Finite Amplitude Ultrasonic Waves Research Articles

Nonlinear reflection coefficient of finite amplitude waves

Ultrasonic waves incident at the interface between two linear media will be reflected based on the linear reflection coefficient, which depends on the linear acoustic impedances of the media. However, propagation of ultrasonic waves in quadratically nonlinear media can lead to nonlinear reflection coefficients which are dependent on the frequency, source amplitude, and nonlinearity of the two mediums. The objective of this work is to explore the existence of a nonlinear reflection coefficient between two mediums using analytical, numerical, and experimental methods. For simplicity and consistency, a water-solid interface was chosen and finite amplitude ultrasonic waves between 2.25 MHz and 25 MHz were used to study the interfacial response. The analytical results show the existence of a nonlinear RC, but whose effect is very minimal. Experimental and numerical results show that the transfer of energy in harmonic generation affects the RC more than other factors. The influence of several factors like interaction zone, shock generation, etc. will be discussed.

Design of Ultrasonic Transducer for Secondary Wave Generations with High Directivity

In this paper, we described a method of designing ultrasonic transducer which simultaneously radiates two finite-amplitude ultrasonic waves to produce the secondary waves with high directivity. For nonlinear effects, it is necessary that the frequencies of two primary waves are coincident with natural frequencies of the ultrasonic transducer. The main problem here is to predict the resonance frequencies of the first mode as well as higher modes. While the first resonance frequency of the transducer can be estimated easily, it is not trivial to do higher ones. When the length of transducer is much greater than its diameter, this problem is reduced to one-dimensional and higher mode frequencies are nothing but multiples of the first mode frequency. However, such a case is seldom encountered. Using the transfer matrix method, we obtained the resonance frequencies of the transducer analytically and compared these with numerical results from the simulation. The theoretical and simulation results are in good agreement with the difference of 3--6 kHz.

Diffraction of Light by Finite Amplitude Ultrasonic Waves

In the late 1950s the Ultrasonic Group at Michigan State University introduced light di raction to study distortion of ultrasonic waves in liquids under the direction of Prof. Egon Hiedemann. In this paper some results of these studies will be presented with detailed description of the author's measurements of B/A. The rate at which the harmonics are developed (the wave distortion is an indication of the harmonics present) during the propagation of the initially sinusoidal ultrasonic wave depends on the nonlinearity of the medium. The light which is di racted by the distorted wave results in asymmetric pattern contrary to the Raman Nath theoretical prediction. From the light intensity measurements due to the generated second harmonics ltered out by a metal plate the nonlinearity parameter B/A was determined. Developments in studying nite amplitude standing ultrasonic waves in a liquid lled cavity will be also discussed. Using light di raction measurements it was observed that above a threshold amplitude, fractional harmonics of the driver transducer are also generated in addition to the generated harmonics. It was recently observed that above a second threshold value of the driver amplitude the system undergoes chaotic behavior. Further increase of the driver amplitude returns the system from chaos to stable oscillation.

Chaos and beyond in a water filled ultrasonic resonance system

Finite amplitude ultrasonic wave resonances in a one dimensional liquid-filled cavity are reported. The resonances are observed to include not only the expected harmonic and subharmonic signals but chaotic signals as well. The nonlinear features of this system were recently investigated and are the focus of this presentation. An ultrasonic interferometer having optical precision was constructed. The transducers having the frequency range 1-10 MHz, driven by a high power amplifier. Both an optical diffraction system and a receiving transducer were used to assess the generated resonance response in the cavity. Five regions of excitation are identified: 1. Linear region: at low intensity of the ultrasonic wave the diffraction pattern of a light beam is symmetric. 2. Nonlinear region: with increased sound amplitude the diffraction pattern becomes asymmetrical. 3. Subharmonic region: further increase of the amplitude above a threshold value leads to the generation of subharmonics. 4. Chaos: increasing the drive amplitude to a second threshold level the diffraction pattern is smeared out indicating a time-chaotic region. 5. Beyond chaos: further increase of the amplitude results again a stable diffraction pattern. A first-principle-based explanation is presented. This work is supported by the Aircraft Aging Program, at NASA Langley Research Center.

