Model equation for strongly focused finite-amplitude sound beams

A model equation that describes the propagation of sound beams in a fluid is developed using the oblate spheroidal coordinate system. This spheroidal beam equation (SBE) is a parabolic equation and has a specific application to a theoretical prediction on focused, high-frequency beams from a circular aperture. The aperture angle does not have to be small. The theoretical background is basically along the same analytical lines as the composite method (CM) reported previously [B. Ystad and J. Berntsen, Acustica 82, 698-706 (1996)]. Numerical examples are displayed for the amplitudes of sound pressure along and across the beam axis when sinusoidal waves are radiated from the source with uniform amplitude distribution. The primitive approach to linear field analysis is readily extended to the case where harmonic generation in finite-amplitude sound beams becomes significant due to the inherent nonlinearity of the medium. The theory provides the propagation and beam pattern profiles that differ from the CM solution for each harmonic component.     

Waveguide propagation of sound beams in a nonlinear medium

A possibility of a waveguide propagation of sound beams in the case of compensation of the diffraction divergence by the nonlinear refraction is demonstrated theoretically. A stationary (with respect to the longitudinal coordinate) solution is obtained to the nonlinear equation for a sound beam (the Khokhlov—Zabolotskaya equation); the solution describes the characteristic bow-shaped profile of the beam and the self-localized (with respect to the transverse coordinate) distribution of the peak values of this profile. The physical and mathematical features of this phenomenon belonging to nonlinear acoustics are discussed and compared with those of the well-known analog from nonlinear optics. A scheme of an experimental realization of the waveguide propagation of acoustic beams is proposed.     

Mathematically trivial control of sound using a parametric beam focusing source

By exploiting a case regarded as trivial, this paper presents global active noise control using a parametric beam focusing source (PBFS). As with a dipole model, one is used for a primary sound source and the other for a control sound source, the control effect for minimizing a total acoustic power depends on the distance between the two. When the distance becomes zero, the total acoustic power becomes null, hence nothing less than a trivial case. Because of the constraints in practice, there exist difficulties in placing a control source close enough to a primary source. However, by projecting a sound beam of a parametric array loudspeaker onto the target sound source (primary source), a virtual sound source may be created on the target sound source, thereby enabling the collocation of the sources. In order to further ensure feasibility of the trivial case, a PBFS is then introduced in an effort to meet the size of the two sources. Reflected sound wave of the PBFS, which is tantamount to the virtual sound source output, aims to suppress the primary sound. Finally, a numerical analysis as well as an experiment is conducted, verifying the validity of the proposed methodology.     

Harmonic propagation of finite amplitude sound beams: experimental determination of the nonlinearity parameter B/A

We propose a method to determine the nonlinearity parameter B/A of a liquid from the spatial evolution of harmonic components. We describe an analytical model, in the parabolic and quasi-linear approximations, that predicts the continuous finite amplitude sound beam propagation radiated by a plane piston source. This model takes into consideration attenuation, diffraction and nonlinear effects. The fundamental and second harmonic ultrasonic fields are expressed as the superposition of Gaussian beams. Axial propagation curves are then compared with those obtained by direct numerical solution of the transformed beam equation using the finite difference method, and with experimental results. Accurate measurements of pressure levels for the nonlinearly generated harmonics in water are performed along and across the propagation axis for different pressure values delivered at the piston surface. Experimental results, for water and ethanol, are in agreement with those of our model, which allows us to obtain the expected value of the nonlinearity parameter B/A.     

Time-domain modeling of finite-amplitude sound beams in three-dimensional Cartesian coordinate system

A three-dimensional time-domain algorithm,which is based on the augmented KZK (Khokhlov-Zabolotskaya-Kuznetsov) equation,is proposed to simulate the nonlinear field of the parametric array.First,KZK equation is transformed into TBE(Transformed beam equation). Then,the effects of diffraction(in parabolic approximation),thermoviscous absorption,relaxation, and nonlinearity are solved with finite difference methods.The numerical results of this code agree well with the theoretical and experimental results presented in previous studies, which demonstrates the validity of the three-dimensional algorithm.Using this code to calculate the nonlinear field of the parametric array in air,it is found that the small time interval is important to the accuracy of the simulation results of the difference frequency wave in the case of high sound pressure level,and the errors caused by taking relaxation absorption for thermoviscous absorption are influenced by the characteristic frequency.     

