Yurii A. Ilinskii

Nonlinear standing waves in an acoustical resonator

A one-dimensional model is developed to analyze nonlinear standing waves in an acoustical resonator. The time domain model equation is derived from the fundamental gasdynamics equations for an ideal gas. Attenuation associated with viscosity is included. The resonator is assumed to be of an axisymmetric, but otherwise arbitrary shape. In the model the entire resonator is driven harmonically with an acceleration of constant amplitude. The nonlinear spectral equations are integrated numerically. Results are presented for three geometries: a cylinder, a cone, and a bulb. Theoretical predictions describe the amplitude related resonance frequency shift, hysteresis effects, and waveform distortion. Both resonance hardening and softening behavior are observed and reveal dependence on resonator geometry. Waveform distortion depends on the amplitude of oscillation and the resonator shape. A comparison of measured and calculated wave shapes shows good agreement.

Acoustic streaming generated by standing waves in two-dimensional channels of arbitrary width

An analytic solution is derived for acoustic streaming generated by a standing wave in a viscous fluid that occupies a two-dimensional channel of arbitrary width. The main restriction is that the boundary layer thickness is a small fraction of the acoustic wavelength. Both the outer, Rayleigh streaming vortices and the inner, boundary layer vortices are accurately described. For wide channels and outside the boundary layer, the solution is in agreement with results obtained by others for Rayleigh streaming. As channel width is reduced, the inner vortices increase in size relative to the Rayleigh vortices. For channel widths less than about 10 times the boundary layer thickness, the Rayleigh vortices disappear and only the inner vortices exist. The obtained solution is compared with those derived by Rayleigh, Westervelt, Nyborg, and Zarembo.

Separation of compressibility and shear deformation in the elastic energy density (L)

A formulation of the elastic energy density for an isotropic medium is presented that permits separation of effects due to compressibility and shear deformation. The motivation is to obtain an expansion of the energy density for soft elastic media in which the elastic constants accounting for shear effects are of comparable order. The expansion is carried out to fourth order to ensure that nonlinear effects in shear waves are taken into account. The result is E≃E0(ρ)+μI2+13AI3+DI22, where ρ is density, I2 and I3 are the second- and third-order Lagrangian strain invariants used by Landau and Lifshitz, μ is the shear modulus, A is one of the third-order elastic constants introduced by Landau and Lifshitz, and D is a new fourth-order elastic constant. For processes involving mainly compressibility E≃E0(ρ), and for processes involving mainly shear deformation E≃μI2+13AI3+DI22.

Linear and nonlinear frequency shifts in acoustical resonators with varying cross sections

The frequency response of a nonlinear acoustical resonator is investigated analytically and numerically. The cross-sectional area is assumed to vary slowly but otherwise arbitrarily along the axis of the resonator, such that the Webster horn equation provides a reasonable one-dimensional model in the linear approximation. First, perturbation theory is used to derive an asymptotic formula for the natural frequencies as a function of resonator shape. The solution shows that each natural frequency can be shifted independently via appropriate spatial modulation of the resonator wall. Numerical calculations for resonators of different shapes establish the limits of the asymptotic formula. Second, the nonlinear interactions of modes in the resonator are investigated with Lagrangian mechanics. An analytical result is obtained for the amplitude-frequency response curve and nonlinear resonance frequency shift for the fundamental mode. For a resonator driven at its lowest natural frequency, it is found that whether...

Thermal effects on acoustic streaming in standing waves.

Acoustic streaming generated by standing waves in channels of arbitrary width is investigated analytically. In a previous paper by the authors [J. Acoust. Soc. Am. 113, 153-160 (2003)], a purely viscous fluid in a two-dimensional channel was considered. That analysis is extended here to a gas in which heat conduction and dependence of the viscosity on temperature are taken into account. Calculations are presented for typical working gases used in thermoacoustic engines at standard temperature and pressure. In channels that are very wide in comparison with the viscous penetration depth, which is the Rayleigh streaming regime, the influence of the two thermal effects is comparable but small. The same is true in very narrow channels, having widths on the order of the viscous penetration depth. In channels having intermediate widths, 10-20 times the viscous penetration depth, the effect of heat conduction can be substantial. The analysis is performed for cylindrical tubes as well as two-dimensional channels, and the results are found to be qualitatively the same.

