High-speed water jet cutting has important industrial applications. To further improve the cutting performance it is critical to understand the theory behind the onset of instability of the jet. In this paper, instability of a water jet flowing out from a nozzle into ambient air is studied. Capillary forces and compressibility of the liquid caused by gas bubbles are taken into account, since these factors have shown to be important in previous experimental studies. A new dispersion equation, generalizing the analogous Rayleigh equation, is derived. It is shown how instability develops because of aerodynamic forces that appear at the streamlining of an initial irregularity of the equilibrium shape of the cross-section of the jet and how instability increases with increased concentration of gas bubbles. It is also shown how resonance phenomena are responsible for strong instability. On the basis of the theoretical explanations given, conditions for stable operation are indicated.
The results of numerical and experimental analysis of the parameters of a singlefrequency micro wave thinfilm electroacoustic resonator based on an (0001)AlN piezofilm with an acoustic reflector operat ing at a frequency of 10 GHz are presented. The effect of the reflector design on the resonator characteristics is considered. Using the modified Butterworth–Van Dyke model, it was shown that the ohmic resistance of electrodes and entrance paths substantially decreases the Qfactor at the resonance frequency of series and the acoustic losses in the resonator deteriorate the Qfactor at the parallel resonance frequency
The observed nonclassical power-law dependence of the amplitude of the second harmonic wave on the amplitude of a harmonic pump wave is explained as a phenomenon associated with two types of nonlinearity in a structurally inhomogeneous medium. An approach to solving the inverse problem of determining the nonlinearity parameters and the exponent in the above-mentioned dependence is demonstrated. To describe the effects of strongly pronounced nonlinearity, equations containing a double nonlinearity and generalizing the Hopf and Burgers equations are proposed. The possibility of their exact linearization is demonstrated. The profiles, spectral composition, and average wave intensity in such doubly nonlinear media are calculated. The shape of the shock front is found, and its width is estimated. The wave energy losses that depend on both nonlinearity parameters—quadratic and modular—are calculated. © 2018, Pleiades Publishing, Ltd.
We study the interaction of noise and regular signals with a front whose steepness increases or decreases owing to nonlinear distortion of the profile of an intense pumping wave. Projective transformation is used, which is a result of one of the Burgers equation symmetries. Signal interaction with the pumping wave at its leading edge results in an increase in signal amplitude, a decrease in its time scale, an increase in the signal evolution rate, and earlier merging of discontinuities. At the trailing edge, an increase in signal amplitude, an increase in the time scale, and deceleration of the evolution rate occur. Formulas are obtained that describe the transformation of the spectrum and the correlation function of noise. Laws of the change in noise energy for both small and large Reynolds numbers are found. We study the interaction of weak noise with a nonstationary shock front in a medium with a finite viscosity. It is shown that, owing to competition between amplification at the shock front and high-frequency attenuation, the dependence on the noise intensity on distance has a nonmonotonic character, and at large distances, the intensity tends to zero, while the correlation time tends to a finite value.
Wave propagation in a near-bottom layer containing gas bubbles is analyzed. Evolution equations are derived for intense acoustic waves and wave beams in a medium with an inhomogeneous bubble distribution. The field of an intense beam along the axis of the focusing sound channel is calculated. The coefficients of reflection and passage of sound from a homogeneous medium into a bubble layer and back again are calculated. It is shown that the near-bottom layer can effectively trap rays incident on it and ensure a waveguide propagation character. The presence of bubbles increases both the interval of angles at which the wave penetrates the layer and the interval of angles at which rays undergo total internal reflection and do not depart the layer. The acoustic field in the layer from a point source is calculated.
The principle of forming a special form of powerful acoustic signals is proposed, which makes it possible to ensure precise spatiotemporal beam focusing. The introduction of a transverse-coordinate-dependent local wave frequency is suggested, due to which the equality of the formation lengths of a discontinuity for all rays is achieved. This thereby ensures an increase in nonlinear absorption; as a result, the temperature and radiation action of focused ultrasound on the medium increase.