Introduction to nonlinear acoustics

A brief review of the basic principles of fluid mechanics needed for development of linear and nonlinear ultrasonic concepts will be given. The fundamental equations of nonlinear ultrasonics will be derived and their physical properties explained. It will be shown how an originally monochromatic finite-amplitude ultrasonic wave, due to nonlinear effects, will distort during its propagation in time and space to form higher harmonics to its fundamental frequency. The concepts of shock formation will be presented. The material nonlinearity, described by the nonlinearity parameter B/A of the material, and the convective nonlinearity, described by the ultrasonic Mach Number, will be explained. Two procedures for determination of B/A will briefly be described and some B/A-values characterizing biological materials will be presented. Shock formation, described by use of the Goldberg Number,and Ultrasonic Saturation will be discussed.. An introduction to focused ultrasonic fields will be given and it will be shown how the ultrasonic intensity will vary axially and laterally in and near the focal region and how the field parameters of interest to biomedical applications may be described by use of the KZK-Model. Finally, an introduction will be given to the parametric acoustic array formed by mixing and interaction of two monochromatic, finite-amplitude ultrasonic waves in a liquid and the potentials of this mixing process in biomedical ultrasound will briefly be mentioned.

On preprocessing techniques for bandlimited parametric loudspeakers

The self-demodulation property of finite-amplitude ultrasonic waves can be applied with parametric loudspeaker to produce audible sound. A special characteristic of the reproduced sound waves using parametric loudspeaker is its high directivity. However, the demodulated signal from parametric loudspeaker suffers from high distortion. To reduce the distortion in the demodulated signal, preprocessing of the modulating signal is usually employed. To determine the effectiveness of the preprocessing technique, an important practical constraint on the bandwidth of the ultrasonic transducer of the parametric loudspeaker should be accounted. In this paper, we shall discuss a class of preprocessing techniques that is based on quadrature amplitude modulation. As compared to the conventional preprocessing methods used with bandlimited ultrasonic transducer, the demodulated signal from our proposed preprocessing techniques exhibits lower distortion.

Plane wave source with minimal harmonic distortion for investigating nonlinear acoustic properties

The objective of this investigation is to introduce and validate a practical ultrasound source to be used in the investigation of the nonlinear material properties of liquids and soft tissues studied in vitro. Methods based on the progressive distortion of finite amplitude ultrasonic waves in the low megahertz frequency range are most easily implemented under the assumption of plane wave propagation. However, achieving an approximately planar ultrasonic field over substantial propagation distances can be challenging. Furthermore, undesired harmonic distortion of the ultrasonic field prior to insonification of the specified region of interest represents another serious limitation. This paper introduces an approach based on the use of the ultrasonic field emanating from a stainless-steel delay line. Both simulation and direct experimental measurement demonstrate that such a field exhibits relatively planar wave fronts to a good approximation (such that a 3-mm-diam receiver would be exposed to no more than 3 dB of loss across its face) and is free from the significant harmonic distortion that would occur in a conventional water path.

Using psycho-acoustics frequency masking to reduce distortion in a parametric array speaker

Using a parametric array of ultrasonic transducers, directional sound can be generated from inaudible ultrasonic as a result of nonlinear interaction of ultrasonic wave in air. A finite amplitude ultrasonic wave that can be amplitude modulated by any audio signal is radiated from the array [Yoneyama et al., J. Acoust. Soc. Am. 73, 1532–1536 (1983)]. As a result, an audio signal is produced because of the self-demodulation effect of the modulated sound wave. This, however, generates a large distortion that is impractical. Another method [T. Kite et al., J. Acoust. Soc. Am. 103, 2871 (1998)] uses a preprocessing technique, which utilizes double integration of the square-rooted input. The resulted distortion is greatly reduced but at the expense of very wide bandwidth requirement due to the square-root operation. Various techniques to reduce the distortion by preprocessing have been discussed in previous literature. In this paper, we propose a psycho-acoustic masking scheme, coupled with a suitable modulation that will provide further reduction in distortion. The psycho-acoustic masking focus on the distortion occurred within the critical bands. Simulation and subjective testing have been done to verify the system.

Acoustic nonlinearity parameter measurement using optcal beam deflection by ultrasonic waves.