Modeling of nonlinear shear waves in soft solids

An evolution equation for nonlinear shear waves in soft isotropic solids is derived using an expansion of the strain energy density that permits separation of compressibility and shear deformation. The advantage of this approach is that the coefficient of nonlinearity for shear waves depends on only three elastic constants, one each at second, third, and fourth order, and these coefficients have comparable numerical values. In contrast, previous formulations yield coefficients of nonlinearity that depend on elastic constants whose values may differ by many orders of magnitude because they account for effects of compressibility as well as shear. It is proposed that the present formulation is a more natural description of nonlinear shear waves in soft solids, and therefore it is especially applicable to biomaterials like soft tissues. Calculations are presented for harmonic generation and shock formation in both linearly and elliptically polarized shear waves.

Nonlinear two-dimensional model for thermoacoustic engines

A two-dimensional model and efficient solution algorithm are developed for studying nonlinear effects in thermoacoustic engines. There is no restriction on the length or location of the stack, and the cross-sectional area of the resonator may vary with position along its axis. Reduced model equations are obtained by ordering spatial derivatives in terms of rapid variations across the pores in the stack, versus slow variations along the resonator axis. High efficiency is achieved with the solution algorithm because the stability condition for numerical integration of the model equations is connected with resonator length rather than pore diameter. Computation time is reduced accordingly, by several orders of magnitude, without sacrificing spatial resolution. The solution algorithm is described in detail, and the results are verified by comparison with established linear theory. Two examples of nonlinear effects are investigated briefly, the onset of instability through to saturation and steady state, and nonlinear waveform distortion as a function of resonator shape.

Motion of a solid sphere in a viscoelastic medium in response to applied acoustic radiation force: Theoretical analysis and experimental verification.

The motion of a rigid sphere in a viscoelastic medium in response to an acoustic radiation force of short duration was investigated. Theoretical and numerical studies were carried out first. To verify the developed model, experiments were performed using rigid spheres of various diameters and densities embedded into tissue-like, gel-based phantoms of varying mechanical properties. A 1.5 MHz, single-element, focused transducer was used to apply the desired radiation force. Another single-element, focused transducer operating at 25 MHz was used to track the displacements of the sphere. The results of this study demonstrate good agreement between theoretical predictions and experimental measurements. The developed theoretical model accurately describes the displacement of the solid spheres in a viscoelastic medium in response to the acoustic radiation force.

Gas bubble and solid sphere motion in elastic media in response to acoustic radiation force.

The general approach to estimate the displacement of rounded objects (specifically, gas bubbles and solid spheres) in elastic incompressible media in response to applied acoustic radiation force is presented. In this study, both static displacement and transient motion are analyzed using the linear approximation. To evaluate the static displacement of the spherical inclusion, equations coupling the applied force, displacement, and shear modulus of the elastic medium are derived. Analytical expressions to estimate the static displacement of solid spheres and gas bubbles are presented. Under a continuously applied static force, both the solid sphere and the initially spherical gas bubble are displaced, and the bubble is deformed. The transient responses of the inclusions are described using motion equations. The displacements of the inclusion in elastic incompressible lossless media are analyzed using both frequency-domain and time-domain formalism, and the equations of motion are derived for both a solid sphere and a gas bubble. For a short pulsed force, an analytical solution for the equations of motion is presented. Finally, transient displacement of the gas bubble in viscoelastic media is considered.

Energy losses in an acoustical resonator

A one-dimensional model has recently been developed for the analysis of nonlinear standing waves in an acoustical resonator. This model is modified to include energy losses in the boundary layer along the resonator wall. An investigation of the influence of the boundary layer on the acoustical field in the resonator and on the energy dissipation in the resonator is conducted. The effect of the boundary layer is taken into account by introducing an additional term into the continuity equation to describe the flow from the boundary layer to the volume. A linear approximation is used in the development of the boundary layer model. In addition to the viscous attenuation in the boundary layer, the effect of acoustically generated turbulence is modeled by an eddy viscosity formulation. Calculatons of energy losses and a quality factor of a resonator are included into the numerical code. Results are presented for resonators of three different geometries: a cylinder, a horn cone, and a bulb-type resonator. A comparison of measured and predicted dissipation shows good agreement.