Finite Amplitude Acoustic Waves Research Articles

Layer contributions to the nonlinear acoustic radiation from stratified media

This study presents the thorough investigation of the second harmonic generation scenario in a three fluid layer system. An emphasis is on the evaluation of the nonlinear parameter B/A in each layer from remote measurements. A theoretical approach of the propagation of a finite amplitude acoustic wave in a multilayered medium is developed. In the frame of the KZK equation, the weak nonlinearity of the media, attenuation and diffraction effects are computed for the fundamental and second harmonic waves propagating back and forth in each of the layers of the system. The model uses a gaussian expansion to describe the beam propagation in order to quantitatively evaluate the contribution of each part of the system (layers and interfaces) to its nonlinearity. The model is validated through measurements on a water/aluminum/water system. Transmission as well as reflection configurations are studied. Good agreement is found between the theoretical results and the experimental data. The analysis of the second harmonic field sources measured by the transducers from outside the stratified medium highlights the factors that favor the cumulative effects.

Calomel-made crystalline acousto-optical cell designed for an advanced regime of noncollinear two-phonon light scattering

We study the potentials of a wide-aperture crystalline calomel-made acousto-optical cell. Characterizing this cell is nontrivial due to the chosen regime based on an advanced noncollinear two-phonon light scattering. Recently revealed important features of this phenomenon are essentially exploited in the cell and are investigated in more detail. These features can be observed more easily and simply in tetragonal crystals, e.g., calomel, exhibiting specific acousto-optical nonlinearity caused by the acoustic waves of finite amplitude. This parametric nonlinearity manifests itself at low acoustic powers in calomel possessing linear acoustic attenuation. The formerly identified additional degree of freedom, unique to this regime, is exploited for designing the cell with an eye to doubling the resolution due to two-phonon processes. We clarify the role of varying the central acoustic frequency and acoustic attenuation using that degree of freedom. Then the efficiency of calomel is exploited to expand the cell’s bandwidth with a cost of its efficiency. Proof-of-principle experiments confirm the developed approaches and illustrate their applicability to innovative techniques of optical spectrum analysis with the improved resolution. The achieved spectral resolution of 0.205 A at 405 nm and the resolving power 19,800 are the best for acousto-optical spectrometers dedicated to space or airborne operations to date as far as we know.

Advanced collinear LiNbO3acousto-optical filter for astrophysical spectroscopy in the near-ultraviolet: exploring high-spectral resolution

We explore the potentiality of using an advanced collinear LiNbO3 acousto-optical filter as a dispersive element for a high-resolution optical spectrograph. Our analysis is focused on weak optical signals in the blue to near-ultraviolet range accessible to ground-based facilities. We examine the phenomenon affecting the filter transmission efficiency and its spectral resolution, namely, the light-induced absorption and photorefraction. A new nonlinear approach is used to determine the performance of this collinear LiNbO3 filter governed by acoustic waves of finite amplitude. The highest available spectral resolution attains δλ=0.15 Å at λ=370 nm (the resolving power R∼25,000), with an efficiency of 11%, or δλ=0.18 Å at λ=532 nm (R∼30,000), with an efficiency of 33%. A slight decrease in the spectral resolution would imply a significant increase in transmission efficiency. Then, we carried out proof of principle experiments with the collinear filter based on the congruent LiNbO3 crystal of 6.3-cm length at λ=405 and 440 nm to verify our analysis and estimations. Potential applications are tackling many issues in astronomy, from detailed abundance analysis in a variety of targets to precise radial velocity measurement.

On measurement of the acoustic nonlinearity parameter using the finite amplitude insertion substitution (FAIS) technique

The acoustic nonlinearity parameter, B/A, is an important parameter which defines the way a propagating finite amplitude acoustic wave progressively distorts when travelling through any medium. One measurement technique used to determine its value is the finite amplitude insertion substitution (FAIS) method which has been applied to a range of liquid, tissue and tissue-like media. Importantly, in terms of the achievable measurement uncertainties, it is a relative technique. This paper presents a detailed study of the method, employing a number of novel features. The first of these is the use of a large area membrane hydrophone (30 mm aperture) which is used to record the plane-wave component of the acoustic field. This reduces the influence of diffraction on measurements, enabling studies to be carried out within the transducer near-field, with the interrogating transducer, test cell and detector positioned close to one another, an attribute which assists in controlling errors arising from nonlinear distortion in any intervening water path. The second feature is the development of a model which estimates the influence of finite-amplitude distortion as the acoustic wave travels from the rear surface of the test cell to the detector. It is demonstrated that this can lead to a significant systematic error in B/A measurement whose magnitude and direction depends on the acoustic property contrast between the test material and the water-filled equivalent cell. Good qualitative agreement between the model and experiment is reported. B/A measurements are reported undertaken at (20 ± 0.5) °C for two fluids commonly employed as reference materials within the technical literature: Corn Oil and Ethylene Glycol. Samples of an IEC standardised agar-based tissue-mimicking material were also measured. A systematic assessment of measurement uncertainties is presented giving expanded uncertainties in the range ±7% to ±14%, expressed at a confidence level close to 95%, dependent on specimen details.