Time-reversed sound beams of finite amplitude.

Numerical simulations based on the nonlinear parabolic wave equation are used to investigate time reversal of sound beams radiated by unfocused and focused sources. Emphasis is placed on nonlinear propagation distortion in the time-reversed beam, and specifically its effect on field reconstruction. Distortion of this kind, due to amplification during time reversal, has been observed in recent experiments [A. P. Brysev et al., Acoust. Phys. 44, 641-650 (1998)]. Effects of diffraction introduced by time-reversal mirrors with finite apertures are also considered. It is shown that even in the presence of shock formation, the ability of time reversal to retarget most of the energy on the source or focal region of the incident beam is quite robust.     

Formation of a weakly diverging caustic beam in an underwater sound channel open to the bottom

The paper studies the patterns manifesting themselves in the formation and propagation of caustic and weakly diverging beams using the geometric acoustic approximation and mode theory applied to the simplest model of an underwater sound channel open to the bottom in the form of the bilinear dependence of the square of the refractive index on depth. It is established that when a vertical array emitting a tonal sound signal is located at a certain critical depth, the multimode caustic beam which forms near the reference ray horizontally departing from its center and which predominates in intensity is simultaneously the most weakly diverging beam. It is shown that the reference ray of such a weakly diverging caustic beam corresponds to the smoothest minimum for the dependence of the length of the ray cycle on their angle of departure from the center of the array and to the depth level of revolution at the critical depth characteristic of the given oceanic waveguide.     

Stochastic differential equation analysis for sound scattering by random internal waves in the ocean

The scattering of a weakly divergent narrow sound beam by random inhomogeneities of a fluctuating ocean is considered in the coupled-mode approximation. The random index of sound refraction is described using the Garrett-Munk internal wave spectrum. The problem is solved using the stochastic differential equations for the first-and second-order statistical moments of the acoustic field. The equations are formulated according to the cumulant expansion method. The existence of weakly divergent narrow sound beams in long-range sound propagation was one of the last discoveries of L.M. Brekhovskikh, to which he attached much importance. The concentration of sound into narrow beams away from the axis of the underwater sound channel was first observed experimentally and then explained by Brekhovskikh and his former students Goncharov, Kurtepov, and Petukhov. In the present paper, the scattered field intensity of a sound beam is calculated for different frequencies and source depths. Analytical expressions are obtained for the coefficients of the differential equation. The intermode energy transfer that accompanies the long-range propagation of a weakly divergent sound beam is analyzed. A comparison with the conventionally used Monte Carlo simulation in the parabolic equation approximation is performed.     

Acoustic nonlinearity parameter tomography for biological tissues via parametric array from a circular piston source--theoretical analysis and computer simulations

The acoustic nonlinearity parameter B/A describes the nonlinear features of a medium and may become a novel parameter for ultrasonic tissue characterization. This paper presents a theoretical analysis for acoustic nonlinear parameter tomography via a parametric array. As two primary waves of different frequencies are radiated simultaneously from a circular piston source, a secondary wave at the difference frequency is generated due to the nonlinear interaction of the primary waves. The axial and radial distributions of sound pressure amplitude for the generated difference frequency wave in the near field are calculated by a superposition of Gaussian beams. The calculated results indicated that the difference frequency component of the parametric array grows linearly with distance from the piston source. It therefore provides a better source to do the acoustic nonlinearity parameter tomography because the fundamental and second harmonic signals both have a near field that goes through many oscillations due to diffraction. By using a finite-amplitude insert substitution method and a filtered convolution algorithm, a computer simulation for B/A tomography from the calculated sound pressure of the difference frequency wave is studied. For biological tissues, the sound attenuation is considered and compensated in the image reconstruction. Nonlinear parameter computed tomography (CT) images for several biological sample models are obtained with quite good quality in this study.     

