B. O. Enflo

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.

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.

Saturation of a nonlinear cylindrical sound wave generated by a sinusoidal source

The propagation of a cylindrical sound wave generated by a sinusoidal source is studied in a nonlinear case by means of a generalized Burgers’ equation with two dimensionless parameters. The amplitude of the wave at a large distance from the source, where all higher harmonics are absorbed, is determined in a region for the two parameters where the amplitude is given by a single number not depending on the parameters. The method is to find a series expansion of the solution of the generalized Burgers’ equation. The series converges both far from the source and in the Taylor shock region nearer to the source. Despite the slow convergence of the series in the latter region it is possible to find the desired amplitude by means of an asymptotic expression for the terms in the series and some numerical work.

Large‐Time Asymptotics for Periodic Solutions of Some Generalized Burgers Equations

In this paper, we construct large-time asymptotic solutions of some generalized Burgers equations with periodic initial conditions by using a balancing argument. These asymptotics are validated by a careful numerical study. We also show that our asymptotic results agree with the approximate solutions obtained by Parker [1] in certain limits.

Saturation of nonlinear spherical and cylindrical sound waves

Nonlinear acoustic waves with spherical and cylindrical symmetry, described by a generalized Burgers’ equation, are studied. A sine oscillation at the source is distorted into a sawtooth wave profile because of nonlinearity. Very far from the source (old‐age region) the wave is again a sine wave solution of a wave equation in which the nonlinear term has grown so small that it can be neglected. In this paper the amplitude of the old‐age wave, which is independent of the amplitude at the source (saturation), is determined for nonlinear sinusoidal spherical and cylindrical waves by analytical methods for a certain range of parameter values. The old‐age amplitude in this range of parameter values was determined earlier by an approximate method, which overestimates the amplitude by a factor 1.36 in the plane‐wave case. It is shown in this paper that this overestimation is still greater for spherical and cylindrical waves.

Standing and propagating waves in cubically nonlinear media

The paper has three parts. In the first part a cubically nonlinear equation is derived for a transverse finite‐amplitude wave in an isotropic solid. The cubic nonlinearity is expressed in terms of elastic constants. In the second part a simplified approach for a resonator filled by a cubically nonlinear medium results in functional equations. The frequency response shows the dependence of the amplitude of the resonance on the difference between one of the resonator’s eigenfrequencies and the driving frequency. The frequency response curves are plotted for different values of the dissipation and differ very much for quadratic and cubic nonlinearities. In the third part a propagating N‐wave is studied, which fulfils a modified Burgers’ equation with a cubic nonlinearity. Approximate solutions to this equation are found for new parts of the wave profile.

Behavior of intense acoustic noise at large distances

Propagation of intense acoustic noise waves is investigated in the case of a nonplanar geometry. It is shown that, at large distances from the source, where the nonlinear effects become negligible, the spectrum of such waves has a universal self-similar shape. The amplitude of the spectrum is determined by a single constant D∞ = D∞(ɛ, R0) (the spectrum steepness at zero-valued argument) whose value depends on two dimensionless parameters: the inverse acoustic Reynolds number ɛ and the dimensionless radius R0. It is shown that the plane of dimensionless parameters (ɛ, R0) can be divided into four regions, so that, within each of them, the quantity D∞ is described by a universal function of these parameters. The numerical factors of these parameters are found from numerical simulations.

Old‐age stage evolution of intense cylindrical and spherical acoustic noise

The investigation of the old‐age behavior of an intense periodic spreading wave was one of the significant contributions of Crighton in nonlinear acoustics. By him and his co‐workers a classification of different regimes of evolution was done and it was shown at which condition the saturation of the amplitude takes place. Here the evolution of intense spreading acoustic noise is considered on the base of generalized Burgers’ equations. It is shown that at the old‐age (linear) stage the energy spectrum of the noise has a universal structure: it is proportional to the second power of the frequency at small frequencies and decreases exponentially at high frequencies. The proportionality coefficient does not depend on the distance and is determined by the nonlinear effects on the initial stage of the propagation. It is shown that even for a small acoustical Reynolds number the energy of the noise at the old‐age stage is proportional to the third power of the initial energy for cylindrical waves and to the pow...

Nonlinear sound waves from a uniformly moving point source

Nonlinear sound waves from a uniformly moving source with dimensions smaller than the wavelength of the emitted sound are investigated. They are described by spherical Burgers’ equations with parameters depending on the source velocity V and the direction angle θ from the source to the point of observation. It is seen that for certain V and θ values, both for V less than and greater than the sound velocity in the medium, shock waves occur, which do not occur in nonlinear waves from a fixed sound source.