Collinear dissipative weakly coupled acousto-optical states governed by the acoustic waves of finite amplitude in a two-mode medium with linear optical losses

The components of dissipative weakly coupled states, originating due to collinear Bragg light scattering by the acoustic wave of finite amplitude, are investigated in the square-law nonlinear medium with the linear optical losses. A novel theoretical model is developed for three-wave weakly coupled states of various pulse profiles, propagating in the quasi-stationary regime with the phase mismatch. The availability of both compact and infinite support is analyzed and compared with one another. Two limiting cases for cnoidal profiles are considered in detail and estimated. Their optical components are observed during the acousto-optical experiments in an α-quartz crystalline cell with calibrated optical losses.

Transmission function of collinear acousto-optical interaction occurred by acoustic waves of finite amplitude

New physical aspects of collinear acousto-optical interaction, occurred by acoustic waves of finite amplitude, are revealed and analyzed in crystalline materials exhibiting moderate linear acoustic losses. The analysis is performed in the regime of continuous traveling waves allowing a specific mechanism of the acousto-optic nonlinearity. Our consideration has shown that such nonlinearity together with linear acoustic losses is able to affect the transmission function inherent in collinear interaction. In particular, the mere presence of linear acoustic losses by themselves leads to broadening the width of the transmission function beginning already from very low levels of the applied acoustic power. Moreover, the transmission function exhibits a marked and quasi-periodical dependence on the applied acoustic power density, and that periodicity is governed by the linear acoustic losses. As a result, the transmission function can be significantly narrowed near isolated points at the cost of decreasing the interaction efficiency. These novelties related to collinear acousto-optical interaction accompanied by moderate linear acoustic losses have been studied and confirmed experimentally with an advanced acousto-optical cell based on calcium molybdate (CaMoO4) single crystal and controlled by acoustic waves of finite amplitude.

Nonlinear and diffraction effects in propagation of N-waves in randomly inhomogeneous moving media

Finite amplitude acoustic wave propagation through atmospheric turbulence is modeled using a Khokhlov-Zabolotskaya-Kuznetsov (KZK)-type equation. The equation accounts for the combined effects of nonlinearity, diffraction, absorption, and vectorial inhomogeneities of the medium. A numerical algorithm is developed which uses a shock capturing scheme to reduce the number of temporal grid points. The inhomogeneous medium is modeled using random Fourier modes technique. Propagation of N-waves through the medium produces regions of focusing and defocusing that is consistent with geometrical ray theory. However, differences up to ten wavelengths are observed in the locations of fist foci. Nonlinear effects are shown to enhance local focusing, increase the maximum peak pressure (up to 60%), and decrease the shock rise time (about 30 times). Although the peak pressure increases and the rise time decreases in focal regions, statistical analysis across the entire wavefront at a distance 120 wavelengths from the source indicates that turbulence: decreases the mean time-of-flight by 15% of a pulse duration, decreases the mean peak pressure by 6%, and increases the mean rise time by almost 100%. The peak pressure and the arrival time are primarily governed by large scale inhomogeneities, while the rise time is also sensitive to small scales.

A novel DRBEM application for nonlinear wave propagation

This paper presents a two-dimensional numerical procedure based on the boundary integral equations to model acoustic waves of finite-amplitude. The analysis is performed in the frequency domain. By applying the perturbation technique up to the second-order term, the governing differential equations are written as a system of two Helmholtz equations, one is homogeneous and the other one is inhomogeneous. Both are transformed into integral equations, which can be numerically solved without domain discretization by the use of the dual reciprocity boundary element method (DRBEM). To the authors' knowledge, this is the first application of DRBEM to nonlinear acoustics. The numerical procedure can be applied to predict the propagation of finite but of moderate amplitude acoustic waves in domains of any geometry. The final formulation is validated by comparison with an analytical solution derived by the authors. Copyright © 2009 John Wiley & Sons, Ltd.

Effect of finite-amplitude acoustic wave nonlinear interaction on farfield directivity of sound source

The acoustic radiation pressure expression of a multi-frequency sound source is obtained on the basis of the Fenlon theory. Then using the solution method of a monochromatic source harmonic wave directivity, farfield directivity of second-order approximate is obtained in the case of the dual-frequency sound source interaction. Subsequently, the effect of two-wave interaction on the farfield of first-order and second-order wave of either wave is studied and discussed when the initial radiation pressures and frequencies are different. The conclusion is that there are some different effects of sound wave interaction on the farfield directivity of a sound source, when the relative initial radiation pressure and frequency between the sound waves is changed.

