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.

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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.

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.

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.

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.

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.

Two models of interacting bubble dynamics are presented, a coupled system of second-order differential equations based on Lagrangian mechanics, and a first-order system based on Hamiltonian mechanics. Both account for pulsation and translation of an arbitrary number of spherical bubbles. For large numbers of interacting bubbles, numerical solution of the Hamiltonian equations provides greater stability. The presence of external acoustic sources is taken into account explicitly in the derivation of both sets of equations. In addition to the acoustic pressure and its gradient, it is found that the particle velocity associated with external sources appears in the dynamical equations.