A self-silenced sound beam

Parametric loudspeakers are transmitting two high power ultrasound frequencies. During propagation through the air, nonlinear interaction creates a narrow sound beam at the difference frequency, similar to a light beam from a torch. In this work is added the physical phenomenon of propagation cancellation, leaving a limited region within which the sound can be heard—a 1 meter long cylinder with diameter 8 cm. It is equivalent to a torch which would only illuminate objects within 1 meter. The concept is demonstrated both in simulation and in experiment.     

高精度球面阵聚焦声源定位方法研究

提出了一种高精度高空间分辨率球面阵聚焦声源定位方法——虚拟源法。该方法通过球面阵波束扫描获得实际声源的空间聚焦谱,并假定各扫描点为虚拟声源,将实际声源聚焦谱看作是全体虚拟源共同作用的结果,由此得到各虚拟源对声场的贡献量,从而可实现声源精确定位。仿真研究分析了频率,阵列孔径,声场模态阶数,信噪比等参数对声源定位性能的影响,并与常规算法进行对比。结果显示,该方法不受频率和阵列孔径的限制,避免了空间“混淆”,能够进行高精度高分辨率声源定位,并具有良好的背景噪声抑制能力。      

线阵波束形成声强缩放方法估算声源的辐射声功率

为了解决波束形成声源识别过程中声源辐射声功率定量计算的问题,给出了阵型简洁、便于组合的线阵声强缩放模型。通过推导线阵的声强缩放系数,建立起线阵波束输出结果与声源辐射声功率之间的换算关系。无论是线阵还是平面阵的声强缩放方法,对于偏离阵列中心位置较远处的声源进行辐射声功率估算时都存在较为明显的误差。通过理论推导和仿真模拟计算,研究了同一单极子点声源在不同位置处的声功率估算偏差随频率、幅度的变化规律,发现该估算偏差只与声源偏离位置有关,而与声源自身的强度信息无关的结论,据此给出了相应的声功率估算修正方法。半消声室实验结果和声压法测量结果对比表明:修正后的线阵声强缩放方法用于中高频声源的辐射声功率计算时,单频声源的估算误差不超过1.0 dB,宽带声源的估算误差不超过1.8 dB。      

Modeling the propagation of nonlinear three-dimensional acoustic beams in inhomogeneous media

A three-dimensional model of the forward propagation of nonlinear sound beams in inhomogeneous media, a generalized Khokhlov-Zabolotskaya-Kuznetsov equation, is described. The Texas time-domain code (which accounts for paraxial diffraction, nonlinearity, thermoviscous absorption, and absorption and dispersion associated with multiple relaxation processes) was extended to solve for the propagation of nonlinear beams for the case where all medium properties vary in space. The code was validated with measurements of the nonlinear acoustic field generated by a phased array transducer operating at 2.5 MHz in water. A nonuniform layer of gel was employed to create an inhomogeneous medium. There was good agreement between the code and measurements in capturing the shift in the pressure distribution of both the fundamental and second harmonic due to the gel layer. The results indicate that the numerical tool described here is appropriate for propagation of nonlinear sound beams through weakly inhomogeneous media.     

Fringed sound fields and their interaction with liquid-solid interfaces

On the one hand, it is well known that Gaussian beams possess the ability to stimulate Rayleigh waves, resulting in the Schoch effect, a lateral beam displacement. This effect, often characterized by a reflected sound pattern consisting of two anti-phase beams, is due to the re-radiation of sound because of the stimulation of leaky Rayleigh waves. On the other hand, fringed sound beams are characterized by the fact that they consist of a number of neighboring anti-phase narrow beams. They are a first approximation of a sound field originating from a phased array of harmonic vibrating crystals in which each crystal vibrates in anti-phase compared to its neighbor. The individual lobes within the fringed sound pattern diverge much less than standard Gaussian beams of the same size. The current study investigates the interaction of fringed beams with a liquid-solid interface. It is found that under certain conditions, a fringed beam, incident at the Rayleigh angle, produces a reflected sound pattern that contains a wide lobe that is not fringed. It is also shown that under other conditions, contrary to the famous forward displacement of the reflected sound for incident Gaussian beams, a strong backward displacement occurs for fringed beams.     

