Nonlinear Acoustic Problems Research Articles

Non-linear Acoustic Problems in Loudspeaker Design

The article deals with the issues of non-linear acoustics. A mathematical model of non-linear distortions of the acoustic path has been compiled in relation to the operation of a loudspeaker in a piston mode. The results of experimental measurements are presented and analyzed. Recommendations are given to reduce distortion in the construction of high-quality acoustic systems.

1D Numerical Study of Nonlinear Propagation of Finite Amplitude Waves in Traveling Wave Tubes with Varying Cross Section

The nonlinear acoustic problem of a finite amplitude plane wave propagating along the axial direction in a traveling wave tube is studied. Based on the one-dimensional Westervelt equation, a one-dimensional nonlinear wave equation is derived in which the cross section of the traveling wave tube is considered. The two-order finite difference scheme is used to solve the nonlinear wave equation. The nonlinear propagation characteristics of a finite amplitude wave in the traveling wave tube is analyzed. In the expanding transition section, the acoustic pressure amplitude of the acoustic wave decreases with the increase of the cross-sectional area of the pipeline. The nonlinear characteristics of the acoustic wave show waveform distortion and harmonic growth. The waveform distortion becomes more serious in the rear of traveling wave tube than in the front of the tube. Considering the acoustic reflection condition at the mouth, the influence of differently shaped diffusion sections on the acoustic pressure distribution in the test section is investigated. The larger the change rate of the diffusion section in an area, the less amplitude of the sound pressure, and the nonlinear effect of the sound wave propagation is weakened. These nonlinear wave propagation characteristics in a travelling wave tube provide important guidance for both designing a uniform sound pressure distribution in the test section and determining the optimal measuring points for different sizes of structures in spacecraft.

New numerical iterative method for nonlinear wave equations

<sec>Nonlinear acoustics is an important branch of acoustics and has important applications in some areas, such as high-intensity focused ultrasound, ultrasonic suspension, acoustic cavitation, acoustic harmonic imaging, and parametric emission array. The solving of nonlinear equations in these fields is very important. </sec><sec>Regarding the solution of the wave equation of a nonlinear acoustic system, the methods used at this stage generally include complete numerical calculation method, strict analytical method, and perturbation method. 1) For the complete numerical calculation method, it covers the finite element method and the finite difference method. The physical meaning of the solution obtained by this kind of method is not clear, and it is difficult to reveal the physical nature of nonlinear event. And in many cases it will lead to the numerical divergence problems, and it is not suitable for all nonlinear problems. 2) For the strict analytical method, it can only deal with nonlinear acoustic problems of very few systems, such as the propagation of nonlinear acoustic waves in an ideal fluid. 3) For perturbation method, its advantage is that the method is simple and the physical meaning of the solution is clear, but it is only suitable for dealing with nonlinear effects at low sound intensity. And it takes into consideration only the effect of low-order harmonics on higher-order harmonics, with ignoring its reaction, so it does not satisfy the law of conservation of energy.</sec><sec>In this paper, we propose a new, semi-analytical numerical iterative method of solving nonlinear wave equations. It is a form of expanding the sound field into a Fourier series in the frequency domain, realizing the separation of time variables from space coordinates. Then, according to the specific requirements for the calculation accuracy, the high frequency harmonics are cut off to solve the equation. Compared with the results from the complete numerical methods (such as finite element method and finite difference method), the solution from this iterative method has a very clear physical meaning. That is, its solution is a combination of harmonics of all orders. Compared with the perturbation method, it can consider the interaction of various harmonics and can satisfy the law of conservation of energy (provided that the system has no dissipation). It is used to study the propagation properties of nonlinear acoustic waves in liquids. The results show that the iterative method is also applicable in the range of sound intensity where the perturbation method is applicable. In a wide range of sound intensity where the perturbation method is unapplicable, the iterative method is still applicable and satisfies the law of conservation of energy (provided that the system has no dissipation). It is unapplicable only if the sound intensity is extremely loud and strong. And when more high-order harmonics are involved, the calculation time by using the numerical method proposed in this paper does not increase sharply.</sec>

Simulating energy cascade of shock wave formation process in a resonator by gas kinetic scheme