The acoustic field and the field of radiative forces that are formed in a liquid layer on a solid substrate are calculated for the case of wave propagation along the interface. The calculations take into account the effects produced by surface tension, viscous stresses at the boundary, and attenuation in the liquid volume on the field characteristics. The dispersion equations and the velocities of wave propagation are determined. The radiative forces acting on a liquid volume element in a standing wave are calculated. The structure of streaming is studied. The effect of streaming on small size particles is considered, and the possibilities of ordered structure formation from them are discussed.
The interconnection between variations of elasticity and dielectric permittivity of mesoscopic solid systems under exposure to ultrasound is experimentally observed. A phenomenological theory generalizing Debye’s approach for polar fluids is developed to explain the measured data. The substitution of acoustic measurements by dielectric ones not only simplifies the procedure, but offers new possibilities to remotely evaluate the mechanical properties of materials and natural media.
Slow Dynamics is a specific material property, which for example is connected to the degree of damage. It is therefore of importance to be able to attain proper measurements of it. Usually it has been monitored by acoustic resonance methods which have very high sensitivity as such. However, because the acoustic wave is acting both as conditioner and as probe, the measurement is affecting the result which leads to a mixing of the fast nonlinear response to the excitation and the slow dynamics material recovery. In this article a method is introduced which, for the first time, removes the fast dynamics from the process and allows the behavior of the slow dynamics to be monitored by itself. The new method has the ability to measure at the shortest possible recovery times, and at very small conditioning strains. For the lowest strains the sound speed increases with strain, while at higher strains a linear decreasing dependence is observed. This is the first method and test that has been able to monitor the true material state recovery process.
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.
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. This approach often provides additional symmetries, and hence opens possibilities to find new analytical solutions. The potentialities of the proposed approach are illustrated by applying to problems of nonlinear acoustics.
We calculate the field of radiation forces in a cylindrical fluid layer on a solid substrate formed as a result of the action on a fluid of a capillary wave propagating from the axis along a free surface. We study the structure of acoustic flows excited by the radiation forces. We discuss the action of flows on small-sized particles and the possibilities of these particles to form ordered structures. Â© 2015, Pleiades Publishing, Ltd.
The field of radiation forces in a fluid layer on a solid substrate is calculated. This field is formed during propagation of surface capillary wave along a free surface. The wave is excited by substrate vibrations as a result of instability development. The structure of acoustic flows is studied. Their effect on small size particles and the possibilities of generating ordered structures from these particles are discussed.
We have studied the dynamics of an artificial nonlinear element representing a flexible membrane with oscillation limiters and a static pressing force. Such an element has the property of “bimodularity” and demonstrates “modular” nonlinearity. We have constructed a mathematical model that describes these oscillations. Their shapes have been calculated. We follow the analogy with a classical object—Galileo’s pendulum. We demonstrate that for a low-frequency excitation of the membrane, the level of the harmonics in the spectrum is higher than in the vicinity of the resonance frequency. We have established a strong dependence of the level of the harmonics on the magnitude of the pressing force for a weak perturbation. We propose a design scheme for a device in the quasi-static approximation possessing the property of bimodularity. We perform an experiment that confirms its operability. We show a qualitative coincidence of the experimental results and calculations when detecting an amplitude-modulated signal. © 2018, Pleiades Publishing, Ltd.
We study experimentally the behavior of a nonlinear element, a light plate pressed to the opening in the cavity of an acoustic resonator. Measurements of field oscillations inside and outside the cavity have shown that for large amplitudes, they become essentially anharmonic. The time dependences of displacement of the plate with increasing amplitude of the exciting voltage demonstrates a gradual change in the shape of vibrations from harmonic to half-period oscillation. A constant component appears in the cavity: rarefaction or outflow of the medium through the orifice. We construct a theory for nonlinear oscillations of a plate taking into account its different elastic reactions to compression and rarefaction with allowance for monopole radiation by the small-wave-size plate or radiation of a plane wave by the plate. We calculate the amplitudes of the harmonics and solve the problem of low-frequency stationary noise acting on the plate. We obtain expressions for the correlation function and mean power at the output given a normal random process at the input.