Non-line-of-sight sound source localization using matched-field processing

Acoustic diffraction allows sound to travel around opaque objects and therefore may allow beyond-line-of-sight sensing of remote sound sources. This paper reports simulated and experimental results for localizing sound sources based on fully shadowed microphone array measurements. The generic geometry includes a point source, a solid 90° wedge, and a receiving array that lies entirely in the shadow defined by the source location and the wedge. Source localization performance is assessed via matched-field (MF) ambiguity surfaces as a function of receiving array configuration, and received signal-to-noise ratio for the Bartlett and minimum variance distortionless (MVD) MF processors. Here, the sound propagation model is developed from a Green's function integral treatment. A simple 16 element line array of microphones is tested in three mutually orthogonal orientations. The experiments were conducted using an approximate 50-to-1-scaled tabletop model of a blind city-street intersection and produced ambiguity surfaces from source frequencies between 17.5 and 19 kHz that were incoherently summed. The experimental results suggest that a sound source may be localized by the MVD processor when using fully shadowed arrays that have significant aperture parallel to the edge of the wedge. However, this performance is reduced significantly for signal-to-noise ratios below 40 dB.     

高斯叠代法研究相控阵非线性声场

相控阵在聚焦超声治疗应用中不可避免地受到非线性影响,提出了采用高斯叠代法计算相控阵的非线性声场。在该方法中,利用预设焦点参数并应用伪逆矩阵算法得到阵元的激励参数;然后将阵元近似拟合成一组高斯声束的叠加,通过高斯声束叠代计算非线性声场。数值计算中以64阵元一维相控阵为研究对象;线性条件下,高斯叠代法结果与菲涅耳积分结果的误差低于0.5%,验证了该方法的可行性;单焦点及双焦点模式的相控阵非线性声场结果表明非线性效应能提高焦点聚焦性能,并且非线性效应与激励声压及激励频率成正比。      

大气湍流和热晕效应对列阵合成激光传输特性和光束质量影响的理论研究进展

介绍了大气湍流效应和热晕效应对列阵合成激光传输特性和光束质量影响的理论研究进展。主要介绍了合成激光在大气中传输的解析和数值模拟研究方法,大气湍流效应对列阵合成光束的光强分布、远场发散角、方向性、曲率半径和湍流距离的影响,以及大气热晕效应对列阵合成光束的光强分布、传输效率、重心偏移、热晕时间尺度和焦移的影响。研究结果表明,大气湍流效应和热晕效应对合成激光光束质量的影响与光束合成方式、合成光束参数以及大气参数密切相关。     

Formation of a narrow sound beam in the underwater sound channel by a vertical array

The possibility of using a vertical array for the generation of a narrow wave beam that propagates in the underwater sound channel along a given reference ray is discussed. The variational problem of choosing the initial field at the array aperture to provide the minimal possible average beam width along the propagation path of a fixed length is solved.     

An alternative method to measure the on-axis difference-frequency sound in a parametric loudspeaker without using an acoustic filter

The self-demodulation characteristic of finite-amplitude ultrasonic sound waves can be applied with parametric loudspeaker to reproduce audible sound with highly directivity. But measuring the difference-frequency sound is still a problem due to the spurious sound generated as a result of nonlinearity caused by the product of the primary waves at the receiving system. In this paper, based on the phase-cancellation method and the Gaussian beam expansion technique, an alternative method is proposed to measure the on-axis difference-frequency sound accurately without using any traditional acoustic filter, where the spurious sound can be greatly reduced or even eliminated. The proposed method is more suitable for the case where the piston source in the parametric loudspeaker comprises multiple small piezoelectric transducers (PZTs) and each transducer element in the array may have different frequency response. The validity of the proposed method is confirmed both by simulations and experiments.     

识别运动声源随机阵列生成法研究

针对随机阵列结构设计方面欠缺高效的生成方法这一问题,在保证阵列对声源识别精度的前提下,提出一种在极坐标下用于识别运动声源的随机阵列生成方法。根据声阵列识别运动声源的原理,推导了非等间距基本同轴圆环的半径,再通过降列分区、极坐标下条件筛选和模拟评价三个步骤,生成适用于识别运动声源的目标随机阵列,最后进行数值模拟和运动汽车噪声源识别实验进行验证。研究结果表明,用该方法能够高效地生成目标随机阵列,与常用规则阵列相比具有更良好的声源识别特性,并且具有准确的运动声源识别性能。