A technique of optical beam deflection by ultrasonic waves was developed in order to measure acoustic nonlinearity parameters B/A in liquids. Finite-amplitude ultrasonic waves of 6.65 MHz propagating in a liquid sample deflect a laser beam of normal incidence. The deflection angle was detected with a photodiode having double elements, yielding a distorted waveform. Spectral analysis of the waveform yielded the ratio of the amplitude of the second-harmonic wave to that of the fundamental wave. The absolute value of the fundamental sound-pressure amplitude was obtained independently from the Raman-Nath parameter measured using the same experimental system under the condition that a laser beam waist is larger than sound wavelength. From these measurements, the nonlinearity parameters were evaluated to be 5.3 for water and 10.3 for ethyl alcohol. These results are in good agreement with established values.

Acoustic nonlinearity parameter tomography for biological specimens via measurements of the second harmonics

The acoustic nonlinearity parameter B/A is a new parameter in ultrasound tissue characterization. In this paper, on the basis of a theoretical analysis of finite amplitude ultrasonic wave propagation in fluidlike media, an experimental system for nonlinearity parameter tomography is developed. Using this system, the amplitude of the second harmonic wave cumulated in the propagation path of the primary wave is measured and the finite amplitude insert-substitution method is used in the conventional computer tomography (CT) technique. The filtered convolution algorithm is implemented using the amplitude of the second harmonic as the projection data, then the B/A tomography can be reconstructed. The B/A images for several biological samples are obtained. Besides, the image of porcine pathological liver tissue is studied and is compared with that of porcine normal liver tissue. These results show the potential applications of the nonlinearity parameter B/A in medical diagnosis.

Nonlinear distortion of finite-amplitude ultrasonic waves investigated by optical heterodyne interferometry

The study of asymmetrical patterns of the light diffracted by ultrasonic waves has been utilized in the last decades to investigate ultrasonic wave distortions caused by the nonlinearity of the medium. In contrast to the investigation of diffraction patterns in space, a novel optical method is proposed in this work to analyze, in the frequency domain, the phase modulation (related to the Raman–Nath parameter) of the probe beam. Using a heterodyne interferometric method, the effects of ultrasonic waves are described as sideband generations (Doppler shift) about a central frequency (optical heterodyning) in the frequency spectrum of the photocurrent. Asymmetry resulting from nonlinear effects was observed upon the spectrum envelope. This effect increases both with ultrasonic intensity and with propagation distance. Experimental results obtained in water confirmed qualitatively the theoretical analysis which includes the second and third harmonics in the formulation.

Physics and Engineering Principles of Nonlinear Acoustics

The nonlinear differential equation describing propagation of finite amplitude ultrasonic waves in gases, liquids, and solids is shown to have the same mathematical form. The nonlinearity parameter in each case is defined in terms of fundamentally significant physical quantities. Waveform distortion is described, and the discontinuity distances are compared. Recent results are given for gases, liquids, and solids and their significance is evaluated in terms of modern condensed matter physics and engineering.

Propagation of finite-amplitude ultrasonic waves in air—II. Plane waves in a tube

This paper is concerned with an investigation of the nonlinear behavior of sinusoidal plane progressive acoustic waves of finite amplitude in air at a frequency of about 21 kHz. The various stages in the distortion of the wave have been experimentally studied and the results compared with theoretical models. The experiments were done in a tube 5.5 m long and 8-mm internal diameter with an absorbing termination. The acoustic source used was a prestressed piezoelectric sandwich transducer with a titanium stepped mechanical amplifier, the small end of which was the piston that produced the high-amplitude motion inside the tube. This source was capable of generating a nearly pure sinusoidal wave at sound pressure levels up to about 155 dB (re: 0.0002 μbar). Interestingly, as compared to previous reports on this field, the use of a high frequency has allowed the study of all the propagation regions (including the old age region) of the wave using a short length tube.

Propagation of finite-amplitude ultrasonic waves in air—I. Spherically diverging waves in the free field

An investigation of the free-field propagation in air of sinusoidal spherical waves of finite amplitude is described. It includes experiments made at 20.4 kHz and a theoretical interpretation using weak shock theory. The acoustic source used was a new high-power and highly directional ultrasonic transducer which is able to generate a nearly pure sinusoidal wave at sound pressure levels over 160 dB. The experiments were done in an anechoic chamber, using a cw mode. They consisted essentially of measuring the amplitude of the fundamental component of the wave and its first three harmonics for different source levels and at various distances along the axis of the source. In addition, oscillograms and directivity patterns were recorded. In general, the experimental data confirm the predictions of the theoretical model.