The temporal-spatial evolution of gas oscillation was simulated by gas kinetic scheme (GKS) in a cylindrical resonator, driven by a piston at one end and rigidly closed at the other end. Periodic shock waves propagating back and forth were observed in the resonator under finite amplitude of gas oscillation. The studied results demonstrated that the acoustic pressure is a saw-tooth waveform and the oscillatory velocity is a square waveform at the central position of the resonant tube. Moreover, it was found by harmonic analysis that there was no presence of obvious feature for pressure node in such a typical standing wave resonator, and the distribution of acoustic fields displayed a one-dimensional feature for the acoustic pressure while a quasi-one-dimensional form for oscillatory velocity, which demonstrated the nonlinear effects. The simulation results for axial distribution of acoustic intensity showed a good consistency with the published experimental data in the open literature domain, which provides a verification for the effectiveness of the GKS model proposed. The influence of displacement amplitude of the driving piston on the formation of shock wave was numerically investigated, and the simulated results revealed the cascade process of harmonic wave energy from the fundamental wave to higher harmonics. In addition, this study found that the acoustic intensity at the driving end of the resonant tube would increase linearly with the displacement amplitude of the piston due to nonlinear effects, rather than the exponential variation by linear theory. This research demonstrates that the GKS model is strongly capable of simulating nonlinear acoustic problems.

Computation of nonlinear acoustic waves using smoothed particle hydrodynamics

A Lagrangian approach for solving nonlinear acoustic wave problems is presented with direct computation from smoothed particle hydrodynamics. The traditional smoothed particle hydrodynamics method has been applied to solve linear acoustic wave propagations. However, nonlinear acoustic problems are common in medical ultrasonography, sonic boom research, and acoustic levitation. Smoothed particle hydrodynamics is a Lagrangian meshfree particle method that shows advantages in modeling nonlinear phenomena, such as the shock tube problem, and other nonlinear problems with material separation or deformable boundaries. The method is used to solve the governing equations of fluid dynamics for simulating nonlinear acoustics. The present work also tests the method in solving the nonlinear simple wave equation based on Burgers’ equation. Effects of initial particle spacing, kernel length, and time step are then discussed based on the wave propagation simulation. Different kernel functions are also evaluated. The res...

Inverse problem of nonlinear acoustics: Synthesizing intense signals to intensify the thermal and radiation action of ultrasound

Inverse problems of nonlinear acoustics have important applied significance. On the one hand, they are necessary for nonlinear diagnostics of media, materials, manufactured articles, building units, and biological and geological structures. On the other hand, they are needed for creating devices that ensure optimal action of acoustic radiation on a target. However, despite the many promising applications, this direction remains underdeveloped, especially for strongly distorted high-intensity waves containing shock fronts. An example of such an inverse problem is synthesis of the spatiotemporal structure of a field in a radiating system that ensures the highest possible energy density in the focal region. This problem is also related to the urgent problems of localizing wave energy and the theory of strongly nonlinear waves. Below we analyze some quite general and simple inverse nonlinear problems.

Analytical extension of Finite Element solution for computing the nonlinear far field of ultrasonic waves scattered by a closed crack

Nonlinear scattering of ultrasonic waves by closed cracks subject to contact acoustic nonlinearity (CAN) is determined using a 2D Finite Element (FE) coupled with an analytical approach. The FE model, which includes unilateral contact with Coulomb friction to account for contact between crack faces, provides the near-field solution for the interaction between in-plane elastic waves and a crack of different orientations. The numerical solution is then analytically extended in the far-field based on a frequency domain near-to-far field transformation technique, yielding directivity patterns for all linear and nonlinear components of the scattered waves. The proposed method is demonstrated by application to two nonlinear acoustic problems in the case of tone-burst excitations: first, the scattering of higher harmonics resulting from the interaction with a closed crack of various orientations, and second, the scattering of the longitudinal wave resulting from the nonlinear interaction between two shear waves and a closed crack. The analysis of the directivity patterns enables us to identify the characteristics of the nonlinear scattering from a closed crack, which provides essential understanding in order to optimize and apply nonlinear acoustic NDT methods.

On the Inverse Problems of Nonlinear Acoustics and Acoustic Turbulence

We consider the problem of retrieval of the radiated acoustic signal parameters from the measured wave field in some cross section of the nonlinear medium. The possibilities of solving regular and statistical inverse problems are discussed on the basis of the solution of the Burgers equation for zero and infinitesimal viscosities.