The paper discusses a model for a screen with dissipative and nonlinear elastic properties that can be used in acoustic sound absorption and frequency conversion systems. Calculation and estimation schemes are explained that are necessary for understanding the functional capabilities of the device. Examples of the nonlinear elements in the screen and promising applications are described.
The paper discusses a universal scheme for constructing nonlinear integro-differential models to describe intense waves in media with a complex internal relaxation-type dynamics. Examples of such media are presented. Various forms of kernels are described. Situations are shown in which the models can be simplified by reducing them to differential or differential-difference equations with partial derivatives. Integral relations for the linear momentum and energy transferred by the wave are obtained. Exact solutions are found. The mapping method is used to obtain approximate solutions and analyze them in the form of difference relations.
The 40th anniversary of the Khokhlo-Zabolotskaya equation was marked by a special session of the 158th Meeting of the Acoustical Society of America (October 2009, San Antonio, Texas, United States). A response on the part of Acoustical Physics to this date is also quite appropriate, all the more so because Russian scientists were the main players involved in formulating and using this equation during the period of time between the middle 1960s and middle 1980s. In this article, the author—a participant and witness of those events—presents his view of the dramatic history of the formulation of this equation and related models in the context of earlier and independent work in aerodynamics and nonlinear wave theory. The main problems and physical phenomena described by these mathematical models are briefly considered.
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. Â© 2016, Pleiades Publishing, Ltd.
Mathematical models are formulated that discribe linear and nonlinear wave propagation in biological tissues. The basis of the method is evolutionary integro-differential equations with a kernel that takes into account the specific properties of tissue. An equation is obtained for the correlation function of acoustic noise in a medium with memory. The procedure for calculating the correlation function by the given type of kernel and noise spectrum at the entrance to the medium is described. It is shown that in different tissue, there is a difference in the laws of decrease in full intensity of a wideband signal with distance. It is demonstrated that the nonlinear equation in the limiting cases of "short-" and "long-term" memory reduces to equations that have been well studied in statistical nonlinear acoustics.
The study of intense waves in soft biological tissues is necessary both for diagnostics and therapeutic aims. Tissue represents an inherited medium with frequency-dependent dissipative properties, in which waves are described by nonlinear integro-differential equations. The equations for such waves are well known. Their group analysis has been performed, and a number of exact solutions have been found. However, statistical problems for nonlinear waves in tissues have hardly been studied. As well, for medical applications, both intense noise waves and waves with fluctuating parameters can be used. In addition, statistical solutions are simpler in structure than regular solutions; they are useful for understanding the physics of processes. Below a general approach is described for solving nonlinear statistical problems applied to the considered mathematical models of biological tissues. We have calculated the dependences of the intensities of the narrowband noise harmonics on distance. For wideband noise, we have calculated the dependence of the spectral integral intensity on distance. In all cases, wave attenuation is determined both by the specific dissipative properties of the tissue and the nonlinearity of the medium.
An integro-differential equation is written down that contains terms responsible for nonlinear absorption, visco-heat-conducting dissipation, and relaxation processes in a medium. A general integral expression is obtained for calculating energy losses of the wave with arbitrary characteristics-intensity, profile (frequency spectrum), and kernel describing the internal dynamics of the medium. It is shown that for weak waves, the general integral leads to well-known results of a linear approximation. Profiles of stationary solutions are constructed both for an exponential relaxation kernel and for other types of kernels. Energy losses at the front of week shock waves are calculated. General integral formulas are obtained for energy losses of intense noise, which are determined by the form of the kernel, the structure of the noise correlation function, and the mean square of the derivative of realization of a random process
The stationary profile in the focal region of a focused nonlinear acoustic wave is described. Three models following from the Khokhlov-Zabolotskaya (KZ) equation with three independent variables are used: (i) the simplified one-dimensional Ostrovsky-Vakhnenko equation, (ii) the system of equations for paraxial series expansion of the acoustic field in powers of transverse coordinates, and (iii) the KZ equation reduced to two independent variables. The structure of the last equation is analogous to the Westervelt equation. Linearization through the Legendre transformation and reduction to the well-studied Euler-Tricomi equation is shown. At high intensities the stationary profiles are periodic sequences of arc sections having singularities of derivative in their matching points. The occurrence of arc profiles was pointed out by Makov. These appear in different nonlinear systems with low-frequency dispersion. Profiles containing discontinuities (shock fronts) change their form while passing through the focal region and are non-stationary waves. The numerical estimations of maximum pressure and intensity in the focus agree with computer calculations and experimental measurements. Â© 2015, Pleiades Publishing, Ltd.