Diffraction of a finite amplitude ultrasonic wave in solids

In order to use the harmonic generation technique to measure the nonlinearity parameters of solids, one must determine the amplitudes of both the fundamental and the second harmonic of an initially sinusoidal ultrasonic wave. In our experiment performed at 30 MHz, a typical value of the fundamental amplitude is 1 A with a second harmonic amplitude of 0.001 A. Since interesting samples often are smaller than those in which the infinite plane‐wave model is sufficiently accurate, we consider the effect of diffraction on the measurements. Ten transducers having diameters between 1.9 mm and 12.7 mm (∼10λ and ∼90λ) were used with the same [100] copper sample. The resulting data show that diffraction does influence the calculated amplitude with the smaller transducers. Interpretation of the data in terms of a nonlinear diffraction model is attempted. Theoretical considerations are discussed. [Research supported by the Office of Naval Research.]

Ultrasonic measurement of third‐order elastic constants of fused silica

A finite amplitude ultrasonic wave propagating through a solid generates harmonics whose magnitudes are determined by some of the third‐order elastic constants. Other third‐order elastic constants determine the magnitude of the variation of ultrasonic wave velocity as the sample is subjected to hydrostatic pressure. We have combined results of harmonic generation experiments and pressure variation of ultrasonic wave velocity to evaluate third‐order elastic constants of fused silica. We used four different samples having different hydroxyl ion content and different homogeneities and obtained the following results.

Higher harmonics of finite amplitude ultrasonic waves in solids

Absolute measurements of the amplitude of the first four harmonicof 30-MHz finite amplitude ultrasonic waves are reported for fused silica samples of various lengths and for aluminum single crystals with different amounts of cold work. The harmonically distorted motion of the end of the sample is detected by a capacity microphone and the individual harmonics are then selected and amplified by a heterodyne receiver with flat frequency response from 30 to 250 MHz. The fused silica data are found to be in excellent agreement with a model of Fubini, originally developed for gases, which for solids depends on the second- and third-order elastic constants but none of the higher constants. A discussion is presented of the reasons for the insensitivity of the measurements to the values of the fourth- and higher-order elastic constants. Agreement with the model of Fubinin is not observed in single crystal aluminum. The quantitative differences are not fully consistent with existing models for dislocation contributions to harmonic generation. Subject Classification: [43]25.15, [43]25.20; [43]35.26; [43]25.25.

Nonlinearity Considerations in Bragg Imaging of Ultrasonic Waves

Two fundamentally different techniques exist for the study of the diffraction of light by finite amplitude ultrasonic waves: (1) Diffraction of light at normal incidence can be used to determine the ultrasonic waveform. (2) Diffraction of light incident at the Bragg angle gives detailed information of a completely different nature. Each diffraction order contains an image of the ultrasonic wavefront. Therefore, objects placed within the ultrasonic field are imaged in the diffraction orders. When a finite amplitude ultrasonic wave is used, an ultrasonic wave transmission filter allows one to distinguish the images produced by the second harmonic from those produced by the fundamental. The image produced by the second harmonic is twice as large as that produced by the fundamental and appears at the position of the fundamental's second order. In addition, at large amplitudes, multiple scattering of light by the fundamental ultrasonic wave can produce higher Bragg diffraction orders. These higher orders contain multiple images. Two images appear in the second order, three in the third, etc. [Research supported by the Office of Naval Research.]

Bragg Imaging with Finite Amplitude Ultrasonic Waves

Light passing through ultrasonic waves of finite amplitude at the angle at which Bragg diffraction is dominant produces diffraction orders which contain images of the ultrasonic wavefront, with multiple images appearing in the higher orders. These multiple images are attributed to multiple scattering of light by the fundamental ultrasonic wave. The harmonics of the ultrasonic wave arising from nonlinearity of the propagating medium are isolated by use of a transmission filter plate. The images produced by them are larger than those produced by the fundamental.

Dispersion of Finite Amplitude Ultrasonic Waves in InSb

The dispersion of finite-amplitude ultrasonic standing waves in single crystal InSb is reported. The observed dispersion was found to be proportional to the square of the voltage applied to the transducer, in agreement with the predictions of a simple model which is proposed to account for the effect. An enhanced sensitivity to the detection of harmonics of the ultrasonic wave results from the use of a standing wave technique.