Numerical simulation of nonlinear acoustic streaming in a resonator using gas-kinetic scheme

The acoustic streaming motion in a compressible air-filled two-dimensional cylindrical resonator driving by a piston is simulated by using the gas-kinetic scheme, and the effects of acoustic field intensity on the formation process of flow structure as well as streaming pattern are investigated numerically for the practical applications of high-intensity acoustic devices. Therefore, five cases with different excitation amplitudes are considered in simulation ranging from the linear to highly nonlinear regions. The validation of the developed model is verified by comparing the numerical results of streaming velocities with the theoretical ones for the linear case. The wave form of pressure and velocity and the transient flow field structure as well as the resulting streaming pattern are found to be strongly correlated to the excitation amplitude. Be observed for the linear case is a sine wave and a uniform of quasi-one-dimensional flow field as well as classical Rayleigh streaming. Periodic shock waves and strongly two-dimensional flow fields as well as the irregular acoustic streaming fields have been observed for the nonlinear cases. And some new physical phenomena have also been revealed for the highly nonlinear case, and these facts are that the wave form will deviate from typical “shock wave”-type toward a more distorted type, and at the same time, the two-dimensional transient flow fields are characterized by the changing flow direction and circulatory flow pattern, which are all correlated to a turbulent streaming with various scales of irregular and unsteady vortices throughout the resonator. Detailed explanations are provided for these nonlinear phenomena, and a critical Reynolds number for the transition to turbulence is also numerically obtained which is consistent with the experiment result from the previous publication. The present work has demonstrated that the gas-kinetic scheme is capable of resolving large Reynolds number of nonlinear acoustic problems with no any restriction on the nonlinearity level.

Erratum to: “Classical problems of nonlinear acoustics and wave theory”

Erratum to: “Classical problems of nonlinear acoustics and wave theory”

Classical problems of nonlinear acoustics and wave theory

Focusing of a vibration field in finite sized elastic systems using time reversal of signals from a point’s wideband source is theoretically analyzed. It is shown that the result of space and time focusing for a classical reversal algorithm in limited systems with small losses depends of the positional relationship of the source and the signal receiver. A focusing algorithm eliminating this dependence and increasing the efficiency of the focusing of the vibration field is proposed. Theoretical estimates are confirmed with the results of numerical experiments with the models of elastic bodies.

The problems of nonlinear acoustics which seem to be the most important and interesting today

The material of the correlated report delivered at the 18th International Symposium on Nonlinear Acoustics (ISNA-18) in July 2008 (Stockholm, Sweden) [1] is presented in a more comprehensive form [1]. These results have also been partially reported on in the lectures given at the 12th and 13th “Nonlinear Waves“ schools in Nizhni Novgorod [2, 3]. The aim of the paper is to briefly consider the directions of the development of nonlinear acoustics and its successes which have been at the center of attention lately.

Hemodynamics and nonlinear acoustics: General approaches and solutions

Related problems of nonlinear acoustics and hemodynamics are discussed. Equations describing the propagation of pulse waves in elastic tubes taking into account the dependence of the cross-section, linear and nonlinear elasticity on the distance along the axis are obtained. It is shown that in narrowing tubes with their elasticity increasing shock fronts may be formed with the peak pressure increasing, and the peak speed decreasing. The analogous behavior was observed in experiments. The expression of the nonlinearity coefficient as a sum of the “geometrical” and “physical” nonlinearities having different signs is obtained. It is supposed that in a normally functioning system these nonlinearities compensate each other. The formation of the shock waves is the evidence of vascular pathology. It is shown that it is possible to restore the local heterogeneity of a vessel (the change of the cross-section or rigidity) on the basis of measurements of the reflected or transmitted wave. The principle of the action of the acoustic pump not containing mechanically moving parts is described.

Nonlinear Generation of Airborne Sound by Waves of Ultrasonic Frequencies

The results of research into the design and optimization of laboratory sources of intense airborne ultrasound are reported. Two types of sources are studied: multielement arrays of small-size piezoelectric radiators and single membrane transducers of a capacitor type. The measured characteristics of the ultrasound fields and the audible sound fields generated in air due to the nonlinear interaction of high-frequency waves are presented. Applications of nonlinear acoustic problems in air are discussed.

Principle of an a priori use of symmetries in the theory of nonlinear waves

The principle of an a priori use of symmetries is proposed as a new approach to solving nonlinear problems on the basis of a reasonable complication of mathematical models. Such a complication often causes an additional symmetry and, hence, opens up possibilities for finding new analytical solutions. The application of group analysis to the problems of nonlinear acoustics is outlined. The potentialities of the proposed approach are illustrated by exact solutions, which are of interest for wave theory.