The difference between strong and weak nonlinear systems is discussed. A classification of strong nonlinearities is given. It is based on the divergence or inanity of series expansions of the equation of state commonly used in the study of weak nonlinear phenomena. Such power or functional series cannot be used in three cases: (i) if the equation of state contains a singularity; (ii) if the series diverges for strong disturbances; (iii) if the linear term is absent, and higher nonlinearity dominates. Strong nonlinearities are known in acoustics, optics, mechanics and in quantum field theory. Mathematical models, solutions and observed phenomena are presented. For example, an equation of Heisenberg type and its generalization for strongly nonlinear wave system are given. In particular, exact solutions of new “quadratically cubic” Burgers and Riemann–Hopf equations are discovered.
The phenomenon of “wave resonance” which occurs at excitation of traveling waves in dissipative media possessing modular, quadratic and quadratically-cubic nonlinearities is studied. The mathematical model of this phenomenon is the inhomogeneous (or “forced”) equation of Burgers type. Such nonlinearities are of interest because the corresponding equations admit exact linearization and describe real physical objects. The presence of “accompanying sources” (traveling with the wave) on the right-hand side of the inhomogeneous equations ensures the inflow of energy into the wave, which thereafter spreads throughout the wave profile, flows to emerging shock fronts, and then dissipates due to linear and nonlinear losses. As an introduction, the phenomenon of wave resonance in ideal and dissipative media is described and physical examples are given. Exact expressions for nonlinear steady-state wave profiles are derived. Non-stationary processes of wave generation, spatial “beating” of amplitudes with different relationship between the speed of motion of the sources and the natural wave velocity in the medium are studied. Resonance curves are constructed that contain a nonlinear shift of the absolute maxima to the “supersonic” region. The features of the resonance in each of the three types of nonlinearity are discussed. © 2018, Pleiades Publishing, Ltd.
The behavior of the wave field in a resonator containing a cubically nonlinear medium is studied. The field is constructed as a linear superposition of two counter-propagating and strongly distorted waves. As distinct from a quadratic nonlinear medium the waves traveling in opposite directions are connected through their averaged (over the period) intensities. Both free and forced standing waves are studied. Profiles of discontinuous vibrations containing shock fronts of both compression and rarefaction are constructed. Resonant curves depicting the dependence of mean intensity on the difference between the frequency of vibration of the boundary and the natural frequency of one of the resonator’s mode are calculated. The structure of temporal profiles of strongly distorted forced waves is analyzed. It is shown, that shocks can appear only if the difference between the mean intensity and the discrepancy takes on definite negative values. The discontinuities are studied as jumps between the different solutions of a nonlinear integro-differential equation degenerating at weak dissipation to a third order algebraic equation with an undetermined coefficient. The dependence of the intensity of shocked vibrations on the frequency of vibration of the boundary is found. Nonlinear saturation is shown to appear: the intensity of wave field inside the resonator does not depend on the amplitude of boundary vibration when the amplitude is large. If the amplitude tends to infinity, the intensity tends to its limiting value determined by the nonlinear absorption at shock fronts. This maximum can be reached by smooth increase in frequency above the linear resonance. A hysteresis area and bistability appears, in analogy with the nonlinear resonance phenomena in localized vibration systems described by ordinary differential equations.