The use of a hybrid model to compute the nonlinear acoustic performance of silencers for the finite amplitude acoustic wave

In the present study, a hybrid method is proposed for predicting the acoustic performance of a silencer for a nonlinear wave. This method is developed by combining two models: (i) a frequency-domain model for the computation of sound attenuation due to a silencer in a linear regime and (ii) a wavenumber space model for the prediction of the nonlinear time-evolution of finite amplitudes of the acoustic wave in a uniform duct of the same length as the silencer. The present method is proposed under the observation that the physical process of the nonlinear sound attenuation phenomenon of a silencer may be decoupled into two distinct mechanisms: (a) a linear acoustic energy loss that owes to the mismatch in the acoustic impedance between reactive elements and/or the sound absorption of acoustic liners in a silencer; (b) a nonlinear acoustic energy loss that is due to the energy-cascade phenomenon that arises from the nonlinear interaction between components of different frequencies. To establish the validity of the present model for predicting the acoustic performance of silencers, two model problems are considered. First, the performance of simple expansion mufflers with nonlinear incident waves has been predicted. Second, proposed method is applied for computing nonlinear acoustic wave propagation in the NASA Langley impedance duct configuration with ceramic tubular liner (CT57). Both results obtained from the hybrid models are compared with those from computational aero-acoustic techniques in a time–space domain that utilize a high-order finite-difference method. Through these comparisons, it is shown that there are good agreements between the two predictions. The main advantage of the present method is that it can effectively compute the nonlinear acoustic performance of silencers in nonlinear regimes without time–space domain calculations that generally entail a greater computational burden.

Flow through orifices caused by large-amplitude asymmetric sound waves: Theory and experiment.

When the displacement amplitude of an acoustic wave consisting of equal parts fundamental and second harmonic exceeds the diameter of a circular orifice, steady flow is generated whose direction is determined by the phase of th etwo wave components [P.J. Westervelt, Ph.D. thesis "The interaction of a finite amplitude acoustic wave with small obstacles and orifices," MIT, 1951)]. Some applications to semipermeable biological membranes are discussed. [Work supported by the Office of Naval Research].

Nonlinear acoustic phenomena in viscous thermally relaxing fluids: Shock bifurcation and the emergence of diffusive solitons.

In this talk, we will consider the propagation of finite-amplitude acoustic waves in fluids that exhibit both viscosity and thermal relaxation. Under the assumption that the thermal flux vector is given by the Maxwell–Cattaneo law, which is a well known generalization of Fourier's law that includes the effects of thermal inertia, we derive the weakly nonlinear equation of motion in terms of the acoustic potential. We then use singular surface theory to determine how an input signal in the form of a shock wave evolves over time and for different values of the Mach number. Then, numerical methods are used to illustrate our analytical findings. In particular, it is shown that the shock amplitude exhibits a transcritical bifurcation; that a stable nonzero equilibrium solution is possible, and that a Taylor shock (i.e., a diffusive soliton), in the form of a “tanh” profile, can emerge from behind the input shock wave. Finally, an application related to the kinematic-wave theory of traffic flow is noted. [Work supported by ONR/NRL funding (PE 061153N)].

Finite-amplitude acoustic traveling waves in a fluid that saturates a porous medium: Acceleration wave formation

An analytical study of finite-amplitude acoustic waves in a homentropic fluid that saturates a rigid porous medium is presented. The exact traveling wave solution (TWS) is obtained in terms of the Lambert W-function. It is shown that the acceleration profile assumes the form of an asymmetric diffusive soliton and that a jump discontinuity in the acceleration forms as the TWSs propagation speed approaches the sound speed of the fluid. Additionally, a connection between this TWS and results from singular surface theory is established and the findings of the present study are compared/contrasted with those for the thermoviscous, nonporous case.

Numerical and experimental analysis of second-order effects and loss mechanisms in axisymmetrical cavities

This paper deals with the analysis of finite amplitude acoustic waves in three-dimensional resonant cavities. A specific finite element model is proposed which includes: (i) the pressure field nonlinearity using a perturbative method; (ii) the loss mechanisms using an experimentally determined effective bulk attenuation or modeling viscous and thermal losses at the walls and acoustic radiation through the aperture with complex impedances. An axisymmetrical cavity with transversal dimension larger than the wavelength has been experimentally studied. The acoustic field is generated by a high-power flexurally vibrating transducer which generates high pressures. Measurements are performed for the fundamental and second-order pressure components for several cavity resonance modes. An important standard deviation is observed. Experimental data are compared to predicted pressure field distributions. Numerical models describe the pressure distribution correctly but overestimate the amplitude.

