonic system. Reversal phase-inversion pulse was used for the first time to separate the contrast harmonics from the harmonics in the emission signal to improve the detection of contrast micro-bubbles. Based on the nonlinear acoustic theory of finite-amplitude effects and the associated distortion of the propagating wave, the Bessel-Fubini series model was applied to describe the nonlinear propagation effects of the reversal phase-inversion pulse, and the Church’s equation for zero-thickness encapsulation model was used to produce the scattering-pulse of the bubble. For harmonic imaging, the experiment was performed using a 64-element linear array, which was simulated by Field II. The results show that the harmonic components from the emission signal can be completely cancelled, and the harmonics generated by the nonlinear propagation of the wave through the tissue, can be reduced by 15–30 dB. Compared with the short pulse, the reversal phase-inversion pulse can improve the contrast and definition of the harmonic image significantly.

Generation of the Vorticity Mode by Sound in a Relaxing Maxwell Fluid

This study develops a new theory of nonlinear acoustics investigating interactions between acoustical and other non-acoustical modes, such as vorticity modes, in a fluid. The ideas proposed by the authors make possible to derive instantaneous equations describing interaction between different modes in a relaxing Maxwell fluid. The procedure of deriving of a new dynamic equation governing the vorticity mode which is generated by sound, is discussed in details. It uses only instantaneous quantities and does not include averaging over sound period. The resulting equation applies to both periodic and aperiodic sound of any waveform as the origin of the vorticity mode. The theory is illustrated by two representative examples of generation of the vorticity mode in a relaxing Maxwell fluid, caused by periodic sound beam and a sound beam with a stationary but aperiodic waveform.

Experimental and numerical study on microcrack detection using contact nonlinear acoustics

This paper introduces a non-classical nonlinear acoustic theory for microcrack detection in materials, comparing contact nonlinearity with material nonlinearity. The paper’s main work concentrates on the experimental and numerical verification of the effectivity of contact nonlinear acoustic detection by using the contact nonlinear parameter β′, which can be represented by the ratio of the second-harmonic amplitude to the square of the first-harmonic amplitude. Both experiments and numerical tests are performed. The results show that β′ is sensitive to the initiation of microcracks and varies with the development of the microcracks. The numerical test illustrates the decline of β′ when microcracks penetrate each other.

The parametric array in Berea sandstone: definitive experiments.

Previous measurements of the characteristics of the parametric array in sandstone by Johnson and Shankland [J. Geophys. Res. 94, 17729–17734 (1989)] were difficult to perform and only qualitative. Scanning laser vibrometers (Polytec) now make parametric array measurements in rock easier. However, hysteresis and memory effects play a strong role in the dynamic behavior of rocks and have the potential to mess up the creation of the parametric array in the medium. Thus, an experimental study was performed to find out just how well the “classical” theory of nonlinear acoustics works for a granular solid, a sandstone. An array of alternating PZT disk transducers was epoxied to a large block of Berea sandstone (1.2, 0.4, 0.4 m), every other one broadcasting with separate frequency generators and amps. Primary frequencies were around 100 kHz; difference frequencies of 10 to 20 kHz were observed. Two‐D beampattern scans were taken and the results compared with calculations based on classical theory. The agreement is surprisingly good with a beta of around 200 and a Q (inverse attenuation) of 50.

The Parametric Array in Berea Sandstone, Revisted

Previous measurements of the characteristics of the parametric array in sandstone by Johnson and Shankland [J. Geophys. Res., 94, 17729-17734 (1989)] were difficult to perform and, given the apparatus of the day, only qualitative. Since then, memory effects have been found to play a strong role in the dynamic behavior of rocks. This experimental study was performed to find out how well the "classical" theory of nonlinear acoustics, i.e., all compressional waves with no memory effects, works at describing the parameteric array in a granular solid. An array of PZT disk transducers (every other one broadcasting one of two frequencies) was epoxied to a large block of Berea sandstone (1.2x0.4x0.4 m). The two primary frequencies were near 100kHz and chosen to produce a difference frequency around 15kHz. Using a scanning laser vibrometer (Polytec), several scans of the beampatterns were made at the opposite end of the rock and the results compared with calculations based on those commonly used for parametric arrays in water. A narrow beam generated from the two primaries was measured and the beampattern compared with those of the primaries. The agreement is surprisingly good using a beta of around 200 and a Q of about 50.

Static and dynamic nonlinear behaviors of unconsolidated granular media

Unconsolidated granular media such as sands, soils, and glass beads, exhibit astonishing static and dynamic nonlinear behaviors that are quite different from those of elastic materials such as liquids and crystalline solids. They can generally classified as mesoscopic hysteretic nonlinear materials since hysteresis is a universal phenomenon for these materials. In order to understand the mechanism and to quantify the nonlinearity of unconsolidated granular materials, a series of studies have been conducted on re-molded unconsolidated soils. The systematic research consists of three parts: (1) a triaxial cell test with pre-defined stress-paths and sound speed measurement, focusing on the exploration of the role of hysteresis in both linear and nonlinear acoustics; (2) the phase shift method, using non-classical nonlinear acoustic theory to measure the hysteretic nonlinearity parameter; (3) the study of fluid effects on the hysteretic nonlinearity parameter. The study reveals many static and dynamic nonlinear behaviors of soils, including: nonlinear stress/strain relationship, hysteresis loop, end-point memory, much greater dynamic elastic modulus than static elastic modulus, strain-amplitude dependence of elastic modulus, linear resonant frequency shift and phase shift relationships against strain, extremely high values of the hysteretic nonlinearity parameter, harmonics generation, and moisture effects.

Weak sparks, N waves, and the calibration of microphones at ultrasonic frequencies

A fortuitous situation involving the availability of wide-band condenser microphones, curiosity about acoustic waves generated by sparks, and his own development of nonlinear acoustic theory provided two significant contributions to David Blackstock’s continuing research. Theoretical understanding of the nonlinear distortion experienced by intense acoustic impulses in air, such as those generated by relatively weak sparks, led to a technique for calibration of microphones that have essentially flat frequency response up to almost 1 MHz. These sparks and calibrated microphones, in turn, provided the means to carry out several graduate student thesis projects that were supervised by David. Diffraction at edges, apertures, and discs was studied in the time domain. Model studies related to sonic boom phenomena were carried out, as was the propagation of N waves in waveguides. The most recent application has been a model study of focusing by an ellipsoidal reflector, such as is typically used in lithotripsy. In each case, the various contributions to the received signals were readily identified in the time-domain presentation.

On the use of Helmholtz resonators as sound attenuators

In the present paper a nonlinear acoustic theory is proposed, to accurately describe the properties of a generalized type of Helmholtz resonators. The theory may be used as a layout tool to design sound attenuators for combustion facilities. The present investigation shows that, in addition to the nonlinear effects, the effects due to thermoacoustic boundary layers and the first frequency correction to the “Helmholtz approximation” should be taken into account, in order to predict the frequency-dependent attenuation properties of Helmholtz attenuators with the accuracy required by typical technical applications. A series of experiments is presented which is used to validate the theoretical predictions.