A new analytical approach is developed for the description of standing waves caused by arbitrary periodic vibration of a boundary. The approach is based on the nonlinear evolution equation written for an auxiliary function. This equation offers the possibility to study not only the steady-state acoustic field, but also its evolution in time. One can take into account the dissipative properties of the medium and the difference between one of the resonant frequencies and the fundamental frequency of the driving motion of the wall. An exact non-steady-state solution is derived corresponding to the sawtooth-like periodic vibration of the boundary. The maximal amplitude? values of the particle velocity and the energy of standing waves are calculated. The temporal profiles of standing waves at different points of the layer are presented. A new possibility of pumping a high acoustic energy into a resonator is indicated for the case of a special type of wall motion having the form of an ?inverse saw?. Theoretically, such a vibration leads to an ? explosive instability? and an unlimited growth of the standing wave. For a harmonic excitation, the exact non-steady-state solution is derived as well. The standing wave profiles are described by Mathieu functions, and the energy characteristics by their eigen-values.
The meaning of the experimentally measured nonlinear parameters of a medium is discussed. The difference in meaning between the local nonlinearity, which is measured in the vicinity of a single defect and depends on the size of the region of averaging, and the effective volume nonlinearity of the medium containing numerous defects is emphasized. The local nonlinearity arising at the tip of a crack is calculated; this nonlinearity decreases with an increase in the region of second harmonic generation. The volume nonlinearity is calculated for a solid containing spherical cavities. The volume nonlinearity is also calculated for a medium containing infinitely thin cracks in the form of circular disks, which assume the shape of ellipsoids in the course of the crack opening. The nonlinear acoustic parameter is calculated with the use of the exact classical results of the theory of cracks.
The paper presents the theory of shear wave propagation in a "soft solid" material possessing anisotropy of elastic and dissipative properties. The theory is developed mainly for understanding the nature of the low-frequency acoustic characteristics of skeletal muscles, which carry important diagnostic information on the functional state of muscles and their pathologies. It is shown that the shear elasticity of muscles is determined by two independent moduli. The dissipative properties are determined by the fourth-rank viscosity tensor, which also has two independent components. The propagation velocity and attenuation of shear waves in muscle depend on the relative orientation of three vectors: the wave vector, the polarization vector, and the direction of muscle fiber. For one of the many experiments where attention was distinctly focused on the vector character of the wave process, it was possible to make a comparison with the theory, estimate the elasticity moduli, and obtain agreement with the angular dependence of the wave propagation velocity predicted by the theory.
Phenomena arising in the course of wave propagation in narrow pipes are considered. For nonlinear waves described by the generalized Webster equation, a simplified nonlinear equation is obtained that allows for low frequency geometric dispersion causing an asymmetric distortion of the periodic wave profile, which qualitatively resembles the distortion of a nonlinear wave in a diffracted beam. Tunneling of a wave through a pipe constriction is investigated. Possible applications of the phenomenon are discussed, and its relation to the problems of quantum mechanics because of the similarity of the basic equations of the Klein– Gordon and Schrödinger types is pointed out. The importance of studying the tunneling of nonlinear waves and broadband signals is indicated.
We analyze nonlinear oscillations and waves in a simple model of a granular medium containing inclusions in the form of fluid layers and gas cavities. We show that in such a medium, the velocity of one of the wave modes is low; therefore, the nonlinearity is high and the effects of interaction are more strongly expressed than usual.
The formation of structured films consisting of ensembles of micro- or nanoparticles and possessing preset functional characteristics is studied both experimentally and theoretically. The films are obtained by drying out droplets of colloidal solutions on a solid substrate under the acoustic effect produced by a standing SAW field.
We have calculated the nonstationary flow of a viscous liquid in a narrow tube under the action of pressure variations with time. Such a flow accompanies venipuncture the procedure of taking a sample from a vein with a hypodermic needle. We show how the changes in the flow characterstics during venipuncture make it possible to actively estimate viscosity. This method is "nonperturbative" for blood in the sense that the measurement process weakly affects the measured quantity. It may find application in medicine.