Claes Hedberg

Nonlinear standing waves in a layer excited by the periodic motion of its boundary

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 nonsteady-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 a standing wave 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 the 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 eigenvalues.

Resonant properties of a nonlinear dissipative layer excited by a vibrating boundary: Q-factor and frequency response

Simplified nonlinear evolution equations describing non-steady-state forced vibrations in an acoustic resonator having one closed end and the other end periodically oscillating are derived. An approach based on a nonlinear functional equation is used. The nonlinear Q-factor and the nonlinear frequency response of the resonator are calculated for steady-state oscillations of both inviscid and dissipative media. The general expression for the mean intensity of the acoustic wave in terms of the characteristic value of a Mathieu function is derived. The process of development of a standing wave is described analytically on the base of exact nonlinear solutions for different laws of periodic motion of the wall. For harmonic excitation the wave profiles are described by Mathieu functions, and their mean energy characteristics by the corresponding eigenvalues. The sawtooth-shaped motion of the boundary leads to a similar process of evolution of the profile, but the solution has a very simple form. Some possibilities to enhance the Q-factor of a nonlinear system by suppression of nonlinear energy losses are discussed.

Pulse response of a nonlinear layer

A simple analytical theory is developed for the description of the non-steady-state response of a thin nonlinear layer, which differs markedly in its linear properties from the surrounding medium. Such a layer can model the behavior of real inhomogeneities like a cloud of gas bubbles in a liquid, a crack or split plane in a solid, or the contact between two slightly tightened rough surfaces. Both weakly nonlinear pulse and harmonic responses are calculated and the general properties of the spectral and temporal structures of the scattered field are discussed. Exact strongly nonlinear solutions are derived for a special type of stress-strain relationship corresponding to the behavior of real condensed media under strong loads. Profiles and spectra shown conform with experimental results. The pulse response on the short delta-pulse shaped incident wave is calculated for arbitrary nonlinear properties of the layer. The possibilities to apply the sets of data on measured characteristics of pulse response in the solution of inverse problems are briefly discussed.

Strong and weak nonlinear dynamics: Models, classification, examples

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.

Finite-Amplitude Standing Acoustic Waves in a Cubically Nonlinear Medium

The acoustic field in a resonator filled with a cubically nonlinear medium is investigated. The field is represented as a linear superposition of two strongly distorted counterpropagating waves. Unlike the case of a quadratically nonlinear medium, the counterpropagating waves in a cubically nonlinear medium are coupled through their mean (over a period) intensities. Free and forced standing waves are considered. Profiles of discontinuous oscillations containing compression and expansion shock fronts are constructed. Resonance curves, which represent the dependences of the mean field intensity on the difference between the boundary oscillation frequency and the frequency of one of the resonator modes, are calculated. The structure of the profiles of strongly distorted “forced” waves is analyzed. It is shown that discontinuities are formed only when the difference between the mean intensity and the detuning takes certain negative values. The discontinuities correspond to the jumps between different solutions to a nonlinear integro-differential equation, which, in the case of small dissipation, degenerates into a third-degree algebraic equation with an undetermined coefficient. The dependence of the intensity of discontinuous standing waves on the frequency of oscillations of the resonator boundary is determined. A nonlinear saturation is revealed: at a very large amplitude of the resonator wall oscillations, the field intensity in the resonator ceases depending on the amplitude and cannot exceed a certain limiting value, which is determined by the nonlinear attenuation at the shock fronts. This intensity maximum is reached when the frequency smoothly increases above the linear resonance. A hysteresis arises, and a bistability takes place, as in the case of a concentrated system at a nonlinear resonance.

Method for monitoring slow dynamics recovery

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.

Nonlinear model of a granulated medium containing viscous fluid layers and gas cavities

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.

Dissipative and hysteresis loops as images of irreversible processes in nonlinear acoustic fields

Irreversible processes taking place during nonlinear acoustic wave propagation are considered using a representation by loops in a thermodynamic parameter space. For viscous and heat conducting media, the loops are constructed for quasi-harmonic and sawtooth waves and the descriptive equations are formulated. The linear and nonlinear absorptions are compared. For relaxing media, the processes are frequency-dependent. The loops broadens, narrows, and bends. The linear and nonlinear relaxation losses of wave energy are shown. Residual stresses and irreversible strains appear for hysteretic media, and here, a generalization of Rayleigh loops is pictured which takes into account the nonlinearly frequency-dependent hereditary properties. These describe the dynamic behavior, for which new equations are derived.

Parametric Sound Fields Formed by Phase-Inversion Excitation of Primary Waves

Two planar ultrasound projectors having identical rectangular apertures were placed side by side. Both projectors radiated bifrequency primary waves in air. The frequencies were 26 and 28 kHz, and the initial phases were different. Two driving modes were considered, namely, conventional in-phase driving and phase-inversion driving. The spatial profiles of sound pressure fields were measured along and across the sound beam axis for the primary waves and for the difference frequency wave of 2 kHz. The second and third harmonic components of the difference frequency waves were also measured. The pressure levels of the primary waves were considerably suppressed near the beam axis owing to phase cancellation when the driving signals were phase-inversed, i.e., 180 degrees out of phase. The beam pattern of the difference frequency was, however, almost the same as that for the case in which the signals were in phase. Interestingly, the harmonic pressure amplitudes of the difference frequency were reduced by more than 10 dB. The validity of the experimental results were confirmed based on their good agreement with the theoretical predictions based on the Khokhlov-Zabolotskaya-Kuznetsov equation.

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.