Modeling of finite amplitude acoustic waves in closed cavities using the Galerkin method.

Nonlinear resonant gas oscillations in closed ducts are investigated by solving a previously derived, quasi-one-dimensional, nonlinear wave equation that accounts for forcing, gas dynamic nonlinearities, and viscous dissipation. This equation is solved with the approximate Galerkin method to determine the dependence of driven oscillations upon the duct shape, forcing frequency, and forcing amplitude. Initially, the applicability of the developed Galerkin solution approach was studied by investigating oscillations in a straight duct, closed at both ends and periodically oscillated at a single frequency. It is shown that the Galerkin method predictions of shock wave-like oscillations in such ducts are in excellent agreement with results obtained with other numerical solution techniques. Next, this study investigated the forced response of a class of horn-shaped ducts, and it is shown that for a given forcing amplitude, there exists a nonmonotonic increase in compression ratio as the duct's flare constant is increased. Finally, it is shown that oscillations driven in ducts whose shapes were chosen to provide shifting of the second and third natural acoustic mode frequencies exhibit significant waveform distortion and non-negligible increases in compression ratio when compared with oscillations driven in straight ducts.

Solution of the Burgers Equation in the Time Domain

This paper deals with a theoretical description of the propagation of a finite amplitude acoustic waves. The theory based on the homogeneous Burgers equation of the second order of accuracy is presented here. This equation takes into account both nonlinear effects and dissipation. The method for solving this equation, using the well-known Cole-Hopf transformation, is presented. Two methods for numerical solution of these equations in the time domain are presented. The first is based on the simple Simpson method, which is suitable for smaller Goldberg numbers. The second uses the more advanced saddle point method, and is appropriate for large Goldberg numbers.

PROPAGATION OF FINITE AMPLITUDE ACOUSTIC WAVES IN A TWO-DIMENSIONAL RECTANGULAR DUCT

Finite amplitude acoustic waves (FAAWs) that propagate in a two-dimensional rectangular duct of semi-infinite length as a result of periodic excitation are determined by using second-order perturbation, based on the partial wave analysis method. With second-harmonic boundary and initial conditions of excitation, second-harmonic analytical expressions, which are applicable to quantitative analysis, have been derived. In this manner, a physical mechanism of second-harmonic generation and propagation in the process of propagation of FAAWs is clearly displayed. Based on the formula, some numerical calculations are performed. The numerical results clearly exhibit the distortion and symmetry of second-harmonic field pattern for a given source of excitation.

Active control of finite amplitude acoustic waves in a confined geometry

An active control approach was developed to observe and control nonlinear effects for high-amplitude acoustic waves in a cavity. The acoustic medium was air at ambient pressure and temperature. A harmonic cavity was driven using a multi-frequency drive to selectively drive the fundamental mode and suppress nonlinear interactions between the fundamental and other modes. A semi-empirical model was developed using effective quadratic and cubic nonlinear coupling coefficients to predict resonator response. The cavity was also shown to act as an acoustic mixer. Typical acoustic pressure amplitudes were 1000 Pa (SPL 154 dB re: 20 μPa).

Measurement of Acoustic Nonlinearity Parameter of Liquids by the Observation of Waveforms

A method for measuring the acoustic nonlinearity parameter of liquids by the observation of acoustic pressure waveforms has been presented. The crest of a finite amplitude acoustic wave advances from the crest of the sine wave as it propagates due to the nonlinearity of the liquid medium. By measuring the amount of advance, the sawtooth parameter and the nonlinearity parameter are obtained, using acoustic pressure amplitude, acoustic velocity, acoustic wavelength, acoustic frequency and the propagated distance. Measurements on six kinds of liquids were made and the results have been shown. The features of this method are the straightforwardness and the simplicity of the measurement.

Etudes d'interactions acoustiques non-linéaires dans les jets confinés, par la fonction de bicohérence

Self-sustained tones are encountered when a flow passes two orifice plates, sufficiently close together, inside a duct. The upstream orifice plate creates a confined jet which impinges on the edge of the downstream orifice plate. The shear layer oscillations (coherent structures) of the confined jet in the presence of a resonant acoustic field (pipe mode), constitute the main features of cavity tones. The tones are synchronized by the pipe modes. When the flow velocity is varied for a fixed geometry, frequency jumps from one mode to another are observed. Pressure spectra showing intense acoustic energy localised in narrow bands near pipe mode frequencies are measured. We emphasize the existence of transition flow regimes during which quadratic non-linear coupling between spontaneously excited modes can take place. A digital bispectral analysis (bicoherence function) is used to quantify the degree of nonlinear interactions of acoustic waves of finite amplitude in the so-called transition region